diff --git a/.gitattributes b/.gitattributes index c9772d2a3e5dde8c6b450d55209794c25a43b5f9..bb8a2680ea742680ec16c434879f5a20fb2bdd38 100644 --- a/.gitattributes +++ b/.gitattributes @@ -153,3 +153,4 @@ etE2T4oBgHgl3EQfGgZ8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex 4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf filter=lfs diff=lfs merge=lfs -text 9tAyT4oBgHgl3EQfqPjX/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text n9E5T4oBgHgl3EQfjw-t/content/2301.05658v1.pdf filter=lfs diff=lfs merge=lfs -text +o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/1NE0T4oBgHgl3EQf_gL8/content/tmp_files/2301.02829v1.pdf.txt b/1NE0T4oBgHgl3EQf_gL8/content/tmp_files/2301.02829v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7131360ebba80c9ee0f15ccfd0fc8658292f37d0 --- /dev/null +++ b/1NE0T4oBgHgl3EQf_gL8/content/tmp_files/2301.02829v1.pdf.txt @@ -0,0 +1,284 @@ +Dark Matter and MOND: Two sides of the same coin? +D. F. Roscoe (The Open University; D.Roscoe@open.ac.uk) +ORCID: 0000-0003-3561-7425 +1 +arXiv:2301.02829v1 [astro-ph.GA] 7 Jan 2023 + +Abstract +It has recently been reported that the application of convolutional neural-network tech- +niques to infer the dark-matter distribution in the local cosmos has revealed how it follows +the D ≈ 2 hierarchical distribution of galaxies in the locality, rather than exhibiting the +expected homogeneity throughout the IGM. Taken at face value, this implies that the Hub- +ble Law, observed to be followed on scales which are deep inside the observed hierarchical +structures, can no longer be assumed to arise from universal expansion. So, if not universal +expansion, then what? +As a possibility, it has been recognized for a considerable time that if the lower cut-off +scales of a D ≈ 2 hierarchical cosmos are identified with the scales of a typical galaxy, then +gravitational redshift automatically follows the Hubble Law with Hg ≈ 70 km/sec/Mpc. +Inter alia, this suggests a model of galaxy formation in a D ≈ 2 hierarchical IGM in which +all of the material M0 within a sphere R0 coalesces about a unique center so that hierachical +symmetry is broken on the scale (M0, R0). +Putting these things together leads unambiguously to the conclusion that, in an hierachical +cosmos, the Dark Matter hypothesis and Milgrom’s MOND hypothesis are two sides of the +same coin. +2 + +1 +Introduction: +It is now widely accepted that on scales up to about 200 Mpc galaxies are distributed in a quasi- +fractal D ≈ 2 fashion. For fairly recent work see Tekhanovich & Baryshev (2016), but many +others have contributed over the years. +It then becomes a point of considerable significance that the Hubble Law is well established +on scales that are deep inside the accepted fractal structure of the general galaxy distribution. +This very interesting circumstance is the primary evidence supporting the idea that the IGM is +largely populated by an homogeneous distribution of dark matter on the small scales required +- for, without homogeneity, the linear nature of Hubble’s Law cannot be understood within the +context of universal expansion. +It is for this reason that the paper of Hong et al (2021) caused so much consternation: specifically, +the authors used state-of-the-art convolutional neural-network techniques combined with modern +positional and peculiar velocity data to compute and map the local dark matter distribution. +Against expectation, this distribution is found to trace the hierarchical distribution of galaxies +very closely - there is no indication of homogeneity, and hence no indication that the Hubble +Law can be understood in terms of universal expansion. +The only immediately plausible alternative is some form of gravitational redshift: Baryshev +et al (1998) point out that inside a D = 2 hierarchical galaxy distribution (with an assumed +homogeneous distribution of dark matter) the gravitational part of redshift is also purely linear +with distance and cannot be distinguished from the expansion component. But if the results of +Hong et al (2021) are to be taken at face value, then any contribution to redshift from expansion +must manifest itself as a departure from linearity. Since such a departure is not observed then, +according to the results of Hong et al, there can be no expansion effect at all. +This line of argument is reinforced by the further observation of Baryshev et al that if the +lower cut-off mass and length scales of the hierarchy are identified with the mass and length +scales of the typical galaxy, then a gravitational redshift of Hg ≈ 70 km/sec/Mpc is to be ex- +pected. +Inter alia, the foregoing considerations suggest a process of galaxy formation according to which +an isolated galactic object can be modelled as a finite bounded spherically symmetric peturbation +of the hierarchical IGM (assumed in the first instance to be a mix of baryonic and non-baryonic +mass) - this automatically entails that all of the mass M0 within the sphere R0 has coalesced +around a unique centre so that fractal symmetry is broken on the scales of (M0, R0). +2 +Consequences on the lower cut-off scales: +From these general considerations we may conclude: +1. The lower cut-off radial and mass scales (M0, R0) must behave according to +M0 = 4πR2 +0ΣF +(1) +3 + +where ΣF is the mass surface density of the D = 2 hierarchical mass distribution in the +local cosmos; +2. Since galaxies in general appear to be stable structures, there must be an equilibrium +constraint at the lower cut-off scales of the hierarchy. Using simple Newtonian arguments, +we show in appendix §A that equilibrium at these lower cut-off scales requires: +V 2 +0 +R0 += aF ≡ 4πGΣF +(2) +where aF is the characteristic acceleration scale associated with ΣF; +3. The relationship +V 4 +0 = aF GM0 , +(3) +which is formally identical to the Baryonic Tully-Fisher Relationship (BTFR), is now de- +rived directly by eliminating R0 between (1) and (2). +It is to be noted that whilst (3) is formally identical to the BTFR of Milgrom’s MOND, it differs +fundamentally in the assumption (expressed in the last paragraph of §1) that M0 is an unknown +mix of baryonic and non-baryonic mass whereas, by definition, the BTFR asserts that this mass +is purely baryonic. +3 +Empirical support for the BTFR hypothesis +3.1 +The analysis of Lelli, McGaugh & Schombert (2016B) +It has only recently been possible to explore the BTFR hypothesis in a statistically rigorous +fashion. Specifically, the SPARC sample of Lelli, McGaugh & Schombert (2016A) contains high +quality rotation curves and high quality modern surface photometry at 3.6 µm for a sample of 175 +nearby disk galaxies. The high quality of the surface photometry over this sample allowed Lelli, +McGaugh & Schombert (2016B) to construct photometric models of baryonic mass distributions +in that particular subsample of 118 disks which also had rotation curves extending to flatness, +making it ideal for a statistically rigorous testing of the BTFR hypothesis. +Subsequently, the authors used regression analysis techniques to demonstrate how the subsample +really does fit the BTFR with very small scatter. In this way, they argued that the observed +scatter is sufficiently below the instrinsic-scatter expectations of ΛCDM cosmology to present a +fundamental difficulty for that cosmology and for the associated idea of dynamically significant +quantities of non-baryonic matter in the generality of galaxy disks. +Since (3) is derived from the hypothesis that galaxies form by coalescing in a stable way out +of the D = 2 hierarchical IGM, this result implies that the IGM itself consists primarily of +undetected baryonic matter and so, in effect, (3) itself represents a derivation of the hitherto +empirical BTFR from a fundamental theoretical position. +4 + +3.2 +The estimation of (aF, ΣF) +The data used by Lelli, McGaugh & Schombert (2016B) is available as an on-line data-sheet giv- +ing estimates for the photometrically modelled baryonic masses M0 and flat rotation velocities +V0 for the 118 disk galaxies. Given this data, an alternative demonstration supporting the BTFR +hypothesis is provided by showing how the hypothesis, applied differently to the data, yields a +very sharp estimate of the characteristic acceleration parameter aF, thereby demonstrating how, +for all practical purposes, its value is identical to that of Milgrom’s critical acceleration parame- +ter, a0. +In order to estimate aF ≡ 4πGΣF from this data, we rearrange (3) as +V 4 +0 +GM0 += 4πGΣF ≡ constant +and hence form the empirical sample distribution +J ≡ +� V 4 +0i +GM0i +, i = 1...118 +� +. +Then, from J, we generate N = 10000 bootstrapped distributions, ˆJi, i = 1..N in the usual way. +For each ˆJi we then compute its geometric mean, ˆaFi say, to obtain, finally, the distribution +AF ≡ (log ˆaFi, i = 1..N) . +The density distribution of AF is given in figure 1 from which it is clear that the estimate for +aF is very tightly constrained around the modal value of 1.3 × 10−10 mtrs/sec2 which, for all +practical purposes, is identical to Milgrom’s value of MOND’s critical acceleration parameter +a0. This estimate of aF corresponds to ΣF ≈ 0.15 kg/mtr2 for the mass surface density of the +hierarchical cosmos. +The grey curve in figure 1 arises from the same analysis but applied to shuffled velocity and +mass data. It is clear that the signal so powerfully present in the unshuffled data is destroyed +by shuffling. We conclude that, for all practical purposes the signal for the mass surface density +ΣF ≈ 0.15 kg/mtr2 in the hierarchical cosmos is real. +5 + +Dotted = 1.1 × 10−10 mtrs/sec2 +Mode = 1.3 × 10−10 mtrs/sec2 +Dashed = 1.6 × 10−10 mtrs/sec2 +0 +5 +10 +15 +log(aF) +Density +Distribution of bootstrapped geometric means +Figure 1: Solid black curve = density distribution of log ˆaF. Solid grey curve arises when velocity +and mass data are shuffled with respect to each other. The signal represented by the black curve +is destroyed on the shuffled data. +4 +Full circle to MOND +Taking the results of Hong et al (2021) at face value, together with the results of Lelli, McGaugh +& Schombert (2016B) and observations of Baryshev et al (1998), it has been shown that the Dark +Matter of modern astrophysics, rather than being an homogeneous distribution of non-baryonic +matter, can reasonably be identified as a D = 2 hierarchical distribution of undetected baryonic +matter which, as the BTFR (3) shows, provides exactly the dynamical support for galaxies that +the original non-baryonic Dark Matter hypothesis was formulated for in the first place. +6 + +However, hitherto the prominence of the originally empirically derived BTFR has rested en- +tirely upon the fact that it is central to the architecture of MOND Milgrom (1983a,b,c). To see +this, (3) gives directly +V 2 +0 +R0 += +√aF GM0 +R0 +which, in effect, MOND extrapolates by hypothesis to give g = √aF gN, where gN ≡ GM0/R2, as +the effective gravitational force for all R ≥ R0 . Consequently, MOND automatically receives the +direct interpretation as a first-order descriptor of gravitational dynamics in a D = 2 hierarchical +IGM. +In short, in an hierachical cosmos, the Dark Matter hypothesis and Milgrom’s MOND hypothesis +are two sides of the same coin. +5 +The nature of baryonic Dark Matter? +We have argued that the Dark Matter of the IGM is distributed in a D = 2 hierarchy and consists +of undetected baryonic material. So, the immediate question is: how can such a distribution of +baryonic material remain undetected? There are three potential strands to the answer, easily +conceived to be acting in concert. +5.1 +Conventional possibilities +Given that the IGM forms a D = 2 hierarchy, then its volume density in a spherical volume of +radius R tends to zero as R → ∞, making its detection at large radii intrinsically difficult. +Furthermore, it can reasonably be assumed that the IGM is at least close to being in thermal +equilibrium with the general background, again making its detection against the background also +intrinsically difficult. +5.2 +Unconventional possibilities +Until recently, it has always been assumed that there is no such thing in nature as a perfect, or +near perfect, blackbody absorber - the reason being that no such thing had ever been observed. +However, Mizuno et al (2009) showed how to fabricate, from agglomerations of single-walled +carbon nanotubes (SWCNTs), material distributions having specific bulk statistics which act +as near-perfect blackbody absorbers (emissivity > 0.98) across a very wide range of incident +wavelengths from UV at 200nm to the far IR at 200µm. +This behaviour has been shown to be independent of the specific properties of the individual +SWCNTs, but is rather a consequence of the bulk statistical characteristics of the fabricated +SWCNT distributions. We know that many allotropes of carbon exist in interstellar space and +these must to some extent be blown into the IGM from the generality of galactic interiors. It is a +short step to visualizing the existence of clouds of SWCNTs dispersed throughout the hierachical +7 + +IGM containing sub-populations which, when viewed in projection along any given line of sight, +possess the bulk statistical characteristics required to mimic the properties of the fabricated +SWCNT distributions of Mizuno et al (2009). +In this way, it is possible to conceive how SWCNT clouds within the IGM have the potential to +act as ‘dispersed near-perfect blackbody objects’ making them virtually undetectable. +6 +Summary and conclusions +There is general agreement that the distribution of galaxies in particular is quasi-fractal D ≈ 2 +out to about 200 Mpc and Baryshev et al (1998) has pointed out that gravitational redshift in +such an hierarchical cosmos will follow the Hubble Law. Furthermore, these authors point out +that if the lower cut-off scales of the hierarchy are identified with the mass and radial scales of +the typical galaxy, then Hg ≈ 70 km/sec/Mpc is to be expected. +Notwithstanding the hierarchical distribution of galaxies in the local cosmos, the conventional +view holds that the IGM itself consists of non-baryonic Dark Matter, the assumed homogeneous +distribution of which makes the Hubble Law a consequence of universal expansion. It is for this +reason that the paper of Hong et al (2021), which computes and maps the distribution of Dark +Matter in the local cosmos, caused so much consternation: specifically, they report that the +distribution of local Dark Matter shows no indication of homogeneity, but instead closely follows +the fractal structures of the galaxy distribution. +These results, taken together, suggest a model of galaxy formation in a D = 2 hierarchical +IGM according to which all of the matter M0 in a sphere R0 coalesces about a unique center +so that hierachical symmetry is broken, with (M0, R0) then representing the lower cut-off scales +of the hierarchy. Given the results of Lelli, McGaugh & Schombert (2016B) to the effect that, +within any given galaxy, M0 primarily consists of baryonic matter then, given that the resulting +galactic objects are in equilibrium with the general environment, this model of galaxy formation +gives a direct derivation of the Baryonic Tully-Fisher Relationship and, consequently, provides +a natural interpretation of MOND as a first-order descriptor of gravitational dynamics in an +hierachical cosmos. +In conclusion, in an hierchical cosmos, the Dark Matter hypothesis and Milgrom’s MOND hy- +pothesis are seen to be two sides of the same coin. +References +Baryshev, Yu V., Sylos Labini, F., Montuori, M., Pietronero, L., astro-ph/9803142 +Hong, S.E., Jeong, D., Hwang, H.S., Kim, J., 2021. Ap. J.; 913, 76 +Lelli, F., McGaugh, SS, Schombert, JM., ApJ., 152, 6, 2016A +Lelli, F., McGaugh, SS, Schombert, JM., ApJL., 816, L14, 2016B +8 + +Milgrom, M., 1983a, Ap. J. 270: 365. +Milgrom, M., 1983b, Ap. J. 270: 371 +Milgrom, M. 1983c. Ap. J. 270: 384-389 +Mizuno, K., Ishii, J., Kishida, H., Hayamizu, Y., Yasuda, S., Futaba, D., Yumura, M., Hata, K., +2009. PNAS, 106, 15, 6044-6047 +Tekhanovich D.I.I and Baryshev Yu.V., Astro.ph/1610.05206 +A +Equilibrium at the lower cut-off scales of the hierarchy +In the cosmos of our experience, galaxies in general appear to be stable and long-lasting struc- +tures. Since the matter distribution in the D = 2 fractal hierarchy is isotropic (by definition) +about any arbitrarily chosen centre, then the notional gravitational acceleration imparted to +a particle at radius R from the centre, and generated by the material contained within R, is +directed towards the chosen centre and has magnitude given by +M(R) G +R2 += 4πG ΣF ≡ aF, R < ∞. +(4) +On this basis, it is clear that the net actual gravitational acceleration imparted to a material +particle immersed anywhere in the global hierarchy is zero, from which it can be concluded that +a D = 2 fractal distribution of material is in a state of dynamical equilibrium. +It follows that: +• if a finite spherical volume, radius R0, is imagined emptied of all material, then the net +actual gravitational acceleration of any material particle placed on R0 will be aF directed +radially outwards from the centre of the empty volume; +• the empty spherical volume is unstable since all accelerations on R0 are outward. It follows +that stability requires the volume to be occupied by a stablizing mass, a galaxy say, creating +a state of zero net radial acceleration on R0. In other words, the equilibrium condition +g0 ≡ V 2 +0 +R0 += aF +(5) +must be satisfied. +9 + diff --git a/1NE0T4oBgHgl3EQf_gL8/content/tmp_files/load_file.txt b/1NE0T4oBgHgl3EQf_gL8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1ebbf8ee8aba845d0f45586ba62dfb99b286b27 --- /dev/null +++ b/1NE0T4oBgHgl3EQf_gL8/content/tmp_files/load_file.txt @@ -0,0 +1,155 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf,len=154 +page_content='Dark Matter and MOND: Two sides of the same coin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Roscoe (The Open University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='Roscoe@open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='uk) ORCID: 0000-0003-3561-7425 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='02829v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='GA] 7 Jan 2023 Abstract It has recently been reported that the application of convolutional neural-network tech- niques to infer the dark-matter distribution in the local cosmos has revealed how it follows the D ≈ 2 hierarchical distribution of galaxies in the locality, rather than exhibiting the expected homogeneity throughout the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Taken at face value, this implies that the Hub- ble Law, observed to be followed on scales which are deep inside the observed hierarchical structures, can no longer be assumed to arise from universal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' So, if not universal expansion, then what?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' As a possibility, it has been recognized for a considerable time that if the lower cut-off scales of a D ≈ 2 hierarchical cosmos are identified with the scales of a typical galaxy, then gravitational redshift automatically follows the Hubble Law with Hg ≈ 70 km/sec/Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Inter alia, this suggests a model of galaxy formation in a D ≈ 2 hierarchical IGM in which all of the material M0 within a sphere R0 coalesces about a unique center so that hierachical symmetry is broken on the scale (M0, R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Putting these things together leads unambiguously to the conclusion that, in an hierachical cosmos, the Dark Matter hypothesis and Milgrom’s MOND hypothesis are two sides of the same coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 2 1 Introduction: It is now widely accepted that on scales up to about 200 Mpc galaxies are distributed in a quasi- fractal D ≈ 2 fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' For fairly recent work see Tekhanovich & Baryshev (2016), but many others have contributed over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' It then becomes a point of considerable significance that the Hubble Law is well established on scales that are deep inside the accepted fractal structure of the general galaxy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' This very interesting circumstance is the primary evidence supporting the idea that the IGM is largely populated by an homogeneous distribution of dark matter on the small scales required for, without homogeneity, the linear nature of Hubble’s Law cannot be understood within the context of universal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' It is for this reason that the paper of Hong et al (2021) caused so much consternation: specifically, the authors used state-of-the-art convolutional neural-network techniques combined with modern positional and peculiar velocity data to compute and map the local dark matter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Against expectation, this distribution is found to trace the hierarchical distribution of galaxies very closely - there is no indication of homogeneity, and hence no indication that the Hubble Law can be understood in terms of universal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' The only immediately plausible alternative is some form of gravitational redshift: Baryshev et al (1998) point out that inside a D = 2 hierarchical galaxy distribution (with an assumed homogeneous distribution of dark matter) the gravitational part of redshift is also purely linear with distance and cannot be distinguished from the expansion component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' But if the results of Hong et al (2021) are to be taken at face value, then any contribution to redshift from expansion must manifest itself as a departure from linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Since such a departure is not observed then, according to the results of Hong et al, there can be no expansion effect at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' This line of argument is reinforced by the further observation of Baryshev et al that if the lower cut-off mass and length scales of the hierarchy are identified with the mass and length scales of the typical galaxy, then a gravitational redshift of Hg ≈ 70 km/sec/Mpc is to be ex- pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Inter alia, the foregoing considerations suggest a process of galaxy formation according to which an isolated galactic object can be modelled as a finite bounded spherically symmetric peturbation of the hierarchical IGM (assumed in the first instance to be a mix of baryonic and non-baryonic mass) - this automatically entails that all of the mass M0 within the sphere R0 has coalesced around a unique centre so that fractal symmetry is broken on the scales of (M0, R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 2 Consequences on the lower cut-off scales: From these general considerations we may conclude: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' The lower cut-off radial and mass scales (M0, R0) must behave according to M0 = 4πR2 0ΣF (1) 3 where ΣF is the mass surface density of the D = 2 hierarchical mass distribution in the local cosmos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Since galaxies in general appear to be stable structures, there must be an equilibrium constraint at the lower cut-off scales of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Using simple Newtonian arguments, we show in appendix §A that equilibrium at these lower cut-off scales requires: V 2 0 R0 = aF ≡ 4πGΣF (2) where aF is the characteristic acceleration scale associated with ΣF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' The relationship V 4 0 = aF GM0 , (3) which is formally identical to the Baryonic Tully-Fisher Relationship (BTFR), is now de- rived directly by eliminating R0 between (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' It is to be noted that whilst (3) is formally identical to the BTFR of Milgrom’s MOND, it differs fundamentally in the assumption (expressed in the last paragraph of §1) that M0 is an unknown mix of baryonic and non-baryonic mass whereas, by definition, the BTFR asserts that this mass is purely baryonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 3 Empirical support for the BTFR hypothesis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='1 The analysis of Lelli, McGaugh & Schombert (2016B) It has only recently been possible to explore the BTFR hypothesis in a statistically rigorous fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Specifically, the SPARC sample of Lelli, McGaugh & Schombert (2016A) contains high quality rotation curves and high quality modern surface photometry at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='6 µm for a sample of 175 nearby disk galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' The high quality of the surface photometry over this sample allowed Lelli, McGaugh & Schombert (2016B) to construct photometric models of baryonic mass distributions in that particular subsample of 118 disks which also had rotation curves extending to flatness, making it ideal for a statistically rigorous testing of the BTFR hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Subsequently, the authors used regression analysis techniques to demonstrate how the subsample really does fit the BTFR with very small scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' In this way, they argued that the observed scatter is sufficiently below the instrinsic-scatter expectations of ΛCDM cosmology to present a fundamental difficulty for that cosmology and for the associated idea of dynamically significant quantities of non-baryonic matter in the generality of galaxy disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Since (3) is derived from the hypothesis that galaxies form by coalescing in a stable way out of the D = 2 hierarchical IGM, this result implies that the IGM itself consists primarily of undetected baryonic matter and so, in effect, (3) itself represents a derivation of the hitherto empirical BTFR from a fundamental theoretical position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='2 The estimation of (aF, ΣF) The data used by Lelli, McGaugh & Schombert (2016B) is available as an on-line data-sheet giv- ing estimates for the photometrically modelled baryonic masses M0 and flat rotation velocities V0 for the 118 disk galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Given this data, an alternative demonstration supporting the BTFR hypothesis is provided by showing how the hypothesis, applied differently to the data, yields a very sharp estimate of the characteristic acceleration parameter aF, thereby demonstrating how, for all practical purposes, its value is identical to that of Milgrom’s critical acceleration parame- ter, a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' In order to estimate aF ≡ 4πGΣF from this data, we rearrange (3) as V 4 0 GM0 = 4πGΣF ≡ constant and hence form the empirical sample distribution J ≡ � V 4 0i GM0i , i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='118 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Then, from J, we generate N = 10000 bootstrapped distributions, ˆJi, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='.N in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' For each ˆJi we then compute its geometric mean, ˆaFi say, to obtain, finally, the distribution AF ≡ (log ˆaFi, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='.N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' The density distribution of AF is given in figure 1 from which it is clear that the estimate for aF is very tightly constrained around the modal value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='3 × 10−10 mtrs/sec2 which, for all practical purposes, is identical to Milgrom’s value of MOND’s critical acceleration parameter a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' This estimate of aF corresponds to ΣF ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='15 kg/mtr2 for the mass surface density of the hierarchical cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' The grey curve in figure 1 arises from the same analysis but applied to shuffled velocity and mass data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' It is clear that the signal so powerfully present in the unshuffled data is destroyed by shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' We conclude that, for all practical purposes the signal for the mass surface density ΣF ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='15 kg/mtr2 in the hierarchical cosmos is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 5 Dotted = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='1 × 10−10 mtrs/sec2 Mode = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='3 × 10−10 mtrs/sec2 Dashed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='6 × 10−10 mtrs/sec2 0 5 10 15 log(aF) Density Distribution of bootstrapped geometric means Figure 1: Solid black curve = density distribution of log ˆaF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Solid grey curve arises when velocity and mass data are shuffled with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' The signal represented by the black curve is destroyed on the shuffled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 4 Full circle to MOND Taking the results of Hong et al (2021) at face value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' together with the results of Lelli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' McGaugh & Schombert (2016B) and observations of Baryshev et al (1998),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' it has been shown that the Dark Matter of modern astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' rather than being an homogeneous distribution of non-baryonic matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' can reasonably be identified as a D = 2 hierarchical distribution of undetected baryonic matter which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' as the BTFR (3) shows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' provides exactly the dynamical support for galaxies that the original non-baryonic Dark Matter hypothesis was formulated for in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 6 However, hitherto the prominence of the originally empirically derived BTFR has rested en- tirely upon the fact that it is central to the architecture of MOND Milgrom (1983a,b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' To see this, (3) gives directly V 2 0 R0 = √aF GM0 R0 which, in effect, MOND extrapolates by hypothesis to give g = √aF gN, where gN ≡ GM0/R2, as the effective gravitational force for all R ≥ R0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Consequently, MOND automatically receives the direct interpretation as a first-order descriptor of gravitational dynamics in a D = 2 hierarchical IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' In short, in an hierachical cosmos, the Dark Matter hypothesis and Milgrom’s MOND hypothesis are two sides of the same coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 5 The nature of baryonic Dark Matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' We have argued that the Dark Matter of the IGM is distributed in a D = 2 hierarchy and consists of undetected baryonic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' So, the immediate question is: how can such a distribution of baryonic material remain undetected?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' There are three potential strands to the answer, easily conceived to be acting in concert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='1 Conventional possibilities Given that the IGM forms a D = 2 hierarchy, then its volume density in a spherical volume of radius R tends to zero as R → ∞, making its detection at large radii intrinsically difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Furthermore, it can reasonably be assumed that the IGM is at least close to being in thermal equilibrium with the general background, again making its detection against the background also intrinsically difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='2 Unconventional possibilities Until recently, it has always been assumed that there is no such thing in nature as a perfect, or near perfect, blackbody absorber - the reason being that no such thing had ever been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' However, Mizuno et al (2009) showed how to fabricate, from agglomerations of single-walled carbon nanotubes (SWCNTs), material distributions having specific bulk statistics which act as near-perfect blackbody absorbers (emissivity > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='98) across a very wide range of incident wavelengths from UV at 200nm to the far IR at 200µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' This behaviour has been shown to be independent of the specific properties of the individual SWCNTs, but is rather a consequence of the bulk statistical characteristics of the fabricated SWCNT distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' We know that many allotropes of carbon exist in interstellar space and these must to some extent be blown into the IGM from the generality of galactic interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' It is a short step to visualizing the existence of clouds of SWCNTs dispersed throughout the hierachical 7 IGM containing sub-populations which, when viewed in projection along any given line of sight, possess the bulk statistical characteristics required to mimic the properties of the fabricated SWCNT distributions of Mizuno et al (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' In this way, it is possible to conceive how SWCNT clouds within the IGM have the potential to act as ‘dispersed near-perfect blackbody objects’ making them virtually undetectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 6 Summary and conclusions There is general agreement that the distribution of galaxies in particular is quasi-fractal D ≈ 2 out to about 200 Mpc and Baryshev et al (1998) has pointed out that gravitational redshift in such an hierarchical cosmos will follow the Hubble Law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Furthermore, these authors point out that if the lower cut-off scales of the hierarchy are identified with the mass and radial scales of the typical galaxy, then Hg ≈ 70 km/sec/Mpc is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Notwithstanding the hierarchical distribution of galaxies in the local cosmos, the conventional view holds that the IGM itself consists of non-baryonic Dark Matter, the assumed homogeneous distribution of which makes the Hubble Law a consequence of universal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' It is for this reason that the paper of Hong et al (2021), which computes and maps the distribution of Dark Matter in the local cosmos, caused so much consternation: specifically, they report that the distribution of local Dark Matter shows no indication of homogeneity, but instead closely follows the fractal structures of the galaxy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' These results, taken together, suggest a model of galaxy formation in a D = 2 hierarchical IGM according to which all of the matter M0 in a sphere R0 coalesces about a unique center so that hierachical symmetry is broken, with (M0, R0) then representing the lower cut-off scales of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Given the results of Lelli, McGaugh & Schombert (2016B) to the effect that, within any given galaxy, M0 primarily consists of baryonic matter then, given that the resulting galactic objects are in equilibrium with the general environment, this model of galaxy formation gives a direct derivation of the Baryonic Tully-Fisher Relationship and, consequently, provides a natural interpretation of MOND as a first-order descriptor of gravitational dynamics in an hierachical cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' In conclusion, in an hierchical cosmos, the Dark Matter hypothesis and Milgrom’s MOND hy- pothesis are seen to be two sides of the same coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} 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+page_content=', Kishida, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=', Hayamizu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=', Yasuda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=', Futaba, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=', Yumura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=', Hata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' PNAS, 106, 15, 6044-6047 Tekhanovich D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='I and Baryshev Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=', Astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='ph/1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content='05206 A Equilibrium at the lower cut-off scales of the hierarchy In the cosmos of our experience, galaxies in general appear to be stable and long-lasting struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' Since the matter distribution in the D = 2 fractal hierarchy is isotropic (by definition) about any arbitrarily chosen centre, then the notional gravitational acceleration imparted to a particle at radius R from the centre, and generated by the material contained within R, is directed towards the chosen centre and has magnitude given by M(R) G R2 = 4πG ΣF ≡ aF, R < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' (4) On this basis, it is clear that the net actual gravitational acceleration imparted to a material particle immersed anywhere in the global hierarchy is zero, from which it can be concluded that a D = 2 fractal distribution of material is in a state of dynamical equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' It follows that: if a finite spherical volume, radius R0, is imagined emptied of all material, then the net actual gravitational acceleration of any material particle placed on R0 will be aF directed radially outwards from the centre of the empty volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' the empty spherical volume is unstable since all accelerations on R0 are outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' It follows that stability requires the volume to be occupied by a stablizing mass, a galaxy say, creating a state of zero net radial acceleration on R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' In other words, the equilibrium condition g0 ≡ V 2 0 R0 = aF (5) must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQf_gL8/content/2301.02829v1.pdf'} diff --git a/1tAyT4oBgHgl3EQfPfba/content/tmp_files/2301.00027v1.pdf.txt b/1tAyT4oBgHgl3EQfPfba/content/tmp_files/2301.00027v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ebc1e79ee753d4edae892744d810a9e1dcc333fe --- /dev/null +++ b/1tAyT4oBgHgl3EQfPfba/content/tmp_files/2301.00027v1.pdf.txt @@ -0,0 +1,4161 @@ +DRAFT VERSION JANUARY 3, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +CEERS Key Paper IV: Galaxies at 4 < z < 9 are Bluer than They Appear — +Characterizing Galaxy Stellar Populations from Rest-Frame ∼1 micron Imaging +CASEY PAPOVICH,1, 2 JUSTIN W. COLE,1, 2 GUANG YANG,3, 4 STEVEN L. FINKELSTEIN,5 GUILLERMO BARRO,6 VÉRONIQUE BUAT,7 +DENIS BURGARELLA,7 PABLO G. PÉREZ-GONZÁLEZ,8 PAOLA SANTINI,9 LISE-MARIE SEILLÉ,7 LU SHEN,1, 2 +PABLO ARRABAL HARO,10 MICAELA B. BAGLEY,5 ERIC F. BELL,11 LAURA BISIGELLO,12, 13 ANTONELLO CALABRÒ,9 +CAITLIN M. CASEY,5 MARCO CASTELLANO,9 KATHERINE CHWOROWSKY,5, * NIKKO J. CLERI,1, 2 M. C. COOPER,14 +LUCA COSTANTIN,8 MARK DICKINSON,10 HENRY C. FERGUSON,15 ADRIANO FONTANA,9 MAURO GIAVALISCO,16 +ANDREA GRAZIAN,13 NORMAN A. GROGIN,15 NIMISH P. HATHI,15 BENNE W. HOLWERDA,17 TAYLOR A. HUTCHISON,18, † +JEYHAN S. KARTALTEPE,19 LISA J. KEWLEY,20 ALLISON KIRKPATRICK,21 DALE D. KOCEVSKI,22 ANTON M. KOEKEMOER,15 +REBECCA L. LARSON,5, * ARIANNA S. LONG,5, ‡ RAY A. LUCAS,15 LAURA PENTERICCI,9 NOR PIRZKAL,23, 15 +SWARA RAVINDRANATH,15 RACHEL S. SOMERVILLE,24 JONATHAN R. TRUMP,25 STEPHANIE M. URBANO STAWINSKI,14 +BENJAMIN J. WEINER,26 STEPHEN M. WILKINS,27, 28 L. Y. AARON YUNG,18, † AND JORGE A. ZAVALA29 +1Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA +2George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA +3Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands +4SRON Netherlands Institute for Space Research, Postbus 800, 9700 AV Groningen, The Netherlands +5Department of Astronomy, The University of Texas at Austin, Austin, TX, USA +6Department of Physics, University of the Pacific, Stockton, CA 90340 USA +7Aix Marseille Univ, CNRS, CNES, LAM Marseille, France +8Centro de Astrobiología (CAB), CSIC-INTA, Ctra. de Ajalvir km 4, Torrejón de Ardoz, E-28850, Madrid, Spain +9INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy +10NSF’s National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA +11Department of Astronomy, University of Michigan, 1085 S. University Ave, Ann Arbor, MI 48109-1107, USA +12Dipartimento di Fisica e Astronomia "G.Galilei", Universitá di Padova, Via Marzolo 8, I-35131 Padova, Italy +13INAF–Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122, Padova, Italy +14Department of Physics & Astronomy, University of California, Irvine, 4129 Reines Hall, Irvine, CA 92697, USA +15Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA +16University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA +17Physics & Astronomy Department, University of Louisville, 40292 KY, Louisville, USA +18Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA +19Laboratory for Multiwavelength Astrophysics, School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY +14623, USA +20Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA +21Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA +22Department of Physics and Astronomy, Colby College, Waterville, ME 04901, USA +23ESA/AURA Space Telescope Science Institute +24Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY, 10010, USA +25Department of Physics, 196 Auditorium Road, Unit 3046, University of Connecticut, Storrs, CT 06269, USA +26MMT/Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA +27Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK +28Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta +29National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan +ABSTRACT +We present results from the Cosmic Evolution Early Release Survey (CEERS) on the stellar-population pa- +rameters for 28 galaxies with redshifts 4 < z < 9 using imaging data from the James Webb Space Telescope +(JWST) Mid-Infrared Instrument (MIRI) combined with data from the Hubble Space Telescope and the Spitzer +Space Telescope. The JWST/MIRI 5.6 and 7.7 µm data extend the coverage of the rest-frame spectral-energy +Corresponding author: Casey Papovich +papovich@tamu.edu +arXiv:2301.00027v1 [astro-ph.GA] 30 Dec 2022 + +2 +distribution (SED) to nearly 1 micron for galaxies in this redshift range. By modeling the galaxies’ SEDs the +MIRI data show that the galaxies have, on average, rest-frame UV (1600 Å) – I-band colors 0.4 mag bluer +than derived when using photometry that lacks MIRI. Therefore, the galaxies have lower (stellar)-mass–to–light +ratios. The MIRI data reduce the stellar masses by ⟨∆logM∗⟩ = 0.25 dex at 4 < z < 6 (a factor of 1.8) and +0.37 dex at 6 < z < 9 (a factor of 2.3). This also reduces the star-formation rates (SFRs) by ⟨∆logSFR⟩ = 0.14 +dex at 4 < z < 6 and 0.27 dex at 6 < z < 9. The MIRI data also improve constraints on the allowable stellar +mass formed in early star-formation. We model this using a star-formation history that includes both a “burst’ at +zf = 100 and a slowly varying (“delayed-τ”) model. The MIRI data reduce the allowable stellar mass by 0.6 dex +at 4 < z < 6 and by ≈1 dex at 6 < z < 9. Applying these results globally, this reduces the cosmic stellar-mass +density by an order of magnitude in the early universe (z ≈ 9). Therefore, observations of rest-frame ∼>1 µm are +paramount for constraining the stellar–mass build-up in galaxies at very high-redshifts. +1. INTRODUCTION +There is growing evidence that galaxies must have started +forming stars very quickly following the Big Bang. Theory +predicts the first stars should form at z ∼> 20 (e.g., Barkana +& Loeb 2001; Miralda-Escudé 2003; Yoshida et al. 2003; +Wise et al. 2012; Visbal et al. 2020). The ionization from +these sources is needed to explain observations that the +hydrogen-neutral fraction of the intergalactic medium (IGM) +was 50% by z ∼ 8 (Planck Collaboration et al. 2020),1 and +to account for the absorption profile of the 21 cm signal at +z ∼ 20 (e.g., Bowman et al. 2018). +Indeed, early obser- +vations from JWST have already identified candidates for +galaxies at z ∼> 15 (Curtis-Lake et al. 2022; Donnan et al. +2022; Finkelstein et al. 2022a; Robertson et al. 2022). Spec- +troscopy from JWST of galaxies at z ∼ 8−9 shows emission +lines from heavy elements that appear to require metallici- +ties of ≈ 5 − 10% Z⊙ (Arellano-Córdova et al. 2022; Fuji- +moto et al. 2022; Heintz et al. 2022; Langeroodi et al. 2022; +Matthee et al. 2022; Schaerer et al. 2022; Trump et al. 2022; +Curti et al. 2023; Katz et al. 2023), implying these galaxies +have experienced at least one (and probably multiple) gener- +ation(s) of previous stars. This is consistent with earlier de- +tections of metal lines in z > 6 galaxies from ground-based +telescopes (e.g., Stark et al. 2017; Hutchison et al. 2019). All +of these results point to the fact that star formation began +early and those early generations of stars enriched the uni- +verse with heavy elements. +It is then important to consider how we may constrain the +history of star-formation in these early galaxies. The num- +ber of stars (and therefore the stellar mass) in galaxies ap- +pears to rise rapidly. The (co-moving) stellar mass density +in galaxies at z ∼ 5 − 6 (when the age of the Universe is +≈ 1 Gyr) is already 1% of the present value (e.g., Madau & +Dickinson 2014; Finkelstein 2016), where simulations and +theory require the stars in those objects formed at much ear- +lier times. Even early JWST observations find some evidence +for massive galaxies at z > 7 (Labbé et al. 2022) with some +* NSF Graduate Fellow +† NASA Postdoctoral Fellow +‡ NASA Hubble Fellow +1 +The Planck Collaboration et al. (2020) analysis also suggests a non-zero +optical depth of CMB photons scattering off free electrons at z ≈ 15, which +implies ionization of the IGM had begun by this epoch. +candidate objects having masses, logM∗/M⊙ > 11 (as large +as the stellar mass of the Milky Way today, e.g., Papovich +et al. 2015). Such objects would be in tension with galaxy +formation models (Boylan-Kolchin 2022) where there are +not sufficient numbers of massive dark-mater halos to sup- +port these objects, even if all the baryons in the halos are in +the form of stellar mass. However, the uncertainty in these +measurements is in the assumed star-formation histories, the +contributions of emission lines to the photometric measure- +ments from broad-band data (e.g., Endsley et al. 2022; Pérez- +González et al. 2022; Steinhardt et al. 2022), and the effects +young stellar populations “outshining” older stellar popula- +tions in the integrated emission of galaxies (e.g., Giménez- +Arteaga et al. 2022). Clearly there are unknown systematics +in the assumptions of the data analysis, or missing physics +in our theoretical understanding of stellar populations and +galaxy formation, or some combination of all of these things. +It is therefore crucial to understand constraints on the stellar +masses (which are the integral of the star-formation histories) +as much as possible. +Motivated by these issues, in this Paper we use new +data from JWST to better constrain the stellar masses, star- +formation rates (SFRs), and star-formation histories of galax- +ies during the first one and a half billion years after the Big +Bang (z > 4). One of the problems with initial studies from +JWST is that they currently rely entirely on observations from +JWST’s Near-IR Camera (NIRcam), which only probes to +wavelengths ∼<5 µm, or about 6000 Å rest-frame for z = 6−7 +galaxies. This complicates the ability to disentangle massive +galaxies with older stellar populations from younger, dusty +galaxies or galaxies with emission lines with extreme equiv- +alent widths (see discussion in Antwi-Danso et al. 2022, and +recent work by Giménez-Arteaga et al. 2022 who find evi- +dence for older stellar populations mixed with recent bursts +in spatially resolved studies using JWST/NIRCam data). To +better constrain the SEDs of these galaxies requires obser- +vations at longer wavelengths. This is where JWST/MIRI +is important as it has the sensitivity to detect z ≈ 10 galax- +ies at rest-frame 1 µm (see Bisigello et al. 2017). Previous +work on this subject has been limited to data from the Spitzer +Space Telescope, which is primarily sensitive to the emission +such distant galaxies at 3.6 to 8.0 µm. JWST offers immense +gains to Spitzer: JWST has a collecting area that is 45 times +larger than that of Spitzer, and the larger aperture provides + +3 +image quality (i.e., angular resolution) that is improved by a +factor of order 10 (Rigby et al. 2022). These gains are es- +pecially manifest at longer wavelengths, and make JWST 5.6 +and 7.7 µm data vastly more sensitive than Spitzer. +The outline for the Paper is as follows. In Section 2 we +discuss the CEERS dataset and the ancillary datasets used in +this study. We also discuss the processes to create (and vali- +date) the flux densities of galaxies in the CEERS JWST/MIRI +data at 5.6 and 7.7 µm. In Section 2.2 we discuss the sample +of 4 < z < 9 galaxies used in this study, and we present the +MIRI data for these objects. In Section 3 we discuss the anal- +ysis methods to derive constrains on the galaxy stellar popu- +lations. In Section 4 we discuss the resulting improvements +that including the MIRI data provide on constraints on the +galaxies stellar populations (specifically their stellar–masses +and SFRs) derived from fitting stellar population models to +the observed photometry. +In Section 4.4 we discuss con- +straints on the range of allowed stellar masses in these high +redshift galaxies by allowing for an early (“maximally old”) +burst of stars at z = 100, and we show that adding the MIRI +data improves the limit on this hypothetical population of +z = 100 stars by a factor of 6 to 10 for galaxies at 4 < z < 9. In +Section 5 we discuss the implications these constraints have +for our understanding of galaxy colors, stellar populations at +these high redshifts, and the evolution of the galaxy stellar +mass density, in particular during the epoch of reionization, +and what this could mean for future studies of galaxies at +higher redshifts (from JWST). In Section 6 we present our +conclusions and prospects for future studies. +Throughout we use a flat cosmology with Ωm,0 = 0.315, +H0 = 67.4 km s−1 Mpc−1 (Planck Collaboration et al. 2020). +All magnitudes reported here are on the Absolute Bolomet- +ric (AB) system (Oke & Gunn 1983). Throughout we use +Chabrier (2003) initial mass function (IMF) for all stel- +lar masses and SFRs. We denote magnitudes measured in +the MIRI F560W and F770W bands as [5.6] and [7.7], re- +spectively. +Similarly, we denote magnitudes measured in +IRAC Channel 1 (3.6 µm), Channel 2 (4.5 µm), Channel 3 +(5.8 µm), and Channel 4 (8.0 µm) as [3.6], [4.5], [5.8], and +[8.0], respectively. +2. DATA AND SAMPLE +2.1. MIRI Catalog +We use the data release DR0.5 images produced by the +CEERS team for the MIRI 3 and MIRI 6 fields (see Finkel- +stein et al. 2022a)2. These data were acquired in 2022 June +21 and 22. The data properties and its reduction are discussed +elsewhere (Yang et al. 2022 in prep), but we provide a sum- +mary here. The data were processed using the JWST Calibra- +tion Pipeline (v1.7.2) using the default parameters for stage +1 and 2. We then removed the backgrounds with a custom +routine that combines images taken in the same bandpass but +from different fields and/or dither positions (rejecting pix- +2 +https://ceers.github.io/releases.html +els in each image that contain galaxies) in order to create +a “super-background” image. We then removed this back- +ground from each image and applied an astrometric correc- +tion to each image prior to processing them with stage 3 of +the pipeline. This produced the final science images (exten- +sion i2d), rms images (extension rms, which account for +Poisson, readout, and correlated pixel noise; see Yang et al. +2022 in prep), and weight maps (wht) for each field with a +pixel scale of 0.09′′, registered astrometrically to the existing +HST/CANDELS v1.9 WFC3 and ACS images (see Koeke- +moer et al. 2011; Bagley et al. 20222). +For the purpose of this study we are interested in sources +detected in the MIRI data, so we create a catalog of sources +derived from these images. +Prior to object detection we +convolved the 5.6 µm image to match the image quality of +the 7.7 µm image. +For this step, we constructed an “ef- +fective” point source function (ePSF) for each image by +identifying unblended stars using the photutils (v1.5.0) +detection task, and modeling them with the photutils +psf task. This produced model ePSFs with measured full- +width at half maxima (FWHM) of 0.24′′ and 0.28′′, for +the F560W and F770W images, respectively. This is con- +sistent with the expected image quality, but takes into ac- +count the exact dithering and reduction steps for the CEERS +MIRI data. We then used PyPHER (Boucaud et al. 2016b) +to construct a convolution kernel to match the image qual- +ity of the model ePSFs. We applied these kernels to each +F560W image, creating a “PSF-matched” image. Our tests +on point sources in the PSF-matched F560W and F770W im- +ages show that we measure the same fraction of light to better +than 2% in fixed circular apertures of radii larger than 0.′′35. +We then created a detection image constructed from the +sum of the MIRI F560W and F770W science images (us- +ing the extension sci) weighted by the appropriate weight +image (using the extension wht). We also created a detec- +tion weight-map as the sum of the weights for these images. +We then created F560W and F770W catalogs using Source +Extractor (SE, version 2.19.5; Bertin & Arnouts 1996) in +“dual-image” mode using the detection image (and its weight +map) for object detection, where we measured photometry in +the PSF-matched F560W and F770W image. We used the +parameters in Table 1. We then measured fluxes and mag- +nitudes using circular apertures of 0.9′′ diameter, and we +scaled these to a total aperture (MAG_AUTO) measured for +each source in the detection image. Uncertainties for each +object are measured from the rms image in the same aper- +tures, and scaled to a total magnitude in the same way. Fig- +ure 1 shows the distribution of the MIRI sources with signal- +to-noise (SNR) ≥ 3 in F560W or F770W (compared with our +galaxy sample, discussed below in 2.2). +We compared the MIRI flux densities for sources F560W +and F770W against those for bright objects from existing +IRAC 5.8 and 8.0 µm catalogs Stefanon et al. (2017). For +bright objects ([5.8] or [8.0] ≤ 22 AB mag) in the IRAC data, +we measure small offsets of ∆m = [5.8] − [5.6] = 0.16 mag +between the IRAC 5.8 and MIRI 5.6 data, and ∆m = [8.0] − + +4 +Table 1. CEERS MIRI F560W and F770W SExtractor +Parameter Settings +SExtractor Parameter +Value +(1) +(2) +DETECT_MINAREA +10 pixels +DETECT_THRESH +1.3 +ANALYSIS_THRESH +1.3 +FILTER_NAME +gauss_2.5_5x5a +WEIGHT_TYPE +MAP_WEIGHT,MAP_RMS +DEBLEND_NTHRESH +32 +DEBLEND_MINCONT +0.005 +MAG_ZEROPOINT +25.701b +PIXEL_SCALE +0.09 arcsec +BACK_TYPE +AUTO +BACK_FILTERSIZE +5 pixels +BACK_SIZE +32 pixels +BACKPHOTO_THICK +8 +BACKPHOTO_TYPE +LOCAL +SEEING_FWHM +0.3 arcsec +NOTE—SE was run using the weighted sum of the PSF- +matched F560W and F770W images for detection, and +using the images separately for photometry. All other +SE parameters are set to the program defaults (for SEx- +tractor v.2.19.5). +aThis is a Gaussian kernel with σ=2.5 pixels and size 5× +5 pixel2 used to filter the image for source detection. +bThe AB magnitude zeropoint for the images, converting +from the JWST default of MJy sr−1 to µJy pixel−1 at the +0.09′′ pixel−1 scale. +[7.7] = 0.07 mag between the IRAC 8.0 and MIRI 7.7 data +(i.e., the MIRI flux densities are slightly brighter). Most of +these offsets can be explained by differences in the shape of +the MIRI and IRAC passbands and because of differences in +the angular resolution of the instruments (MIRI has a PSF +FWHM smaller by a factor of more than seven). These tests +are discussed more fully in Yang et al. (2022 in prep), but +this gives us confidence that the MIRI data are calibrated to +better than ≃ 0.1−0.2 mag. +2.2. Galaxy Sample +For +this +study +we +use +galaxies +identified +in +the +CEERS/MIRI first epoch fields with redshifts 4.3 < z < 10. +The lower redshift bound is selected to ensure the HST pho- +tometric data (used for galaxy photometric redshifts) probes +the redshifted Lyman-break. +The upper redshift limit in- +cludes the highest redshift galaxies detectable by HST/WFC3 + +All MIRI with [5.6], [7.7] > 3 +2 +4 +6 +8 +redshift +2 +0 +2 +MIRI [5.6] + [7.7] (mag) +Zitrin+15, z=8.683 +constant SFR, 10 Myr, Z = 0.1 Z +, log U = +2 +no nebular emission, A(V) = 1 mag +Figure 1. +MIRI [5.6] − [7.7] colors of the sample studied here +as a function of redshift. The data points (and error bars) denote +the 28 objects studied here (blue-shaded points have 4 < z < 6 and +red-shaded points have 6 < z < 10). The three sources with spec- +troscopic redshifts are indicated with larger symbols (the source +z = 8.683 published by Zitrin et al. 2015 is labeled). The gray his- +togram shows the distribution of MIRI colors for all objects detected +in both bands in CEERS. The bold-dashed line shows the expected +color of a stellar population with nebular emission as discussed in +the text. +data (Bouwens et al. 2019; Finkelstein et al. 2022b). This is +illustrated in Figure 2 which shows that for galaxies around +z ∼ 5 and z ∼ 9, the HST/ACS and WFC3 data constrain this +break. This improves the quality of the sample (as compared +to, for example, using galaxies at z ≈ 3 where the Lyman– +break shifts blueward of the HST/ACS F606W band. +The parent sample for our study is the catalog from Finkel- +stein et al. (2022b), which uses the existing HST/ACS, WFC3 +and Spitzer/IRAC 3.6 and 4.5 µm data to select photomet- +ric samples of galaxies at these high redshifts. +The data +include both the imaging from the original CANDELS sur- +vey (HST/ACS F606W, F814W, WFC3 F125W, F160W) and +additional imaging from WFC3 F140W (see Footnote 2). +Finkelstein et al. (2022b) then use these data to measure +photometric redshifts and redshift probably distribution func- +tions, P(z) for each object. +We then matched objects from the catalog from Finkelstein +et al. (2022b) with objects in our MIRI catalogs that are de- +tected with S/N > 3 in either the MIRI 5.6 or 7.7 µm data +(Section 2.1) using a matching radius of 0.5′′. This sample +includes 29 objects, though one object was later identified +as foreground star and removed. Table 5 which lists the ob- +served properties of the 28 galaxies in the sample, their ID +numbers from Finkelstein et al. (2022b), their HST/F160W +flux densities the MIRI 5.6 and 7.7 µm flux densities. The ta- +ble includes the photometric redshifts derived by Finkelstein +et al. (including the 16th and 84th-percentile range from the + +5 +0.5 +0.7 +1 +2 +3 +5 +7 +10 +wavelength ( m) + + + --------- ACS -------- + --- WFC3 --- +--- IRAC --- +--- MIRI --- + + + + + + + + + + + +Lyman limit +H +H +H +H +[O III] +[O III] +[O II] +[O II] +z=5.1 +z=8.7 +Ly break +Figure 2. +Illustration of galaxy spectra (in relative units of erg s−1 cm−2 Å−1) compared to the broad-band data for the observations in this +work. The bottom panel shows the relative transmission functions for the HST/ACS and WFC3 filters (ACS F606W, F814 and WFC3 F125W, +F160W), Spitzer/IRAC 3.6 µm (Ch1) and 4.5 µm (Ch2), and JWST/MIRI F560W and F770W. The top panel shows model spectra of star- +forming galaxies at z = 5.1 and z = 8.7 (which coincidentally match two galaxies with spectroscopic redshifts in this sample). Key emission +lines and features are labeled. The MIRI data probe the shape of the galaxy spectral energy distributions to 8000 Å rest-frame, even for galaxies +with z = 9. +P(z)) used for object selection). The Table also includes the +amount of the integrated P(z) contained between ∆z = ±0.5 +of the stated redshift, for example +P(z = zc) = +� zc+0.5 +zc−0.5 +P(z′) dz′ +(1) +in bins with central redshifts of zc = 4, 5, 6, 7, 8, and 9 (see +Finkelstein et al. 2022b). These integrated probabilities indi- +cate a likelihood that a given galaxy is within the redshift bin +to which it is assigned. For our analysis we rederive the pho- +tometric redshifts below (from SED modeling that includes +the new 5.6 and 7.7 µm MIRI data) but include this here as +we use the P(z) in Table 5 as a prior likelihood on the SED +fitting (discussed in Section 3 below). +Three of the galaxies in our sample have spectroscopic red- +shifts. This includes a previously known galaxy (ID 6811 +in our catalog) with z = 8.683 from Zitrin et al. (2015), +and two new redshifts obtained by the CEERS and WERLS +collaborations from observations with Keck/DEIMOS and +Keck/LRIS. The latter two sources are ID 37653 with z = +4.899 measured by (Stawinski et al. 2023a, in preparation) +and ID 19180 with z = 5.077 (Stawinski et al. 2023b, in +preparation). In all of these cases the spectroscopic redshifts +are consistent with the photometric redshifts in Table 5, and +we fix the redshift to the value of the spectroscopic redshift +in our analysis of the spectral energy distributions below. +Figure 3 shows the HST/ACS, HST/WFC3, Spitzer/IRAC, +and JWST/MIRI imaging for all the objects in our sample, +with the objects ordered by increasing redshift (the full Fig- +ure Set of all 28 objects is available online). In all cases the +galaxies show prominent “Lyman–breaks” at the location of +the redshifted Lyman-limit and/or Lyman-α. In some cases, +the flux density appears to be much brighter in a given pass- +band compared to the adjacent band (for example, galaxy +ID=12514 at z = 7.6 shows evidence of enhanced emission +at MIRI 5.6 µm, indicative of strong redshifted Hα). Fig- +ure 2 illustrates how the bandpasses are sensitive to different +features in the SED of galaxies (using z = 5.1 and z = 8.7 as +examples as these are similar to two of the objects with spec- +troscopic redshifts in our sample). We will return to these +cases below when we explore constraints on the galaxy stel- +lar populations by modeling their SEDs (Section 3). +Figure 1 shows the MIRI [5.6]−[7.7] colors for galaxies in +our sample, compared to the distribution of MIRI [5.6]−[7.7] +colors for all objects detected in the MIRI images. The three +sources with spectroscopic redshifts are indicated with larger +symbols. The high-redshift sources in our sample have MIRI +[5.6]−[7.7] colors largely consistent with expectations: most +objects have relatively flat ([5.6]−[7.7] ≈ 0 mag) or blue col- +ors ([5.6]−[7.7] ∼< 0 mag). This implies that in most cases the +MIRI data sample the continuum of galaxies. In some cases +the MIRI colors suggest very blue colors, [5.6]−[7.7] ∼< −0.5 +to −1 mag, roughly bounded by the bold-dashed line in the +figure. The dashed line represents a photoionization model + +6 +ACS F606W +ID=7600 +z=4.57 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=13389 +z=4.57 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=37653 +z=4.899 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=42638 +z=4.71 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=19180 +z=5.077 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=41545 +z=5.19 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=7818 +z=5.27 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=12514 +z=7.57 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=7364 +z=8.52 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=6811 +z=8.683 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +ACS F606W +ID=26890 +z=9.16 +ACS F814W +WFC3 F125W WFC3 F140W WFC3 F160W +IRAC 3.6 m +IRAC 4.5 m +MIRI F560W +MIRI F770W +Figure 3. Montage of images of a subset of the galaxies used in this study (ordered by increasing redshift). Each row shows images for one +galaxy (labeled by galaxy ID and redshift). The images are (left to right), ACS F606W, F814W, WFC3 F125, F160W, IRAC Ch1 (3.6 µm) and +Ch2 (4.5 µm), and MIRI F560W and F770W. The images above include the three galaxies with spectroscopic redshifts (ID 37653, 19180, and +6811). The images are 6′′ ×6′′ centered on the galaxy in each bandpass, as labeled along the top row). The complete figure set (28 images) is +available online. +Fig. Set 3. Montage of ACS F606W, F814W, WFC3 F125W, F140W, F160W, IRAC 3.6 and 4.5 µm, and MIRI 5.6 and 7.7 µm images +for all galaxies in the sample (28 figures). + +7 +with a young, metal-poor stellar population (10 Myr; 0.1 Z⊙) +driving strong nebular emission (set by an ionization param- +eter of logU = −2). We will explore evidence for this inter- +pretation below. +3. SPECTRAL ENERGY DISTRIBUTION MODELLING +We model the spectral energy distributions (SEDs) of the +galaxies in our sample using stellar population synthesis +models. Our goal is to test how the inferred properties of +the stellar populations in the high-redshift galaxies change by +including the JWST/MIRI 5.6 and 7.7 µm data, in particular +the stellar masses and the SFRs. Previous work (pre-JWST) +showed the MIRI data is able to recover these quantities ac- +curately (Bisigello et al. 2017), and here we test how they +improve the constraints on the stellar population parameters. +This is critical for galaxies at higher redshifts, z ∼> 4, where +the rest-frame optical features shift to longer wavelengths +(rest-frame 4000 Å corresponds to >2 µm), which probes +light from longer-lived stars. Perhaps more problematic are +the effects of nebular emission lines, which can litter the op- +tical portion of the SED (see Figure 2). There are observa- +tions that z > 2 galaxies have a higher incidence of “extreme” +emission lines with rest-frame EWs up to ≈1000 Å (e.g., van +der Wel et al. 2011; Tang et al. 2019; Tran et al. 2020; Boyett +et al. 2022; Matthee et al. 2022; Pérez-González et al. 2022; +Sun et al. 2022), consistent with inferences made from the +>3 µm colors of z > 6 galaxies (Smit et al. 2015; Roberts- +Borsani et al. 2016; Castellano et al. 2017; Hutchison et al. +2019; Endsley et al. 2021). As the EW scales with redshift as +(1+z) this implies these lines have a stronger impact for high- +redshift galaxies for bandpasses of fixed wavelength width +(e.g. Papovich et al. 2001; Burgarella et al. 2022). +We model each galaxy by fitting HST/ACS and WFC3, +Spitzer/IRAC, and JWST/MIRI data with stellar population +models using BAGPIPES (Carnall et al. 2018). BAGPIPES +is a Bayesian SED-fitting code that models the multiband +photometry (flux densities) with stellar population synthesis +models formed over a wide range of user-defined parame- +ters. The code has flexibility on the type of stellar popula- +tion synthesis models, star-formation history, dust attenua- +tion, and nebular emission. It has the ability to incorporate +prior knowledge on parameters. +The code then computes +a probability density for model parameters (i.e., posteriors) +given the data by calculating a likelihood weighted by priors +on the parameters, and samples the posteriors for the param- +eters using the MultiNest nested sampling algorithm (see +Feroz et al. 2009; Carnall et al. 2018). +Table 2 lists the range of parameters considered for the +SED fitting in this study. +For all models we use stellar +population synthesis models from Bruzual & Charlot (2003) +formed with a Chabrier IMF. The table defines the parameters +and their range of parameter values we explored. BAGPIPES +also can incorporate priors on these parameters. +In most +cases we adopt uniform priors, as listed in Table 2, with +two exceptions. The first is related to the nebular ioniza- +tion parameter, which controls the strength of the nebular +emission features. Current evidence from spectroscopy (e.g., +Oesch et al. 2015; Stark et al. 2015, 2017; Le Fèvre et al. +2015; Sanders et al. 2016, 2020; Laporte et al. 2017; Back- +haus et al. 2022; Papovich et al. 2022), including recent +JWST spectroscopy (Brinchmann 2022; Schaerer et al. 2022; +Trump et al. 2022), shows that emission lines are common +in star-forming galaxies at z ∼> 1 (and the strength appears +to increase with increasing redshift). Therefore we fix the +ionization parameter to a high, physically plausible value of +logU = −2 as this is representative of the values used in previ- +ous studies when fitting the SEDs of galaxies (see for exam- +ple the discussion in Whitler et al. 2022). We plan to explore +how variations in the ionization parameter impact the con- +straints on the stellar populations using future data that can +include spectroscopy, e.g., from JWST/NIRSpec. +The other exception is for galaxies with photometric red- +shifts, where we use the photometric redshift posterior, +P(z), derived from EAZY as the prior on the redshift (see +Chworowsky et al. 2022). For galaxies with spectroscopic +redshifts, we force the fit to the spectroscopic redshift value +listed in Table 5. +For the galaxy star-formation histories (SFHs) we test two +possibilities. First, we primarily employ “delayed-τ” mod- +els, where SFR ∝ (t/τ) × exp(−t/τ) for age, t, and star- +formation e-folding timescale, τ. These models allow SFHs +that rise with time (when t/τ ≪ 1) (as is expected for high- +redshift galaxies, Finlator et al. 2011; Papovich et al. 2011) +and for exponentially declining models (when t/τ ≫ 1) and +these have the flexibility to broadly reproduce the evolution +of galaxies over long time periods (e.g., Larson & Tinsley +1978; Tinsley 1980; Carnall et al. 2019; García-Argumánez +et al. 2022). +Second, following Papovich et al. (2001) we also consider +more extreme SFHs that include both the delayed-τ model +(above) an early burst of stars that formed at zf = 100. The +reason for this is that a burst of stars formed at the earliest +times has the highest mass-to-light ratio possible. As such it +adds the most amount of stellar mass (at least for commonly +assumed IMFs, like the Chabrier one assumed here), with the +minimum impact on the observed galaxy SED. In contrast, +the slowly evolving delayed-τ model provides a fit to the +light from the more-recently formed stellar populations that +dominate the rest-frame UV and optical light. Therefore, the +UV/optical light from the stars formed in the maximally old +burst can be “lost in the glare” of more recently formed stars +(this is also referred to as “outshining”, Conroy 2013). The +choice of zf = 100 is motivated by the fact that current mod- +els expect stars to be forming by z = 20−30 Barkana & Loeb +2001, and references in Section 1). The time from z = 100 to +z = 20 spans less 200 Myr, during which little stellar evolu- +tion occurs for the longer-lived stars that dominate the stellar +mass. By using z f = 100 we allow for a “maximally old” +stellar population and we constrain any star-formation that +may have occurred at the earliest times which could have the + +8 +Table 2. Parameter Settings for BAGPIPES +Model +Parameter +Prior +Limits +Star-Formation History (1): +Delayed-τ, Ψ ∝ (t/τ)exp(−t/τ) +e-folding timescale, τ / Gyr +Uniform +(0.01, 10) +age, t / Gyr +Uniform +(0.01,15) +stellar mass, log(M∗/M⊙) +Uniform +(5, 12) +Star-Formation History (2): +Burst at zf = 100 and delayed-τ model from (1) +burst age, tburst / Gyr +Fixed +tburst = Age(z) - Age(z f = 100) +burst stellar mass, log(M∗/M⊙) +Uniform +(0, 13) +Additional parameters for all models +dust attenuation law +. . . +Calzetti (2001) +dust attenuation, A(V) / mag +Uniform +(0, 3) +metallicity, Z/Z⊙ +Uniform +(0,1) +ionization parameter, logU +Fixed +−2 +redshift†, z +EAZY P(z) +(3, 15) +†For galaxies with photometric redshifts the redshift prior is the posterior from the photometric redshift. For galaxies with spectroscopic +redshifts the redshift is fixed at the spectroscopic redshift. +highest possible M/L (for a canonical stellar populations).3 +By studying the SEDs of galaxies to longer wavelengths we +can constrain the amount of light in this population. For ex- +ample, Papovich et al. (2001) and Dickinson et al. (2003) +found that using K-band data, galaxies at z ∼ 3 could hide as +much as 75–90% of their stellar mass in early bursts formed +at z f = 100. Indeed, at the risk of foreshadowing, we find +that including the MIRI 5.6 and 7.7 micron data reduces the +amount of possible stellar mass formed in such maximally +old bursts by up to an order of magnitude (see Section 4.4). +4. RESULTS +4.1. Analysis of Galaxy SEDs +Figure 4 shows the BAGPIPES SED fits and one- +dimenstional (1D) posterior likelihoods for select parameters +of the SED fit. The Figure shows six galaxies as an example. +The online version of the Paper includes a Figure Set with +these plots for the full sample. For each galaxy, the plots +compare the SED fits with and without the MIRI 5.6 and +7.7 µm data. Tables 6 and 7 provide the medians (50th per- +centiles), 16th and 84th percentiles derived from these poste- +rior likelihoods for the stellar masses, SFRs, and photometric +redshifts for all galaxies in the sample, both with and without +using the MIRI data, respectively. +3 +A redshift of z = 100 is essentially “immediately” in the history of the +Universe as it corresponds to an age of only ≃17 Myr after the Big Bang +for the assumed cosmology. Stars are expected to form by z ≈ 20 − 30 +(Barkana & Loeb 2001 and references in Section 1), and there are reputed +candidates from JWST imaging for galaxies at redshifts as high as z ≈ 15 +(see Section 1). Therefore, zf = 100 seems to be a reasonable upper bound +to ensure we capture the earliest time when stars could plausibly form. +Using a lower formation redshift, zf < 100, would lower the upper limit on +the stellar masses that could form in the bursts as these would be younger, +with lower M/L. +To test the robustness of the stellar masses and SFRs de- +rived from the BAGPIPES fits, we have refit all the galaxies +in our sample using several independent SED-fitting codes ( +CIGALE, Boquien et al. 2019; FAST, Kriek et al. 2009, and +the codes of Santini et al. 2022a and Pérez-González et al. +2008). Comparing the stellar masses and SFRs, we find they +agree in the mean (with bias, µ ≃ 0 dex) and an inter-method +scatter of σ = 0.23 dex in stellar mass (a factor of 1.7) and +σ = 0.27 dex in SFR (a factor of 1.9). This scatter is typical +in comparisons of SED-fitting results (e.g., Mobasher et al. +2015). We therefore interpret the scatter as representative of +the systematic uncertainties on the stellar masses and SFRs +here. +In some cases adding the MIRI data has a small effect on +the median values of the stellar mass and SFR, but it does +tighten the allowable range of these parameters. Figure 4 +panels (a) and (b) show galaxy ID 7364 (at z = 8.1−8.2) and +galaxy ID 6811 (with zsp = 8.683, Zitrin et al. 2015). These +have MIRI data that support the interpretation inferred us- +ing only the HST and Spitzer data. However, in both of these +cases adding the MIRI data tighten the allowed range of mod- +els, and thus improve the constraints on the stellar popula- +tion parameters (this is true for the sample in general). In +both cases shown here, the favored range of stellar mass and +SFR are improved significantly when MIRI data are included +(with improvements in the inter-68%-tile range by more than +a factor of two). Below the SED fit for each galaxy, Figure 4 +shows the posterior probability densities for the SFR, mass- +weighted age (AgeMW), stellar mass, and dust attenuation. +Adding the MIRI data typically produce narrower posteriors +for SFR and stellar mass. This is a result of improved con- +straints on the dust attenuation (A(V)), and this forces the +models to a narrower range of SFR and stellar mass. This is +the case for ID 7364 and 6811. + +9 +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +22 +23 +24 +25 +26 +27 +AB magnitude +log M /M + = 9.9 (+0.1,-0.2) +SFR/M + yr +1 = 52 (+16,-13) +z = 8.1 (+0.5,-0.7) +log M /M + = 9.8 (+0.2,-0.3) +SFR/M + yr +1 = 39 (+24,-16) +z = 8.2 (+0.4,-0.8) +ID 7364 +with MIRI [5.6], [7.7] +without MIRI +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +23 +24 +25 +26 +27 +AB magnitude +log M /M + = 9.8 (+0.1,-0.2) +SFR/M + yr +1 = 40 (+12,-10) +z = 8.683 +log M /M + = 9.8 (+0.3,-0.4) +SFR/M + yr +1 = 47 (+45,-23) +ID 6811 +with MIRI [5.6], [7.7] +without MIRI +0 +100 +SFR/M + yr +1 + + + + + +density +0.0 +0.2 +0.4 +AgeMW/Gyr + + + + + + +9 +10 +log M /M + + + + + + +0 +1 +A(V)/mag + + + + + + +without MIRI +with MIRI + + + + + + + + + + + + + + + + + + + + + + + + +(a) +0 +200 +SFR/M + yr +1 + + + +density +0.0 +0.2 +AgeMW/Gyr + + + + + + +9 +10 +log M /M + + + + + + +0 +1 +A(V)/mag + + + + + + +without MIRI +with MIRI + + + + + + + + + + + + + + + + + + + + + + +(b) +Figure 4. +Examples of SED fits for galaxies in the sample. The green data points show the measured flux densities and uncertainties on the +HST/WFC3, Spitzer/IRAC, and JWST/MIRI bands. The red-shaded regions shows the model fit to all the data points (the shaded region shows +the inner-68% range of models; the solid red line shows the median). The black-shaded region shows the fit to the data points excluding the +MIRI bands. The small red and grey points show the median-model photometry. The inset text gives the median and 68%-tile uncertainties +on the stellar masses and SFRs inferred from the fits. Below each SED plot, the panels show the posteriors (probability density) for the SFR, +mass-weighted (MW) age, stellar mass, and dust attenuation for each galaxy (using MIRI and excluding MIRI data). The dashed lines denote +the 5, 50, and 95% intervals. The examples include galaxies where adding the MIRI data yields similar stellar masses and SFRs, but with tighter +constraints (panels (a) and (b)). Other examples show galaxies where adding the MIRI data greatly reduces both the stellar masses and SFRs. +Typically this results from contamination in the IRAC data or because of strong emission lines impacting the IRAC data (or both; see panels +(c) and (d)), or because the stellar continua appear very blue (see panels (e) and (f)). The complete figure set (28 images) is available online. +Fig. Set 4. SED fits and 1D posteriors for SFRs, stellar masses, mass-weighted ages, and dust attenuation for all galaxies in the sample. +In other cases adding the MIRI data changes the interpre- +tation of the galaxy stellar populations dramatically. Fig- +ure 4 panels (c) and (d) show galaxies with ID 19180 (with +zsp = 5.077) and ID 18441 (at z = 6.5−6.6). In both of these +cases, without MIRI data the HST to IRAC 3.6 and 4.5 µm +data implied very red rest-UV–to–optical colors, leading to +high dust-attenuation values (A(V) ∼> 1 mag) with high SFRs +(∼> 90 − 100 M⊙ yr−1). The stellar masses in these cases are +also elevated primarily because the higher dust attenuation +increases the mass-to-light ratio (M/L) of the models, and +therefore increases the stellar mass. Including the MIRI data +changes the favored stellar population models to ones with +much bluer rest-UV–to–optical colors. As a result the dust +attenuation, SFR and stellar mass are decreased, by an order +of magnitude in some cases (the decrease in stellar mass is +more than that of the SFR, implying the specific SFR declines +slightly as well). This impacts roughly ≈33% of the sample +here (10 of the 28 galaxies based on our visual inspection of +the SEDs and 1D posteriors, see Figure 4). +Yet in other cases, the MIRI data forces the constraints on +the stellar populations to be bluer than expected based on the +HST and Spitzer data. Figure 4 panels (e) and (f) show two +galaxies that demonstrate these situations. In both the cases +of galaxy ID 12514 (at z = 7.4 − 7.7) and ID 26890 (at z = +8.8−8.9) the MIRI data favor very blue UV/optical colors. In +the case of galaxy 12514, there are indications that the IRAC +and MIRI 5.6 µm data are boosted by the presence of strong +emission lines. Having the 7.7 µm photometry favors a lower +stellar continuum. As a result the SFR and stellar mass are +reduced (the presence of redshifted Hα in the MIRI F560W +band is apparent even in the galaxy image in Figure 3 which +shows the 5.6 µm flux density is noticeably brighter than the +7.7 µm flux density). For this reason the SED fit favors a + +10 +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +22 +23 +24 +25 +26 +AB magnitude +log M /M + = 9.7 (+0.2,-0.2) +SFR/M + yr +1 = 34 (+14,-9) +z = 5.077 +log M /M + = 10.5 (+0.2,-0.2) +SFR/M + yr +1 = 120 (+60,-41) +ID 19180 +with MIRI [5.6], [7.7] +without MIRI +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +24 +25 +26 +27 +28 +AB magnitude +log M /M + = 9.3 (+0.1,-0.2) +SFR/M + yr +1 = 10 (+4,-3) +z = 6.5 (+0.6,-0.6) +log M /M + = 10.2 (+0.3,-0.3) +SFR/M + yr +1 = 90 (+67,-45) +z = 6.6 (+0.3,-0.6) +ID 18441 +with MIRI [5.6], [7.7] +without MIRI +0 +200 +400 +SFR/M + yr +1 + + + + + +density +0.0 +0.5 +AgeMW/Gyr + + + + + + +9 +10 +11 +log M /M + + + + + + +0 +1 +2 +A(V)/mag + + + + +without MIRI +with MIRI + + + + + + + + + + + + + + + + + + + + + + +(c) +0 +200 +400 +SFR/M + yr +1 + + + + + +density +0.00 +0.25 +0.50 +AgeMW/Gyr + + + + + + +9 +10 +11 +log M /M + + + + + + +0 +2 +A(V)/mag + + + + + + +without MIRI +with MIRI + + + + + + + + + + + + + + + + + + + + + + + + +(d) +Figure 4 (continued). +slightly higher photometric redshift (to accommodate the Hα +emission in the F560W bandpass), see Section 4.2 below. +Galaxy ID 26890 is noteworthy in itself because it is +the highest redshift galaxy in the sample, and because +the HST/WFC3–to–JWST/MIRI colors are H160 − [5.6] ≈ +−0.5 mag, and indicative of stellar populations with very low +M/L ratios. The IRAC 4.5 µm emission shows indications of +enhancement, possibly owing to redshifted Hβ+[O III]. The +MIRI data reign in the allowable range of stellar population +parameters, favoring models with lower SFRs than the con- +straints that lack MIRI data. +Therefore, the MIRI data favor bluer rest-UV/optical col- +ors compared to the IRAC data. In part this may have a physi- +cal explanation. At the redshifts of the galaxies under consid- +eration, the IRAC data contain strong emission lines, includ- +ing Hβ+[O III] at 5 < z < 8, [O II]+[Ne III] at 7 < z < 12, and +Hα+[N II] at 4 < z < 7 (Labbé et al. 2013; Smit et al. 2015; +De Barros et al. 2019; Roberts-Borsani et al. 2020; Ends- +ley et al. 2021). At certain redshifts both the 3.6 and 4.5 µm +bands can be both be affected (e.g., Smit et al. 2014; Roberts- +Borsani et al. 2020). Without the SED fits, the models that fit +the data may include both models with strong emission lines +and/or models with redder colors (indicative of strong dust +attenuation) or both. The models in Figure 4 show that the +presence of strong emission lines in the bands augments the +flux densities. This could account for part of the difference +between the SED fits with and without MIRI: the MIRI data +are evidence in these cases that the stellar populations have +bluer colors, and the elevated IRAC flux densities then would +require strong emission line EWs to reproduce them. +However, crowding between sources in the IRAC data may +be another reason for the increased IRAC flux densities. The +lower angular resolution of the IRAC data can cause blending +from bright, nearby galaxies, and this can lead to additional +uncertainties in the flux density. As the images in Figure 3 il- +lustrate, some galaxies (e.g., ID 19180 and 6811) have bright +objects within 3 arcseconds. The light from these objects +makes deblending more difficult and could potentially bias +the flux-density measurements (see, e.g., Laidler et al. 2007; +Skelton et al. 2014; Merlin et al. 2016). For this reason the +elevated IRAC flux densities may include systematic mea- +surement uncertainties from blended objects. In Appendix A +we test for the effect of source blending by excluding ob- +jects that have any bright neighbor within 3′′ (we show that +source blending does not bias our interpretation for the stellar +masses nor SFRs, nor in their evolution, that we infer for the +galaxy population). However this does emphasize the benefit +of having data with the enhanced angular resolution available +from JWST imaging (for both NIRCam and MIRI). +4.2. The Impact of MIRI on Redshifts and Implications for +Emission Lines +Figure 5 compares the redshifts for the galaxies in our sam- +ple derived from our BAGPIPES fits for the galaxies exclud- +ing the MIRI data (znoMIRI) and when including the MIRI +data (zMIRI) as a function of the prior photometric redshift + +11 +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +24 +25 +26 +27 +28 +AB magnitude +log M /M + = 8.8 (+0.2,-0.3) +SFR/M + yr +1 = 5 (+2,-2) +z = 7.7 (+0.5,-0.6) +log M /M + = 9.3 (+0.4,-0.4) +SFR/M + yr +1 = 12 (+13,-7) +z = 7.4 (+0.8,-0.6) +ID 12514 +with MIRI [5.6], [7.7] +without MIRI +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +24 +25 +26 +27 +28 +AB magnitude +log M /M + = 8.8 (+0.3,-0.3) +SFR/M + yr +1 = 7 (+2,-3) +z = 8.8 (+0.2,-0.2) +log M /M + = 9.4 (+0.3,-0.3) +SFR/M + yr +1 = 19 (+15,-8) +z = 8.9 (+0.3,-0.2) +ID 26890 +with MIRI [5.6], [7.7] +without MIRI +0 +50 +SFR/M + yr +1 + + + + + +density +0.0 +0.2 +0.4 +AgeMW/Gyr + + + + +8 +10 +log M /M + + + + +0 +1 +2 +A(V)/mag + + + + + + +without MIRI +with MIRI + + + + + + + + + + + + + + + + + + + + +(e) +0 +50 +SFR/M + yr +1 + + + +density +0.0 +0.2 +AgeMW/Gyr + + + + +8 +10 +log M /M + + + + + + +0 +1 +A(V)/mag + + + + + + +without MIRI +with MIRI + + + + + + + + + + + + + + + + + + + + +(f) +Figure 4 (continued). +5 +6 +7 +8 +9 +redshift, z +0.1 +0.0 +0.1 +( znoMIRI +zMIRI) / (1 + z) +35896 +7818 +12514 +Figure 5. Comparison of the redshifts for galaxies in our sample +with and without the MIRI data. The data points show the rela- +tive difference between the photometric redshifts (derived from our +BAGPIPES fits) for the galaxies excluding MIRI (znoMIRI) and when +including MIRI (zMIRI) as a function of the prior photometric red- +shift from Finkelstein et al. (2022b). Annotated points indicate ob- +jects with |znoMIRI −zMIRI| > 0.2. These objects indicate shifts in +their photometric redshift, where at least in part this is because of +nebular emission in one or more bands. +from Finkelstein et al. (2022b). For most galaxies there is +good agreement. (For completeness we include all 28 galax- +ies in this comparison, using the photometric-redshifts even +for galaxies with spectrscopic redshifts.) +In several instances we see the median redshift from the +posterior shifts appreciably when including the MIRI data. +Figure 5 shows three objects where the shift is greater than +∆z = 0.2. In all cases the shift in the redshift probability +density appears to be related to the effects of one or more +emission lines in the IRAC or MIRI passbands. Figure 6 +illustrates the shift in P(z) from the BAGPIPES fits for these +galaxies. +Two of these galaxies lie at 4 < z < 6 (galaxy IDs 35896 +and 7818). +Adding the MIRI data favors having strong +nebular emission in both of the IRAC bands. +This oc- +curs at z ∼> 5 when Hβ+[O III] enters the IRAC Ch1 band- +pass (at 3.6 µm) and Hα enters the IRAC Channel 2 band- +pass (at 4.5 µm; see Figure 2). +Inspection of the galaxy +SEDs (see Fig. Set. 4) shows that the IRAC–to–MIRI col- +ors ([3.6] − [5.6] and [4.5] − [5.6]) are blue for both of these +galaxies. The BAGPIPES fits that include the MIRI data fa- +vor models in which the galaxy has strong nebular emission +in the IRAC bands to account for these color. This increases +(decreases) the redshift probability density at higher (lower) +redshifts for these galaxies. +The remaining galaxy, ID 12514, lies at z ≈ 7.7. Figure 6 +shows the redshift probability densities for this galaxy. The +MIRI F560W image (Fig. 3 shows evidence of “boosted” flux + +12 +4.5 +5.0 +5.5 +6.0 +redshift +0.0 +0.5 +1.0 +1.5 +P(z) +ID 35896 +with MIRI +without MIRI +4.5 +5.0 +5.5 +redshift +0.0 +0.5 +1.0 +1.5 +2.0 +P(z) +ID 7818 +with MIRI +without MIRI +6 +7 +8 +9 +redshift +0.0 +0.2 +0.4 +0.6 +P(z) +ID 12514 +with MIRI +without MIRI +Figure 6. Comparison of redshift posterior probability densities, P(z), derived from the BAGPIPES fits to three of the galaxies in the sample +(as labeled). These three galaxies each have a change of more than 0.2 between the median redshift when including the MIRI F560W and +F770W data in the fit versus when they are excluded. For the case of galaxy IDs 35896 and 7818, the IRAC−MIRI colors are blue, indicating +redshifted emission lines (likely Hβ+[O III] and Hα) inhabit both IRAC bands (while the MIRI data probe the galaxy continua). In the case of +galaxy ID 12514, the MIRI F560W band clearly shows boosted emission, likely from redshifted Hα at z = 7.6−8. +compared to the MIRI F770W image, where the MIRI color +is [5.6]−[7.7] = −1.1±0.3 mag, see Table 5). The most likely +explanation for this boosted emission appears to be from red- +shifted Hα+[N II] at z = 7.6 − 8.0 in the MIRI F560W band- +pass (see Fig. 4). The strength of the flux in the F560W band- +pass decreases the probability density for redshifts with z ∼< 7 +as these would place Hα at wavelengths shorter than covered +by the bandpass (see Figure 2). +Therefore, nebular emission lines appear to be responsible +for the cases where there are larger shifts in the photometric +redshift solutions, However, these cases are generally excep- +tions. For most galaxies in the sample the redshift posteriors +are consistent, where Fig. 5 shows the differences are negligi- +ble in median redshifts with and without including the MIRI +data. We expect that many (even most) of the galaxies in +the sample also exhibit strong emission lines given the pre- +ponderance of evidence from other galaxies at these redshifts +(see Section 4.1) and by inspection of the SEDs in Figure 4. +In these cases adding the MIRI data supports the redshifts, +either because the emission lines make less of an impact on +the redshift likelihoods or because the MIRI data reinforce +them. +4.3. The Impact of MIRI on the Stellar Masses, Dust +Attenuation, and SFRs +Figure 7 compares the stellar masses and SFRs derived for +the galaxies in our sample using the simple delayed-τ models +with BAGPIPES for the case where we include the MIRI +F560W and F770W data and when we exclude it. In general, +including the MIRI data reduces the stellar masses and SFRs +for the galaxies. +We study the offsets in stellar mass and SFR for our galax- +ies in two redshift bins, 4 < z < 6 and 6 < z < 10, where the +median offsets in stellar mass and SFR are indicated in the +Figure as large rectangles, and are listed in Table 3. Formally, +the offsets in stellar mass are ∆log(M∗/M⊙) = 0.25 dex for +4 < z < 6 and 0.38 dex for 6 < z < 10. The inter-68%-tile +intervals are 0.28 and 0.44 dex, respectively (estimated us- +ing the median absolute deviation, σNMAD). For the SFRs the +offsets are ∆log(SFR/M⊙ yr−1) = 0.15 dex for 4 < z < 6 +Table 3. Offsets in Stellar Mass and SFRs for 4 < z < 10 galaxies +when including the MIRI F560W and F770W data +Stellar Mass offsets +SFR offsets +⟨∆logM∗⟩ = +⟨ ∆ log SFR ⟩ = +logM∗,noMIRI/M∗,MIRI +log SFRnoMIRI / SFRMIRI +Sample +Median +Scatter +Median +Scatter +4 < z < 6 +0.25 dex +0.28 dex +0.15 dex +0.12 dex +6 < z < 10 +0.38 dex +0.44 dex +0.29 dex +0.27 dex +NOTE—The quantities with the subscript “noMIRI” denote values +derived without the MIRI data. +Quantities with the subscript +“MIRI” denote the values derived with the 5.6 and 7.7 µm MIRI +data. The scatter is the inter-68-percentile interval derived from the +median absolute deviation. +and 0.29 dex for 6 < z < 10, with an inter-68 percentile +interval of 0.12 dex and 0.27 dex, respectively. +Because +the impact of MIRI is larger on the stellar mass than the +SFR, the specific SFRs will be reduced by approximately by +∆logSFR−∆logM∗ ≈ 0.1 dex (see Table 3). +The reasons for the offsets are similar to that discussed +for the individual galaxies in Section 4.1. +Including the +MIRI data generally favors stellar populations with bluer +rest-UV/optical colors, with lower M/L ratios. This forces +the constraints to lower stellar-mass models. For the SFRs, +because the colors are bluer, there is less dust attenuation +favored in the models, which lowers the SFRs compared to +models with higher dust attenuation (at fixed observed galaxy +luminosity). The MIRI data also remove some of the degen- +eracy between models with redder stellar populations versus +those with strong emission lines impact select bands. These +have the combined effect of favoring lower stellar masses and +SFRs when including the MIRI data. +There is significant scatter in the offsets of stellar mass +and SFR for individual objects. In Figure 7 the error bars on +the large rectangles show the inter-68-percentile range (i.e,. + +13 +8 +9 +10 +11 +log M [M +] +stellar mass with MIRI +8 +9 +10 +11 +stellar mass without MIRI +log M [M +] +4 +6 +8 +redshift +0.5 +0.0 +0.5 +1.0 +1.5 + log M (dex) +0 +1 +2 + log SFR [M + yr +1] +SFR with MIRI +0 +1 +2 +SFR without MIRI +log SFR [M + yr +1] +4 +6 +8 +redshift +0.5 +0.0 +0.5 +1.0 +1.5 + log SFR (dex) +Figure 7. Comparison of stellar masses and SFRs derived from the SED modeling for galaxies including the MIRI F560W and F770W data +and without the MIRI data. The top set of panels show the comparison for the stellar mass. The bottom set of panels show the comparison for +the SFR. In each row the left panel shows the direct comparison, where the dashed line shows the one-to-one relation. The right panel shows the +logarithmic difference, defined as ∆logMwith MIRI +∗ +−logMwithout MIRI +∗ +(similarly for the SFR). The symbols are color-coded by redshift (using the +right panel). In the right panel the large rectangular boxes (and error bars) show the median value (and scatter) in two bins of redshift (4 < z < 6 +and 6 < z < 10). The values for these are given in Table 3. Adding the MIRI data generally lowers the SFRs and stellar masses of these galaxies +(though the scatter is significant). The median offset is larger for galaxies at higher redshifts. +the difference between the 16–84 percentiles) for both the +4 < z < 6 and 6 < z < 10. For some galaxies the offsets are +insignificant, with ∆logM∗ ≈ 0 and ∆logSFR ≈ 0. Two of +these galaxies are shown in Figure 4 (IDs 7364 and 6811; +others are available in Fig. Set. 4). In these cases the MIRI +data tighten the existing constraints on the derived stellar +masses and SFRs, reducing the uncertainties (by 0.1–0.2 dex +in stellar mass and by 0.3 dex in SFR). In other words, for +some cases the MIRI data support the range of stellar popula- +tion parameters favored by the fits to the HST/ACS + WFC3 +and Spitzer/IRAC data, but the MIRI data improve the accu- +racy of the measurements, typically by a factor of order two. +In other cases the MIRI data dramatically change the inter- +pretation of the galaxies. This was noted above for galaxies +18441 and 19180 (Section 4.1 and Figure 4), where adding +the MIRI data reduce the stellar mass and SFRs substantially. +Figure 7 shows that this is typically the case, where the MIRI +data decrease the average stellar mass and SFR for galaxies + +14 +in our sample, typically by a factor of order two. We will ex- +plore the implications this has on the evolution of the galaxy +stellar-mass density in Section 5. +4.4. The Impact of MIRI on the Inferred Star-Formation +History (and the Mass Formed in Bursts) +Arguably, one of the most extreme star-formation histo- +ries imaginable is the case where a galaxy forms in either +one burst at zf → ∞, or (slightly less extreme) in a series +of bursts extending back to that time. When a burst forms, +the stellar population immediately begins aging. A burst at +z f = ∞ has the oldest possible age at any subsequent time, +and the smallest amount of light (at a given mass), and there- +fore it would have a maximal M/L at any observed epoch. +As the stellar population ages, its colors also become redder. +For all these reasons, it is conceivably possible to hide signif- +icant amounts of stellar mass formed at earlier times (this is +the “outshining” problem, e.g., Papovich et al. 2001; Dickin- +son et al. 2003; Papovich et al. 2006; Finkelstein et al. 2010; +Pforr et al. 2012; Conroy 2013). +One advantage of focusing on galaxies at high redshifts +is that the amount of time for discrete episodes of star- +formation (i.e., many individual bursts) is small given the age +of the Universe (the Universe has an age of 1 Gyr at z = 5.7 +and 500 Myr at z = 9.6 for the adopted cosmology). This is +shorter than the lifetimes of stars of spectral type A and later. +The short age of the Universe, combined with longer wave- +lengths probed by the MIRI 5.6 and 7.7 µm data allow us +to place tighter constraints on the mass from earlier bursts in +high redshift galaxies than has been possible previously. +We explored the possibility that such an early burst could +contribute stellar mass to the galaxies in our sample, and how +the JWST/MIRI data can improve the constraints on this pop- +ulation. We used the SED fits to the galaxies in our sample +using a star-formation history that include both the delayed-τ +model (as in Section 3 above) and the burst of stars formed at +z f = 100. The details of the other model parameters are listed +in Table 2. +To quantify the amount of allowed stellar mass formed +in the burst for each galaxy, we use the Bayesian Informa- +tion Criterion (BIC, Bailer-Jones 2017). The BIC provides +a criterion for model selection in the case that one intro- +duces a model with an additional parameter (in our case we +are selecting between between two models, one with a star- +formation history that has a delayed-τ model only, and one +with both the delayed-τ model and an early burst at zf = 100). +When comparing the two models, the BIC applies a penalty +term for the additional parameter (to determine if the addi- +tional parameter improves the fit, or is overfitting the data). +We use the BIC defined as (e.g., Buat et al. 2019), +BIC = χ2 +0 × klnN, +(2) +where χ2 +0 is the minimum chi-squared from the SED-fitting, k +is the number of independently fitted parameters (we use k = +7) and N is the number of data points (N = 8 or 10, depending +on if the MIRI [5.6] and [7.7] data are included or not). +Our goal is to quantify the upper limit on the stellar mass +formed in an early burst that could exist in our galaxies. To +do this, we select models with bursts (zf = 100) that are not +excluded by the BIC criteria. That is, for a given galaxy we +identify all models that satisfy χ2 ≤ BIC, where the BIC is +defined in Equation 2, and χ2 is the fit for a given model (this +is similar to the approach adopted by Buat et al. 2019). For +each galaxy, we select the model with the highest stellar mass +in the zf = 100 burst from the subset of models that satisfy the +BIC. We then take this as upper limit on the stellar mass per- +mitted in the burst. Comparing this upper limit on the stellar +mass formed in bursts to the range of stellar masses from the +fits we find that the limiting stellar mass from models that +satisfy the BIC criteria corresponds roughly to a 99.7% up- +per limit on the mass (i.e., the values we report correspond +approximately to a 3σ upper limit on the stellar mass). Ta- +bles 6 and 7 list the upper limit on the stellar mass in the burst +component for all galaxies in the sample for the case that we +include and exclude the MIRI data, respectively. +Figure 8 shows example SED fits for galaxies in our sam- +ple, both with and without the bursts, for the both the cases +that we include and exclude the MIRI 5.6 and 7.7 µm data. +The complete figure set (28 images) is available online. The +examples shown in the Figure include a galaxy where the +IRAC data are bright relative to the MIRI data (ID 18441). +In this case, without the MIRI data the allowed stellar mass +in the burst can reach logM∗/M⊙ = 11.0, but adding MIRI +reduces this by nearly an order of magnitude. For galaxy ID +7364, the MIRI data favor red IRAC–to–MIRI colors. Never- +theless, because the MIRI data constrain the models at longer +wavelengths, they also lower the amount of stellar mass al- +lowed in the burst: without the MIRI data the burst can in- +clude logM∗/M⊙ = 10.9; when MIRI data are included the +mass in the burst declines by 0.3 dex (a factor of two). For +galaxy ID 6811, the MIRI data show that because they con- +strain the SED at longer wavelengths, the amount of stellar +mass allowed in the burst is reduced by a factor of 0.6 dex +(nearly a factor of five). +Therefore, the addition of the MIRI F560W and F770W +data reduce the amount of stellar mass that can form in early +bursts. Figure 9 shows the change in stellar mass for the case +that the star-formation histories include only delayed-τ mod- +els (labeled “no burst” in the figure) compared to when an +early burst of star formation at zf = 100 is included (labeled +“allowed in burst”) in the figure. Figure 9 shows the results +for both the case that the MIRI data are excluded (top row) +and when the MIRI data are included (bottom row). For the +fits that lack MIRI data, the amount of stellar mass allowed +in the burst is nearly an order of magnitude higher than con- +strained in the delayed-τ models: in this case the median +differences in the log of the stellar mass of the models with +early bursts and those with only delayed-τ models is 0.9 dex +at 4 < z < 6 and 1.1 dex at 6 < z < 9. For the fits that in- +clude the MIRI data, the amount of stellar mass allowed in +the bursts is significantly reduced: the median differences in + +15 + + + + + +allowed in burst component +log M / M +=10.0 + + + + + +allowed in burst component +log M / M +=11.1 +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +24 +25 +26 +27 +28 +AB magnitude +with burst: log M /M + = 10.2 (+0.3,-0.3) +no burst: log M /M + = 10.2 (+0.2,-0.3) +ID 18441, z = 6.44 + -- without MIRI -- +with burst at z = 100 +delayed- model only +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +24 +25 +26 +27 +28 +with burst: log M /M + = 9.3 (+0.1,-0.2) +no burst: log M /M + = 9.3 (+0.1,-0.2) +z = 6.29 + -- with MIRI [5.6], [7.7] -- +with burst at z = 100 +delayed- model only + + + + + +allowed in burst component +log M / M +=10.6 + + + + + +allowed in burst component +log M / M +=10.9 +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +23 +24 +25 +26 +27 +AB magnitude +with burst: log M /M + = 9.8 (+0.3,-0.3) +no burst: log M /M + = 9.8 (+0.2,-0.3) +ID 7364, z = 8.04 + -- without MIRI -- +with burst at z = 100 +delayed- model only +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +23 +24 +25 +26 +27 +with burst: log M /M + = 9.9 (+0.1,-0.2) +no burst: log M /M + = 9.9 (+0.1,-0.2) +z = 7.90 + -- with MIRI [5.6], [7.7] -- +with burst at z = 100 +delayed- model only + + + + + +allowed in burst component +log M / M +=10.3 + + + + + +allowed in burst component +log M / M +=11.2 +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +23 +24 +25 +26 +27 +AB magnitude +with burst: log M /M + = 9.9 (+0.3,-0.4) +no burst: log M /M + = 9.9 (+0.3,-0.4) +ID 6811, z = 8.683 + -- without MIRI -- +with burst at z = 100 +delayed- model only +1 +2 +3 +4 +6 +10 +observed wavelength [ m] +23 +24 +25 +26 +27 +with burst: log M /M + = 9.8 (+0.1,-0.2) +no burst: log M /M + = 9.8 (+0.1,-0.2) +z = 8.683 + -- with MIRI [5.6], [7.7] -- +with burst at z = 100 +delayed- model only +Figure 8. Example fits to SEDs for the galaxies in our sample, comparing the results from fits that include an early burst of star-formation at +z f = 100. Data points and upper limits have the same definitions as in Figure 4. In each panel, the shaded regions show the stellar population fit +to the SED using the total model (cyan-shaded = delayed−τ plus the burst) and delayed-τ model only (gray-shaded, these are identical to those +in Figure 4; there is almost no difference between the cyan- and gray-shaded models). The tan-shaded region shows the light permitted in the +burst component. The labels indicate the amount of stellar mass in each component. Each row shows the results for one galaxy, where the Left +panel shows the results that exclude the MIRI F560W and F770W data, and the tight panel shows the results including the MIRI F560W and +F770W data. The complete figure set (28 images) is available online. +Fig. Set 8. SED fits including the MIRI data and excluding the MIRI data for all galaxies comparing the fits with and without bursts +at zf = 100. + +16 +8 +9 +10 +11 +log M [M +] +no burst, no MIRI +9 +10 +11 +allowed in burst: +log M [M +] +4 +6 +8 +redshift +0.0 +0.5 +1.0 +1.5 +2.0 + log M (dex) +8 +9 +10 +11 +log M [M +] +no burst, with MIRI +9 +10 +11 +allowed in burst: +log M [M +] +4 +6 +8 +redshift +0.0 +0.5 +1.0 +1.5 +2.0 + log M (dex) +Figure 9. Comparison of stellar masses derived for galaxies with and without early bursts (at z f = 100). The top row shows the results that lack +MIRI data. The top-left panel shows the stellar masses derived from the delay-τ models only (labeled “no burst”) compared to the models that +include the burst (labeled “allowed in burst”). The top-right panel shows the difference between the stellar masses as a function of redshift. The +bottom row shows the same results for the galaxies including the MIRI data. In the left panels, the dashed lines show the one-to-one relation +and the solid lines show the median offsets. In the right panels, the large rectangles show the medians in two bins of redshift (4 < z < 6 and +6 < z < 9) these are given in Table 4. Adding the MIRI data reduces the amount of stellar mass allowed in the burst components. + +17 +Table 4. Ratio of the stellar mass allowed in models that include an +early burst of star-formation (at z f = 100) to those that include only a +delayed−τ model. +logM∗(with burst)−logM∗(no burst) +with MIRI data +no MIRI data +Sample +Median +Scatter +Median +Scatter +4 < z < 6 +0.59 dex +0.16 dex +0.87 dex +0.23 dex +6 < z < 10 +0.69 dex +0.04 dex +1.11 dex +0.13 dex +NOTE—The values labeled “with burst” denote the upper limit on +the stellar mass for models that include bursts. The values labeled +“no burst” denote the stellar masses for models that assume only a +delayed-τ star-formation history. The values “with MIRI” include +the 5.6 and 7.7 µm flux densities from MIRI. +this case 0.6 dex at 4 < z < 6 and 0.7 dex at 6 < z < 9. These +values are listed in Table 4. +5. DISCUSSION +5.1. On the Colors, Stellar Masses, and Nebular Emission +in Early Galaxies +One of main findings in this paper is that the MIRI data +favor bluer colors in galaxies at 4 < z < 9. Figure 10 shows +this by comparing the relative SED for each galaxy, both in +the case of including the MIRI data and without the MIRI +data. Including the MIRI data reduces the derived rest-frame +I-band light by approximately ∆m1600 − I ≈ 0.4 mag. In- +specting Figure 10 this appears to result from many galax- +ies favoring bluer SEDs when the MIRI data are included. +In other words, without the MIRI data, the SED is uncon- +strained at longer wavelengths, and this allows for a greater +range of SED shape (where the median favors a solution +which on average is redder). Adding the MIRI data shifts the +likelihood to bluer populations for many galaxies. This has +a major impact on the implied M/L, as the blue rest-frame +colors implies younger ages, lower dust attenuation, or both. +The fact that adding the MIRI data makes the galaxies bluer +largely explains the differences in the derived stellar masses +and SFRs observed in Figure 7, where adding the MIRI data +lower the stellar masses and SFRs compared what the models +favor when the MIRI data are excluded. +Therefore, our interpretation of the MIRI data is that galax- +ies at high redshifts (z > 4) are bluer than inferred from pre- +vious studies. This adds to other studies that find that galax- +ies at high redshifts must have (very) blue colors. Studies +from the pre-JWST era have argued that galaxies at z > 4 +show indications of declining (i.e., steepening) UV spectral +slopes with increasing redshift (e.g., Bouwens et al. 2012; +Finkelstein et al. 2012; Wilkins et al. 2016; Bhatawdekar & +Conselice 2021). These conclusions have been reinforced +by early JWST imaging that shows very blue colors among +UV-selected galaxies (e.g., Nanayakkara et al. 2022; Topping +0 +2 +with MIRI 5.6 and 7.7 m +best (median) SED fits for each galaxy +0 +2 +m1600Å - m (mag) +no MIRI data +median rest-frame colors of samples +1000 +2000 +5000 +10000 +rest wavelength [Å] +1 +0 +1 +(m1600Å - m ) (mag) +redder with MIRI data +bluer with MIRI data +Figure 10. +Comparison of median, relative SED for each galaxy, +both for the case that MIRI data are used (top panel) and when the +MIRI data are excluded (middle panel). The individual lines are +the median SED model fit to each galaxy in the sample, shifted to +the rest-frame. The large data points show the median rest-frame +magnitude at 1600 Å, 2800 Å, and U, B, V, and I. The error bars +show the scatter in the sample. All models have been normalized to +the 1600 Å magnitude (which accounts for the lack of scatter at that +wavelength). The bottom panel shows the difference in color (∆m) +between the models with and without the MIRI data (the error bars +show the range of the 16th-84th percentiles of the sample). The +change in the color at the reddest wavelengths probed (about the +rest-frame I-band) corresponds to a ∆m ≈ 0.4 mag +et al. 2022). We find here that including the MIRI 5.6 and +7.7 µm data show strong evidence that the stellar populations +are very blue in their rest-frame UV–to–I-band colors, seem- +ingly more so than inferred from these previous studies (as in +some cases the galaxies in our sample are identical to those +in these other studies). Therefore, the MIRI data favor lower +stellar masses and lower SFRs in distant galaxies. +Adding the MIRI data also changes the interpretation of +the strength of nebular emission in the galaxies in the sam- +ple. +Many of the galaxies show red colors between the +HST/WFC3 and Spitzer/IRAC data (Figure 4). In the ab- +sence of MIRI data, the SED fitting interprets these colors +as a combination of higher dust obscuration, older ages, and + +18 +strong nebular emission. This has been reported previously +for individual galaxies and for stacked samples at z > 6 (e.g., +Castellano et al. 2017; Stark et al. 2017; De Barros et al. +2019; Hutchison et al. 2019; Endsley et al. 2021; Stefanon +et al. 2022). The fact that without the MIRI data the mod- +els allow for higher dust obscuration and older stellar popu- +lations increases the M/L and the stellar masses. The higher +dust obscuration also leads to higher SFRs. However, includ- +ing the MIRI data changes this interpretation for the galaxy +population. Nebular emission lines appear to be the primary +explanation for the red HST/WFC3 to Spitzer/IRAC colors +(while there are some galaxies where the MIRI data does +not change [e.g., galaxy IDs 6811 and 7364 in Figure 4] this +does not change the main result in general). This means (1) +the nebular emission lines for high redshift galaxies must be +strong and (2) the stellar populations are blue. Early JWST +results show emission lines remain strong in high-redshift +galaxies and impact the reddest (4 − 5 µm) bandpasses in +NIRCam (e.g., Endsley et al. 2022; Giménez-Arteaga et al. +2022; Santini et al. 2022b; Topping et al. 2022; Whitler et al. +2022). The MIRI data appear necessary to extend the rest- +frame wavelength coverage to >7000 Å, past the strongest +of the nebular emission features in the rest-frame optical. +5.2. Implications for Early Star-Formation and Stellar +Masses in Galaxies +The difference between the delayed-τ star-formation his- +tory and the model that includes bursts, while simplistic, ar- +guably span the gamut of available star-formation histories +of galaxies. Simulations show that galaxies experience many +discrete bursts, but when averaged over long time baselines +the evolution is mostly smooth (e.g., Diemer et al. 2017; Iyer +et al. 2019; Leja et al. 2019). Therefore the smoothly evolv- +ing delayed-τ model represents the slowly evolving evolu- +tion of galaxy star-formation histories. This can reproduce +the bluest colors (and lowest M/L) of the stellar population. +Of course, galaxies can experience a host of stochastic bursts +through changes in gas accretion or events that can sudden +changes in the star formation (as a result of mergers, e.g., +Kartaltepe et al. 2022, or the sudden onset of strong feed- +back from an AGN, e.g., Wagner et al. 2016). “Bursts” of +star-formation from these events will add stellar mass, but if +these occur at z < 100 then will be younger, and will have +M/L lower than a model with a burst at zf = 100. As such, +the models with a burst at zf = 100 have the maximum M/L +and the oldest possible ages for a stellar population at the ob- +served redshift of each galaxy. Therefore the models with +this burst represent a maximum stellar mass possibly formed +in these galaxies. +Our results show that including MIRI reduces the amount +of stellar mass allowed in these models, by an order of mag- +nitude in some cases. In itself this is interesting as nearly +all galaxies show no direct evidence for such early star- +formation. Comparing the median stellar masses of galax- +ies when fit by the delayed-τ models only and those with the +delayed-τ models and the early burst at z f = 100 shows they +differ by median(∆M∗) ≈ 0.1 dex (see the plots in Figure 8). +We find no convincing cases in our sample where the galax- +ies require a burst at zf = 100 to better fit their SEDs. This +would imply that galaxies do not experience early bursts of +star-formation (or at least such bursts do not form sufficient +mass that we require them). +Part of our findings could be impacted by biases. First +there is selection bias: all the galaxies studied here were se- +lected in HST/WFC3 data, and therefore required some rest- +frame UV emission above the HST/WFC3 detection limit. +It will be important to test for objects in, for example, fu- +ture JWST/NIRCam-selected populations show evidence for +early bursts (especially JWST-selected samples that lack HST +counterparts, Glazebrook et al. 2022; Pérez-González et al. +2022). Second, very recent work shows evidence for older +stellar populations in the spatially resolved colors of 6 < z < +9 galaxies (Giménez-Arteaga et al. 2022). As these issues +become better constrained, then the ages of the stellar pop- +ulations in distant galaxies could begin to inform us about +when galaxies first form stars (see Whitler et al. 2022). +5.3. Implications for Galaxy Growth +The question of how much mass is contained in galaxies +is related to the integral of the galaxies’ star-formation his- +tories. This is important because it contains the integrated +record of how rapidly galaxies acquire their baryons, and +how efficiently they convert these into stars. This has already +been discussed as an impossibly early galaxy problem, where +galaxies may have acquired too much stellar mass: in typical +JWST surveys galaxies should have less stellar mass than a +few times 1011 M⊙ (Behroozi & Silk 2018; Boylan-Kolchin +2022). +The effect of adding the MIRI data already show that stel- +lar masses and SFRs derived for galaxies tend to be overesti- +mated. The median offsets for the galaxies in our sample are +0.15 dex at 4 < z < 6 and rise to ≈0.3 dex at 6 < z < 9 (Fig- +ure 7 and Table 3). Assuming that these (median) offsets ap- +ply to previous estimates of galaxy stellar masses, then it im- +plies that measurements of the cosmic SFR density (SFRD, +which is the average SFR in all galaxies per co-moving vol- +ume element) are similarly biased to higher values at these +redshifts. +Figure 11 shows the impact of these lower SFRs on the +cosmic stellar-mass density, ρ∗. +Firstly, the figure shows +ρ∗ derived from the integral of the SFRD for two empirical +models calibrated against measurements from the literature +(Madau & Dickinson 2014; Finkelstein 2016). Finkelstein +(2016) shows that these models are consistent with a compi- +lation of measurements of ρ∗ from the literature at 4 < z < 10 +prior to the launch of JWST, (Oesch et al. 2014; Duncan et al. +2014; Grazian et al. 2015; Song et al. 2016). The thick line +in Figure 11 labeled “MIRI corrected” shows the empirical +model of Finkelstein corrected by the offsets of the SFRs de- +rived in Table 3. To derive these corrections we have inter- +polated the results from Table 3 assuming median redshifts +of z = 5 and 8 for the derived offsets in the two redshift bins. +Parenthetically, although we use the offsets derived from the +SFRs, using those for the stellar masses changes the results + +19 +4 +5 +6 +7 +8 +9 +10 +redshift +4 +5 +6 +7 +8 +stellar mass density +log + [M + Mpc +3] +Finkelstein 2016, original +Finkelstein 2016, +MIRI corrected +Madau & Dickinson 2014 + + + + + + + +St21 +Bh21 +Ki20 +Sa22 +0.5 +0.6 +0.8 +1 +1.5 +age of Universe [Gyr] +constrained with +JWST/MIRI +permitted/favored +without JWST/MIRI +-4 +-3 +-2 +-1 +log fraction of +(z = 0) +Figure 11. +Evolution of the cosmic stellar mass density, ρ∗, in +galaxies from 4 < z < 10. The lines show pre-JWST constraints +from Finkelstein (2016) and Madau & Dickinson (2014). The data +points show recent measurements of ρ∗ at z > 6 from the literature +(Kikuchihara et al. 2020 [Ki20], Bhatawdekar & Conselice 2021 +[Bh21], Stefanon et al. 2021 [St21], Santini et al. 2022a [Sa22]), +which largely follow the pre-JWST constraints. The shaded regions +show maximally allowable stellar mass density assuming galaxies +experience a burst at z = 100 followed by “normal” star-formation. +Constraints lacking JWST/MIRI coverage to rest-frame 1 µm allow +for a stellar mass density that is up to 0.8 dex higher at z = 4 and 1.4 +dex higher at z = 10. Including MIRI 5.6 and 7.7 µm data lowers +the maximum allowed by up to a factor of 5. +by ≈0.1 dex). We note, however, that because the MIRI data +imply offsets in the SFRs of galaxies, the similarly lower the +values of the cosmic SFRD at z > 4. +Secondly, the MIRI data improve the constraints the +amount of stellar mass possible in early bursts of star- +formation (Section 4.4). This is illustrated by the shaded re- +gions in Figure 11. To derived the area in the shaded swaths, +we applied the ratio between the mass permitted in early burst +at z f = 100 given to the fiducial value (listed in Table 4), inter- +polated in redshift as above. Therefore, the effects of adding +the MIRI data both lower SFR (and stellar masses) and limit +the total stellar mass allowed in early bursts. The combi- +nation of these effects reduces the upper bound on the total +cosmic stellar mass density allowed by the data by 0.4 dex at +z = 4 and by 1.0 dex at z = 9. As illustrated in Figure 11, this +implies that the JWST/MIRI data have constrained the stel- +lar mass in galaxies at z = 9 to be less than 0.1% that of the +present-day value (ρ∗(z = 0)). +5.4. Random Musings +The fact that galaxies are bluer than previous constraints +(i.e., they are “bluer than they appear”) has other conse- +quences, and likely dovetails with other recent results from +JWST NIRCam imaging. +These results in this Paper are +likely only the first foray into the properties of distant +galaxies using the longer wavelength data available from +MIRI. Future studies will be able to combine both NIRCam +and MIRI imaging, provide JWST–quality data from 0.8 − +−10 µm, which will improve the constraints in this Paper. +Already there are indicates using JWST/NIRCam data only +that the number density of luminous galaxies at z > 10 may +be much higher than predictions (e.g., Bouwens et al. 2022; +Donnan et al. 2022; Finkelstein et al. 2022a; Harikane et al. +2022; Naidu et al. 2022; Robertson et al. 2022). One expla- +nation for these discoveries could be that the UV–luminosity +per unit stellar mass (the UV “efficiency”) may be higher +than our models predict. This could be a result of changes +in the stellar populations (a shift toward bluer/harder ioniz- +ing spectra) or a change in the stellar IMF that is weighted +toward higher-mass stars. +However, a change in the IMF or in the UV efficiency does +not change the conclusion that the light from older stars could +be lost in the glare of the more recently formed stars, nor +would it change the conclusion that there is less light in gen- +eral at longer wavelengths from these galaxies. Even if the +lower-mass cutoff of the IMF is higher, e.g., > 50M⊙, (Raiter +et al. 2010) then after ∼10 Myr the mass left in such stars +would be effectively zero. Therefore, any of these effects +would further lower the galaxy M/L values, and therefore +lower the stellar masses by even more than what we have +measured with the MIRI data. (The only way to add more +stellar mass in galaxies at this epoch is if there is a substan- +tial population of galaxies at z > 6 that are undetected in HST, +which would be an important discovery for JWST). Regard- +less, the MIRI data have better constrained the available light +in stars at these early epochs and shown that galaxies contain +more than three times less “light” at rest-frame 0.7 − 1 µm +than previously known. +6. CONCLUSIONS +In this Paper we have presented results from CEERS on +the stellar population parameters for 28 galaxies with red- +shifts 4 < z < 9 using new imaging data from JWST/MIRI +at 5.6 and 7.7 µm. Our galaxy sample was detected in deep +data from HST/WFC3 and ACS and has observations from +Spitzer/IRAC at 3.6 and 4.5 µm. The MIRI 5.6 and 7.7 µm +data extend the coverage of the rest-frame spectral energy +distribution to nearly 1 micron for galaxies in this redshift +range. We use these data to study the improvements in the +stellar masses and SFRs of the galaxies at these redshifts +when the MIRI data are included. Our main results are the +following. + +20 +1. Galaxies at 4 < z < 9 have bluer rest-frame UV–I-band +colors (m1600 −I). Using the MIRI data we model the +SEDs using stellar population synthesis models (with +BAGPIPES). When we compare the average galaxy +SED (Figure 10) we find that models that include the +MIRI data are (on average) ∆(m1600 − I) ≈ 0.4 mag +bluer in their rest-frame colors compared to models +that exclude the MIRI data. +2. Galaxies generally have lower stellar masses and SFRs +when the MIRI data are included. For the majority of +the galaxies (Figure 7) adding the MIRI data reduces +the derived stellar masses by 0.25 dex at 4 < z < 6 (a +factor of 1.8) and by 0.38 dex at 6 < z < 9 (a factor +of 2.4). Similarly including the MIRI data reduces the +SFRs by 0.15 dex at 4 < z < 6 (a factor of 1.4) and 0.29 +dex at 6 < z < 9 (a factor of 2). +There are multiple reasons the stellar masses and SFRs +are lowered when we include the MIRI data. +The +first reason is that the galaxies are blue, and the fits +favor models with lower dust attenuation and mod- +els with lower M/L in general. +The second reason +is that in many cases the IRAC 3.6 and 4.5 µm data +probe the rest-frame optical, and these show indica- +tions of containing light from strong emission lines +(e.g., redshifted Hβ + [O III], Hα+[N II], [O II], etc.). +These boost the flux in these bands. In the absence +of MIRI data the models can not determine if the red +rest-frame UV-optical colors are a result of dust at- +tenuation, older stellar populations, or strong emission +lines (or all of them). The parameter constraints then +give more weight (probably density) to models with +higher stellar masses and SFRs. When the MIRI data +are included, then probe more of the stellar continuum +at >7000 Å rest-frame. The model fits that include he +MIRI data then show the dominant effect in the rest- +frame optical are strong nebular emission lines. This +problem will persist for models that use NIRCam data +as it also is limited to wavelengths less than 5 µm, but +this can be tested with forthcoming datasets. +3. The amount of stellar mass that could have formed in +early bursts is lower. We estimated the amount of stel- +lar mass formed by using a star-formation history that +includes an early burst (at zf = 100) in addition to a +smoothly evolving component. A stellar population +formed at this early time would fade and redden with +time, and it would have the highest M/L at any sub- +sequent time and therefore representations an upper +limit on the amount of mass that could exist in these +galaxies. The MIRI data improves the constraint on +this stellar population by probing longer wavelengths +(where the impact from this stellar population is more +pronounced). Figure 9 shows that without the MIRI +data, the amount of stellar mass in this population can +be as much as 0.9 dex higher at 4 < z < 6 (a factor of 7) +and as much as 1.1 dex higher at 6 < z < 9 (a factor of +>10). Including the MIRI data, these drop to 0.6 dex +at 4 < z < 6 (a factor of 4) and 0.7 dex at 6 < z < 9 (a +factor of 5). Therefore, adding the MIRI reduces the +amount of mass in early bursts by a factor of order 2 +(compared to when no MIRI data are used). +4. Our analysis of the MIRI 5.6 and 7.7 µm therefore pro- +vides evidence that there is less star-formation in dis- +tant galaxies (because the SFRs and stellar masses are +lowered) than found in previous studies. The MIRI +data also reduce the limits on the amount of stellar +mass possibly formed at early times. The combination +of these results has implications for the evolution of +the cosmic stellar-mass density, ρ∗. We showed (Fig- +ure 11) that applying our results to the galaxy popula- +tion shows that the amount of stellar mass density in +galaxies at z = 9 is less than 0.1% of the present day, +z = 0, value. This is an order of magnitude lower than +implied by previous studies (i.e,. pre-JWST). +We wish to thank everyone that brought JWST to fruition. +We also thank our other colleagues in the CEERS collabora- +tion for their hard work and valuable contributions on this +project. +CP thanks Marsha and Ralph Schilling for gen- +erous support of this research. +Portions of this research +were conducted with the advanced computing resources pro- +vided by Texas A&M High Performance Research Comput- +ing (HPRC, http://hprc.tamu.edu). This work benefited from +support from the George P. and Cynthia Woods Mitchell +Institute for Fundamental Physics and Astronomy at Texas +A&M University. This work acknowledges support from the +NASA/ESA/CSA James Webb Space Telescope through the +Space Telescope Science Institute, which is operated by the +Association of Universities for Research in Astronomy, In- +corporated, under NASA contract NAS5-03127. Support for +program No. JWST-ERS01345 was provided through a grant +from the STScI under NASA contract NAS5-03127. +Software: +AstroPy (Astropy Collaboration et al. 2013), +BAGPIPES (Carnall et al. 2018), matplotlib (Hunter 2007), +NumPy (van der Walt et al. 2011), photutils (Bradley +et al. 2020), PyPHER (Boucaud et al. 2016a), SE (Bertin +& Arnouts 1996), SciPy (Virtanen et al. 2020), Seaborn +(Waskom 2021). +APPENDIX + +21 +. +Table 5. Observed Properties of the Galaxy Sample +ID +R.A. (J2000) +Decl. (J2000) +F160 +E160 +F560 +E560 +F770 +E770 +zphot +z16 +z84 +zspec +P(z = 4) +P(z = 5) +P(z = 6) +P(z = 7) +P(z = 8) +P(z = 9) +(deg) +(deg) +nJy +(nJy) +(nJy) +(nJy) +(nJy) +(nJy) +5090 +215.04973 +52.89656 +102.0 +12.5 +158.8 +17.9 +148.0 +14.4 +4.38 +4.18 +4.56 +-1.000 +0.68 +0.27 +0.00 +0.00 +0.00 +0.00 +11329 +215.04086 +52.90623 +49.4 +6.9 +63.5 +20.4 +44.5 +15.7 +4.47 +4.15 +4.65 +-1.000 +0.55 +0.39 +0.00 +0.00 +0.00 +0.00 +34813 +214.97597 +52.920297 +82.2 +16.2 +67.0 +16.7 +43.9 +19.4 +4.52 +4.17 +4.59 +-1.000 +0.52 +0.36 +0.00 +0.00 +0.00 +0.00 +7600 +215.04415 +52.898731 +564.7 +29.4 +3283.1 +32.4 +4101.7 +24.8 +4.57 +4.13 +4.65 +-1.000 +0.62 +0.38 +0.00 +0.00 +0.00 +0.00 +13389 +215.03793 +52.909329 +453.9 +29.7 +578.3 +26.9 +661.0 +22.0 +4.57 +4.43 +4.61 +-1.000 +0.38 +0.61 +0.00 +0.00 +0.00 +0.00 +37703 +214.99094 +52.924279 +116.7 +11.8 +278.9 +30.4 +61.0 +17.5 +4.60 +4.47 +4.70 +-1.000 +0.22 +0.78 +0.00 +0.00 +0.00 +0.00 +15445 +215.02689 +52.907215 +124.5 +14.4 +187.8 +20.2 +116.6 +14.9 +4.60 +4.21 +4.84 +-1.000 +0.24 +0.62 +0.00 +0.00 +0.00 +0.00 +41564 +214.98708 +52.912734 +266.4 +13.4 +481.6 +20.8 +276.8 +15.7 +4.63 +4.56 +4.71 +-1.000 +0.03 +0.97 +0.00 +0.00 +0.00 +0.00 +45145 +215.00878 +52.919869 +101.2 +15.2 +108.6 +29.6 +69.0 +19.3 +4.68 +4.37 +5.20 +-1.000 +0.15 +0.74 +0.03 +0.00 +0.00 +0.00 +14913 +215.02081 +52.901523 +78.3 +14.1 +72.8 +17.8 +9.4 +8.9 +4.68 +4.47 +4.94 +-1.000 +0.19 +0.81 +0.00 +0.00 +0.00 +0.00 +42638 +214.97762 +52.90349 +547.0 +31.8 +711.4 +28.5 +596.3 +23.4 +4.71 +4.57 +4.77 +-1.000 +0.02 +0.90 +0.00 +0.00 +0.00 +0.00 +41375 +215.00283 +52.924312 +109.2 +15.4 +47.2 +18.1 +119.1 +14.8 +4.74 +4.60 +5.13 +-1.000 +0.07 +0.92 +0.01 +0.00 +0.00 +0.00 +35896 +214.98522 +52.924266 +96.3 +10.2 +114.5 +14.8 +110.0 +15.3 +4.76 +4.35 +5.33 +-1.000 +0.06 +0.73 +0.07 +0.00 +0.00 +0.00 +13179 +215.04260 +52.911968 +736.4 +23.7 +1355.2 +26.8 +1519.1 +21.5 +4.78 +4.72 +4.91 +-1.000 +0.00 +1.00 +0.00 +0.00 +0.00 +0.00 +19180 +215.03170 +52.919632 +773.3 +35.6 +749.1 +26.0 +787.7 +19.0 +4.93 +4.86 +4.95 +5.077† +0.00 +1.00 +0.00 +0.00 +0.00 +0.00 +37653 +214.99190 +52.925053 +103.1 +12.5 +154.0 +22.2 +108.4 +18.5 +4.95 +4.72 +5.59 +4.899‡ +0.06 +0.71 +0.22 +0.00 +0.00 +0.00 +18449 +215.02314 +52.912683 +69.0 +10.0 +185.0 +24.2 +165.3 +17.6 +5.05 +4.30 +5.57 +-1.000 +0.14 +0.59 +0.20 +0.00 +0.00 +0.00 +41545 +215.00312 +52.924103 +384.9 +18.3 +749.2 +21.5 +656.6 +17.5 +5.19 +5.02 +5.44 +-1.000 +0.00 +0.90 +0.10 +0.00 +0.00 +0.00 +7818 +215.02758 +52.887744 +289.9 +21.4 +421.4 +24.5 +399.5 +20.9 +5.27 +4.74 +5.46 +-1.000 +0.02 +0.86 +0.12 +0.00 +0.00 +0.00 +12773 +215.03057 +52.902606 +56.4 +12.2 +17.0 +7.5 +71.0 +14.1 +5.35 +4.87 +5.43 +-1.000 +0.00 +0.90 +0.09 +0.00 +0.00 +0.00 +24007 +214.95185 +52.928275 +47.1 +4.9 +107.8 +18.8 +56.7 +12.6 +5.64 +5.16 +5.79 +-1.000 +0.00 +0.46 +0.51 +0.00 +0.00 +0.00 +49365 +215.00994 +52.910669 +143.9 +14.1 +408.5 +26.8 +541.0 +20.6 +5.64 +5.03 +5.82 +-1.000 +0.01 +0.38 +0.50 +0.00 +0.00 +0.00 +18441 +215.03209 +52.918972 +97.2 +11.5 +199.6 +28.2 +169.9 +22.0 +6.76 +5.88 +7.24 +-1.000 +0.00 +0.01 +0.36 +0.48 +0.07 +0.00 +39096 +214.98901 +52.919652 +64.1 +8.7 +159.5 +17.0 +57.7 +11.2 +6.82 +6.67 +7.08 +-1.000 +0.00 +0.00 +0.02 +0.97 +0.01 +0.00 +12514 +215.03716 +52.906712 +71.6 +13.2 +142.3 +17.8 +50.4 +13.6 +7.57 +6.90 +8.41 +-1.000 +0.00 +0.00 +0.02 +0.39 +0.44 +0.13 +7364 +215.03561 +52.892208 +292.6 +17.5 +575.7 +28.4 +611.4 +23.1 +8.52 +7.47 +8.68 +-1.000 +0.00 +0.00 +0.00 +0.17 +0.52 +0.31 +6811 +215.03538 +52.890666 +314.5 +13.3 +426.8 +21.6 +404.0 +16.9 +8.93 +8.60 +9.05 +8.683∗ +0.00 +0.00 +0.00 +0.00 +0.08 +0.92 +26890 +214.96754 +52.932966 +133.3 +8.9 +101.9 +21.4 +60.8 +16.8 +9.16 +8.46 +9.24 +-1.000 +0.00 +0.00 +0.00 +0.00 +0.06 +0.83 +NOTE—References for spectroscopic redshifts: ‡ Stawinski, S., et al., in prep; † WERLS; ∗Zitrin et al. (2015). + +22 +Table 6. Derived Stellar masses, SFRs, and Redshifts including the MIRI [5.6] and [7.7] data. +redshift +logM∗/M⊙ (dex) +logSFR/M⊙ yr−1 (dex) +“Burst” Mass +ID +z50 +z16 +z84 +logM50 +logM16 +logM84 +log SFR50 +log SFR16 +log SFR84 +logM∗/M⊙ (dex) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +(10) +(11) +5090 +4.33 +4.19 +4.45 +9.01 +8.84 +9.16 +0.59 +0.50 +0.69 +9.55 +6811 +... +... +... +9.76 +9.57 +9.88 +1.60 +1.49 +1.72 +10.34 +7364 +8.07 +7.35 +8.55 +9.91 +9.72 +10.04 +1.70 +1.57 +1.82 +10.61 +7600 +4.41 +4.22 +4.56 +10.54 +10.36 +10.65 +2.09 +1.97 +2.20 +10.93 +7818 +5.19 +4.93 +5.39 +9.51 +9.30 +9.64 +1.19 +1.10 +1.29 +10.23 +11329 +4.45 +4.31 +4.59 +8.48 +8.14 +8.73 +0.27 +0.17 +0.37 +9.17 +12514 +7.69 +7.03 +8.21 +8.78 +8.47 +9.02 +0.70 +0.53 +0.85 +9.45 +12773 +5.08 +4.91 +5.24 +8.01 +7.78 +8.24 +0.08 +-0.18 +0.30 +8.87 +13179 +4.79 +4.73 +4.86 +10.07 +9.86 +10.20 +1.71 +1.60 +1.83 +10.57 +13389 +4.53 +4.45 +4.60 +9.61 +9.42 +9.75 +1.25 +1.15 +1.37 +10.16 +14178 +4.89 +4.76 +5.08 +8.54 +8.29 +8.79 +0.54 +0.35 +0.67 +9.16 +15445 +4.59 +4.46 +4.71 +8.92 +8.67 +9.11 +0.63 +0.54 +0.73 +9.36 +18441 +6.54 +5.90 +7.11 +9.32 +9.12 +9.46 +1.03 +0.90 +1.17 +10.00 +18449 +4.97 +4.43 +5.32 +9.17 +8.93 +9.32 +0.81 +0.67 +0.95 +9.83 +19180 +... +... +... +9.72 +9.48 +9.90 +1.54 +1.38 +1.68 +10.14 +24007 +5.36 +5.17 +5.52 +8.76 +8.48 +8.91 +0.41 +0.32 +0.53 +9.59 +26890 +8.80 +8.61 +8.97 +8.79 +8.46 +9.04 +0.84 +0.50 +0.97 +9.45 +34813 +4.38 +4.27 +4.50 +8.26 +7.97 +8.62 +0.33 +0.02 +0.46 +8.97 +35896 +5.13 +4.82 +5.40 +8.92 +8.67 +9.08 +0.63 +0.51 +0.76 +9.44 +37653 +... +... +... +8.95 +8.58 +9.13 +0.66 +0.53 +0.83 +9.57 +37703 +4.63 +4.53 +4.71 +8.41 +8.20 +8.86 +0.47 +0.23 +0.65 +9.21 +39096 +6.78 +6.68 +6.87 +8.81 +8.48 +8.98 +0.63 +0.52 +0.75 +9.41 +41375 +4.79 +4.66 +4.96 +8.54 +8.27 +8.77 +0.51 +0.34 +0.61 +9.15 +41545 +5.13 +5.00 +5.25 +9.80 +9.62 +9.89 +1.38 +1.29 +1.49 +10.32 +41564 +4.66 +4.59 +4.74 +9.36 +9.08 +9.52 +1.05 +0.93 +1.17 +9.78 +42638 +4.70 +4.64 +4.76 +9.65 +9.39 +9.80 +1.34 +1.20 +1.45 +10.12 +45145 +4.72 +4.55 +4.96 +8.59 +8.28 +8.83 +0.51 +0.34 +0.61 +9.52 +49365 +5.31 +5.14 +5.52 +9.63 +9.43 +9.75 +1.29 +1.18 +1.42 +10.23 +NOTE—(1) Galaxy ID; (2)–(4) redshift median (50%-tile), and 16th and 84th-percentiles; galaxies with no redshift use the +spectroscopic redshift in Table 5; (5)–(7) stellar mass (50th percentile), and 16th and 84th–percentiles in the delayed-τ model; +(8)–(10) SFR median (50th percentile), and 16th and 84th percentiles (all SFRs are averaged over the past 100 Myr) in +the delayed-τ model; (11) maximum stellar mass allowed in the “burst” formed at zf = 100, these satisfy the BIC criteria +(equation 2 in Section 4.4) and correspond approximately to a 3σ upper limit. +A. IMPACT OF CROWDED SOURCES IN IRAC DATA +. +One source of potential bias relates to the photometry of our sources in the Spitzer/IRAC data. As illustrated in the images +(Figure 3) some objects have bright neighbors. In the case of the IRAC images, the light from the wings of these objects can +blend with that for our sources. There is a large body of literature on the subject of performing crowded source photometry +(Laidler et al. 2007; Labbé et al. 2013; Merlin et al. 2015, 2016). We have used the catalog from (Finkelstein et al. 2022b) who +used the HST/F160W image as a prior for the locations of sources and neighbors. Source photometry is then carried out using +TPHOT (Merlin et al. 2015), which estimates the source flux from objects simultaneously when measuring photometry. While +this method is theoretically robust, residuals from poorly modeling ePSFs and changes in galaxy morphology with wavelength +(the “Morphological K-correction”, Papovich et al. 2005) can lead to systematic uncertainties in source photometry. +To test if our results are impacted by blended sources in the IRAC bands, we did the following. We first searched around each +of the galaxies in our sample and identified galaxies that neighbors in the MIRI 5.6 µm catalog within a radius of r ≤ 3′′ and a +magnitude of [5.6] ≤ 26.7 mag (near the flux limit). We selected neighbors in the MIRI 5.6 µm image as the central wavelength +is closest to that of IRAC for our dataset (see Figure 2). The IRAC ePSF has a FWHM of ≈2′′, so any source within 3′′ in the +MIRI data could therefore have IRAC light blended with our source of interest. + +23 +Table 7. Derived Stellar masses, SFRs, and Redshifts excluding the MIRI data. +redshift +logM∗/M⊙ (dex) +logSFR/M⊙ yr−1 (dex) +“Burst” Mass +ID +z50 +z16 +z84 +logM50 +logM16 +logM84 +log SFR50 +log SFR16 +log SFR84 +logM∗/M⊙ (dex) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +(10) +(11) +5090 +4.32 +4.17 +4.46 +9.08 +8.75 +9.26 +0.61 +0.50 +0.74 +10.00 +6811 +-1.00 +-1.00 +-1.00 +9.85 +9.47 +10.12 +1.70 +1.43 +1.98 +11.17 +7364 +8.23 +7.42 +8.61 +9.77 +9.48 +9.97 +1.60 +1.38 +1.80 +10.85 +7600 +4.42 +4.26 +4.57 +10.53 +10.32 +10.68 +2.10 +1.97 +2.24 +11.17 +7818 +4.83 +4.67 +5.17 +9.83 +9.57 +10.02 +1.45 +1.28 +1.61 +10.51 +11329 +4.40 +4.24 +4.56 +8.82 +8.50 +9.05 +0.40 +0.24 +0.55 +9.84 +12514 +7.38 +6.77 +8.13 +9.30 +8.92 +9.64 +1.10 +0.77 +1.44 +10.41 +12773 +4.97 +4.83 +5.15 +8.96 +8.55 +9.22 +0.58 +0.41 +0.78 +10.23 +13179 +4.76 +4.71 +4.81 +10.25 +10.03 +10.38 +1.81 +1.70 +1.96 +10.89 +13389 +4.52 +4.44 +4.59 +9.82 +9.65 +9.97 +1.37 +1.24 +1.51 +10.41 +14178 +4.78 +4.65 +4.92 +9.27 +8.99 +9.45 +0.90 +0.72 +1.07 +10.08 +15445 +4.58 +4.45 +4.70 +9.03 +8.70 +9.25 +0.68 +0.55 +0.85 +10.07 +18441 +6.61 +6.07 +6.87 +10.18 +9.90 +10.42 +1.94 +1.66 +2.16 +11.11 +18449 +5.00 +4.43 +5.41 +9.24 +8.92 +9.47 +0.90 +0.64 +1.10 +10.13 +19180 +-1.00 +-1.00 +-1.00 +10.52 +10.31 +10.70 +2.09 +1.90 +2.25 +11.52 +24007 +5.28 +5.06 +5.47 +8.93 +8.59 +9.27 +0.58 +0.38 +0.88 +10.40 +26890 +8.92 +8.67 +9.17 +9.39 +9.10 +9.67 +1.28 +1.05 +1.51 +10.53 +34813 +4.34 +4.23 +4.47 +8.78 +8.40 +9.05 +0.48 +0.39 +0.59 +9.64 +35896 +4.88 +4.66 +5.20 +9.23 +8.94 +9.47 +0.93 +0.72 +1.12 +10.08 +37653 +-1.00 +-1.00 +-1.00 +9.55 +9.26 +9.75 +1.18 +1.00 +1.36 +10.39 +37703 +4.58 +4.48 +4.68 +9.16 +8.90 +9.35 +0.75 +0.62 +0.88 +9.94 +39096 +6.73 +6.59 +6.84 +9.04 +8.68 +9.46 +0.80 +0.57 +1.21 +10.03 +41375 +4.73 +4.60 +4.87 +9.15 +8.79 +9.34 +0.73 +0.58 +0.91 +10.19 +41545 +5.12 +4.98 +5.26 +9.85 +9.61 +10.03 +1.45 +1.31 +1.60 +10.49 +41564 +4.64 +4.57 +4.71 +9.56 +9.34 +9.72 +1.15 +1.04 +1.29 +10.18 +42638 +4.68 +4.62 +4.75 +9.85 +9.62 +10.01 +1.47 +1.32 +1.64 +10.55 +45145 +4.66 +4.52 +4.86 +9.08 +8.71 +9.30 +0.69 +0.53 +0.88 +9.98 +49365 +5.43 +5.23 +5.63 +9.45 +9.14 +9.65 +1.09 +0.89 +1.29 +10.31 +NOTE—(1) Galaxy ID, (2)–(4) redshift median (50%-tile), and 16th and 84th-percentiles; galaxies with no redshift use the spectro- +scopic redshift in Table 5; (5)–(7) stellar mass (50th percentile), and 16th and 84th–percentiles in the delayed-τ model; (8)–(10) +SFR median (50th percentile), and 16th and 84th percentiles (all SFRs are averaged over the past 100 Myr) in the delayed- +τ model; (11) maximum stellar mass allowed in the “burst” formed at zf = 100, these satisfy the BIC criteria (equation 2 in +Section 4.4) and correspond approximately to a 3σ upper limit. +From our sample, we identified 11 galaxies that have a neighbor within 3′′ in the MIRI 5.6 µm image. To estimate their effect +on our study we removed these objects from the sample and recomputed the offsets in stellar mass and SFR for the results that +include the MIRI 5.6 and 7.7 µm data versus the results that exclude the MIRI data. These results are shown in Figure 12. +Contrasting this figure with the one for the full sample (Figure 7) shows there is little change in the median offsets in stellar mass +and SFR. The galaxy sample used in this Appendix is obviously smaller, but the median values do not change appreciably. For +the sample that excludes blended objects, the offsets in stellar mass are ∆logM∗ = 0.21 dex for 4 < z < 6 and 0.53 for 6 < z < 9 +(though the later now includes only three galaxies). The offsets in SFR are ∆logSFR = 0.13 dex for 4 < z < 6 and 0.42 dex for +6 < z < 9. These are within ≈0.1 dex of the values reported for the full sample (in Figure 7). +Similarly, we also investigated how the IRAC data for sources with close neighbors impact our finding that the stellar pop- +ulations of the galaxies in our sample are generally “bluer” when the MIRI data are included in the analysis (see Section 5.1 +and Figure 10). We repeated our analysis of the rest-frame colors in Section 5.1 with our sample of galaxies that excludes those +objects with a neighboring MIRI 5.6 µm source with [5.6] < 26.7 mag and within r ≤ 3′′. We find that in this case the relative +rest-frame colors change only slightly. The rest-frame far-UV–I color become bluer by 0.015 mag (to have a total rest-frame +(blue) color of ∆(m1600 −I) ≈ 0.42 mag) when the objects with crowded IRAC photometry are excluded. + +24 +4 +6 +8 +redshift +0.5 +0.0 +0.5 +1.0 +1.5 + log M (dex) +excluding IRAC-crowded objects +using the full sample (Fig. 4) +4 +6 +8 +redshift +0.5 +0.0 +0.5 +1.0 +1.5 + log SFR (dex) +excluding IRAC-crowded objects +using the full sample (Fig. 4) +Figure 12. +Testing the impact of sources with “crowded” IRAC photometry. The plots in this figure are similar to those in Figure 7, and +compare the stellar masses and SFRs derived from the SED modeling for galaxies including the MIRI F560W and F770W data and without +the MIRI data. In both panels the results show the difference between the mass (SFR) derived without MIRI data and the mass (SFR) deriving +including the MIRI data. In this figure, we have excluded objects that have a neighbor with r ≤ 3′′ and MIRI [5.6] ≤26.7 mag. This eliminates +11 objects, and allows us to test if crowding in the IRAC data (which has lower angular resolution) impacts object photometry in the IRAC +bands. We do not observe any significant offset compared to the results in Figure 7: the median offsets in stellar mass and SFR change by +≈0.1 dex. Therefore we conclude that blended IRAC photometry does not significantly impact the results here. +Therefore, we conclude that our results are not dominated by photometry from sources crowded in the IRAC data. Obviously, +future studies using JWST/NIRCam will be valuable to testing the IRAC photometry (see, e.g., Bagley et al. 2022). +REFERENCES +Antwi-Danso, J., Papovich, C., Leja, J., et al. 2022, arXiv e-prints, +arXiv:2207.07170. https://arxiv.org/abs/2207.07170 +Arellano-Córdova, K. Z., Berg, D. A., Chisholm, J., et al. 2022, +ApJL, 940, L23, doi: 10.3847/2041-8213/ac9ab2 +Astropy Collaboration, Robitaille, T. P., Tollerud, E. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' ZAVALA29 1Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA 2George P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA 3Kapteyn Astronomical Institute, University of Groningen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Box 800, 9700 AV Groningen, The Netherlands 4SRON Netherlands Institute for Space Research, Postbus 800, 9700 AV Groningen, The Netherlands 5Department of Astronomy, The University of Texas at Austin, Austin, TX, USA 6Department of Physics, University of the Pacific, Stockton, CA 90340 USA 7Aix Marseille Univ, CNRS, CNES, LAM Marseille, France 8Centro de Astrobiología (CAB), CSIC-INTA, Ctra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' de Ajalvir km 4, Torrejón de Ardoz, E-28850, Madrid, Spain 9INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy 10NSF’s National Optical-Infrared Astronomy Research Laboratory, 950 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Cherry Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Tucson, AZ 85719, USA 11Department of Astronomy, University of Michigan, 1085 S.' metadata={'source': 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+page_content=' UK 28Institute of Space Sciences and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' University of Malta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Msida MSD 2080,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Malta 29National Astronomical Observatory of Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2-21-1 Osawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Mitaka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Tokyo 181-8588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Japan ABSTRACT We present results from the Cosmic Evolution Early Release Survey (CEERS) on the stellar-population pa- rameters for 28 galaxies with redshifts 4 < z < 9 using imaging data from the James Webb Space Telescope (JWST) Mid-Infrared Instrument (MIRI) combined with data from the Hubble Space Telescope and the Spitzer Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The JWST/MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data extend the coverage of the rest-frame spectral-energy Corresponding author: Casey Papovich papovich@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='00027v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='GA] 30 Dec 2022 2 distribution (SED) to nearly 1 micron for galaxies in this redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' By modeling the galaxies’ SEDs the MIRI data show that the galaxies have, on average, rest-frame UV (1600 Å) – I-band colors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 mag bluer than derived when using photometry that lacks MIRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, the galaxies have lower (stellar)-mass–to–light ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI data reduce the stellar masses by ⟨∆logM∗⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='25 dex at 4 < z < 6 (a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='37 dex at 6 < z < 9 (a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This also reduces the star-formation rates (SFRs) by ⟨∆logSFR⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='14 dex at 4 < z < 6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='27 dex at 6 < z < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI data also improve constraints on the allowable stellar mass formed in early star-formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We model this using a star-formation history that includes both a “burst’ at zf = 100 and a slowly varying (“delayed-τ”) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI data reduce the allowable stellar mass by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 dex at 4 < z < 6 and by ≈1 dex at 6 < z < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Applying these results globally, this reduces the cosmic stellar-mass density by an order of magnitude in the early universe (z ≈ 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, observations of rest-frame ∼>1 µm are paramount for constraining the stellar–mass build-up in galaxies at very high-redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' INTRODUCTION There is growing evidence that galaxies must have started forming stars very quickly following the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Theory predicts the first stars should form at z ∼> 20 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Barkana & Loeb 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Miralda-Escudé 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Wise et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Visbal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The ionization from these sources is needed to explain observations that the hydrogen-neutral fraction of the intergalactic medium (IGM) was 50% by z ∼ 8 (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2020),1 and to account for the absorption profile of the 21 cm signal at z ∼ 20 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Bowman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Indeed, early obser- vations from JWST have already identified candidates for galaxies at z ∼> 15 (Curtis-Lake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Donnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Spec- troscopy from JWST of galaxies at z ∼ 8−9 shows emission lines from heavy elements that appear to require metallici- ties of ≈ 5 − 10% Z⊙ (Arellano-Córdova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Fuji- moto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Heintz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Langeroodi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Trump et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Curti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2023), implying these galaxies have experienced at least one (and probably multiple) gener- ation(s) of previous stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is consistent with earlier de- tections of metal lines in z > 6 galaxies from ground-based telescopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Hutchison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' All of these results point to the fact that star formation began early and those early generations of stars enriched the uni- verse with heavy elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' It is then important to consider how we may constrain the history of star-formation in these early galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The num- ber of stars (and therefore the stellar mass) in galaxies ap- pears to rise rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The (co-moving) stellar mass density in galaxies at z ∼ 5 − 6 (when the age of the Universe is ≈ 1 Gyr) is already 1% of the present value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Madau & Dickinson 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Finkelstein 2016), where simulations and theory require the stars in those objects formed at much ear- lier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Even early JWST observations find some evidence for massive galaxies at z > 7 (Labbé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022) with some * NSF Graduate Fellow † NASA Postdoctoral Fellow ‡ NASA Hubble Fellow 1 The Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2020) analysis also suggests a non-zero optical depth of CMB photons scattering off free electrons at z ≈ 15, which implies ionization of the IGM had begun by this epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' candidate objects having masses, logM∗/M⊙ > 11 (as large as the stellar mass of the Milky Way today, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Such objects would be in tension with galaxy formation models (Boylan-Kolchin 2022) where there are not sufficient numbers of massive dark-mater halos to sup- port these objects, even if all the baryons in the halos are in the form of stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' However, the uncertainty in these measurements is in the assumed star-formation histories, the contributions of emission lines to the photometric measure- ments from broad-band data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Pérez- González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Steinhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022), and the effects young stellar populations “outshining” older stellar popula- tions in the integrated emission of galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Giménez- Arteaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Clearly there are unknown systematics in the assumptions of the data analysis, or missing physics in our theoretical understanding of stellar populations and galaxy formation, or some combination of all of these things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' It is therefore crucial to understand constraints on the stellar masses (which are the integral of the star-formation histories) as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Motivated by these issues, in this Paper we use new data from JWST to better constrain the stellar masses, star- formation rates (SFRs), and star-formation histories of galax- ies during the first one and a half billion years after the Big Bang (z > 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' One of the problems with initial studies from JWST is that they currently rely entirely on observations from JWST’s Near-IR Camera (NIRcam), which only probes to wavelengths ∼<5 µm, or about 6000 Å rest-frame for z = 6−7 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This complicates the ability to disentangle massive galaxies with older stellar populations from younger, dusty galaxies or galaxies with emission lines with extreme equiv- alent widths (see discussion in Antwi-Danso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022, and recent work by Giménez-Arteaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022 who find evi- dence for older stellar populations mixed with recent bursts in spatially resolved studies using JWST/NIRCam data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' To better constrain the SEDs of these galaxies requires obser- vations at longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is where JWST/MIRI is important as it has the sensitivity to detect z ≈ 10 galax- ies at rest-frame 1 µm (see Bisigello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Previous work on this subject has been limited to data from the Spitzer Space Telescope, which is primarily sensitive to the emission such distant galaxies at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' JWST offers immense gains to Spitzer: JWST has a collecting area that is 45 times larger than that of Spitzer, and the larger aperture provides 3 image quality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', angular resolution) that is improved by a factor of order 10 (Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These gains are es- pecially manifest at longer wavelengths, and make JWST 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data vastly more sensitive than Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The outline for the Paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In Section 2 we discuss the CEERS dataset and the ancillary datasets used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We also discuss the processes to create (and vali- date) the flux densities of galaxies in the CEERS JWST/MIRI data at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 we discuss the sample of 4 < z < 9 galaxies used in this study, and we present the MIRI data for these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In Section 3 we discuss the anal- ysis methods to derive constrains on the galaxy stellar popu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In Section 4 we discuss the resulting improvements that including the MIRI data provide on constraints on the galaxies stellar populations (specifically their stellar–masses and SFRs) derived from fitting stellar population models to the observed photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 we discuss con- straints on the range of allowed stellar masses in these high redshift galaxies by allowing for an early (“maximally old”) burst of stars at z = 100, and we show that adding the MIRI data improves the limit on this hypothetical population of z = 100 stars by a factor of 6 to 10 for galaxies at 4 < z < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In Section 5 we discuss the implications these constraints have for our understanding of galaxy colors, stellar populations at these high redshifts, and the evolution of the galaxy stellar mass density, in particular during the epoch of reionization, and what this could mean for future studies of galaxies at higher redshifts (from JWST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In Section 6 we present our conclusions and prospects for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Throughout we use a flat cosmology with Ωm,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='315, H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 km s−1 Mpc−1 (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' All magnitudes reported here are on the Absolute Bolomet- ric (AB) system (Oke & Gunn 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Throughout we use Chabrier (2003) initial mass function (IMF) for all stel- lar masses and SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We denote magnitudes measured in the MIRI F560W and F770W bands as [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] and [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7], re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Similarly, we denote magnitudes measured in IRAC Channel 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm), Channel 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm), Channel 3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 µm), and Channel 4 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 µm) as [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5], [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8], and [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' DATA AND SAMPLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' MIRI Catalog We use the data release DR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 images produced by the CEERS team for the MIRI 3 and MIRI 6 fields (see Finkel- stein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022a)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These data were acquired in 2022 June 21 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The data properties and its reduction are discussed elsewhere (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022 in prep), but we provide a sum- mary here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The data were processed using the JWST Calibra- tion Pipeline (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) using the default parameters for stage 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We then removed the backgrounds with a custom routine that combines images taken in the same bandpass but from different fields and/or dither positions (rejecting pix- 2 https://ceers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='io/releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='html els in each image that contain galaxies) in order to create a “super-background” image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We then removed this back- ground from each image and applied an astrometric correc- tion to each image prior to processing them with stage 3 of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This produced the final science images (exten- sion i2d), rms images (extension rms, which account for Poisson, readout, and correlated pixel noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' see Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022 in prep), and weight maps (wht) for each field with a pixel scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='09′′, registered astrometrically to the existing HST/CANDELS v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 WFC3 and ACS images (see Koeke- moer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Bagley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 20222).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For the purpose of this study we are interested in sources detected in the MIRI data, so we create a catalog of sources derived from these images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Prior to object detection we convolved the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm image to match the image quality of the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For this step, we constructed an “ef- fective” point source function (ePSF) for each image by identifying unblended stars using the photutils (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0) detection task, and modeling them with the photutils psf task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This produced model ePSFs with measured full- width at half maxima (FWHM) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='24′′ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='28′′, for the F560W and F770W images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is con- sistent with the expected image quality, but takes into ac- count the exact dithering and reduction steps for the CEERS MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We then used PyPHER (Boucaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2016b) to construct a convolution kernel to match the image qual- ity of the model ePSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We applied these kernels to each F560W image, creating a “PSF-matched” image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Our tests on point sources in the PSF-matched F560W and F770W im- ages show that we measure the same fraction of light to better than 2% in fixed circular apertures of radii larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='′′35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We then created a detection image constructed from the sum of the MIRI F560W and F770W science images (us- ing the extension sci) weighted by the appropriate weight image (using the extension wht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We also created a detec- tion weight-map as the sum of the weights for these images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We then created F560W and F770W catalogs using Source Extractor (SE, version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Bertin & Arnouts 1996) in “dual-image” mode using the detection image (and its weight map) for object detection, where we measured photometry in the PSF-matched F560W and F770W image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We used the parameters in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We then measured fluxes and mag- nitudes using circular apertures of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9′′ diameter, and we scaled these to a total aperture (MAG_AUTO) measured for each source in the detection image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Uncertainties for each object are measured from the rms image in the same aper- tures, and scaled to a total magnitude in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Fig- ure 1 shows the distribution of the MIRI sources with signal- to-noise (SNR) ≥ 3 in F560W or F770W (compared with our galaxy sample, discussed below in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We compared the MIRI flux densities for sources F560W and F770W against those for bright objects from existing IRAC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 µm catalogs Stefanon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For bright objects ([5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8] or [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0] ≤ 22 AB mag) in the IRAC data, we measure small offsets of ∆m = [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8] − [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='16 mag between the IRAC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 and MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 data, and ∆m = [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0] − 4 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' CEERS MIRI F560W and F770W SExtractor Parameter Settings SExtractor Parameter Value (1) (2) DETECT_MINAREA 10 pixels DETECT_THRESH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 ANALYSIS_THRESH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 FILTER_NAME gauss_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5_5x5a WEIGHT_TYPE MAP_WEIGHT,MAP_RMS DEBLEND_NTHRESH 32 DEBLEND_MINCONT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='005 MAG_ZEROPOINT 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='701b PIXEL_SCALE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='09 arcsec BACK_TYPE AUTO BACK_FILTERSIZE 5 pixels BACK_SIZE 32 pixels BACKPHOTO_THICK 8 BACKPHOTO_TYPE LOCAL SEEING_FWHM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 arcsec NOTE—SE was run using the weighted sum of the PSF- matched F560W and F770W images for detection, and using the images separately for photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' All other SE parameters are set to the program defaults (for SEx- tractor v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' aThis is a Gaussian kernel with σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 pixels and size 5× 5 pixel2 used to filter the image for source detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' bThe AB magnitude zeropoint for the images, converting from the JWST default of MJy sr−1 to µJy pixel−1 at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='09′′ pixel−1 scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='07 mag between the IRAC 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 and MIRI 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', the MIRI flux densities are slightly brighter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Most of these offsets can be explained by differences in the shape of the MIRI and IRAC passbands and because of differences in the angular resolution of the instruments (MIRI has a PSF FWHM smaller by a factor of more than seven).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These tests are discussed more fully in Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2022 in prep), but this gives us confidence that the MIRI data are calibrated to better than ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Galaxy Sample For this study we use galaxies identified in the CEERS/MIRI first epoch fields with redshifts 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 < z < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The lower redshift bound is selected to ensure the HST pho- tometric data (used for galaxy photometric redshifts) probes the redshifted Lyman-break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The upper redshift limit in- cludes the highest redshift galaxies detectable by HST/WFC3 All MIRI with [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] > 3 2 4 6 8 redshift 2 0 2 MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] (mag) Zitrin+15, z=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='683 constant SFR, 10 Myr, Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 Z , log U = 2 no nebular emission, A(V) = 1 mag Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] − [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] colors of the sample studied here as a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The data points (and error bars) denote the 28 objects studied here (blue-shaded points have 4 < z < 6 and red-shaded points have 6 < z < 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The three sources with spec- troscopic redshifts are indicated with larger symbols (the source z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='683 published by Zitrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015 is labeled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The gray his- togram shows the distribution of MIRI colors for all objects detected in both bands in CEERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The bold-dashed line shows the expected color of a stellar population with nebular emission as discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' data (Bouwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is illustrated in Figure 2 which shows that for galaxies around z ∼ 5 and z ∼ 9, the HST/ACS and WFC3 data constrain this break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This improves the quality of the sample (as compared to, for example, using galaxies at z ≈ 3 where the Lyman– break shifts blueward of the HST/ACS F606W band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The parent sample for our study is the catalog from Finkel- stein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2022b), which uses the existing HST/ACS, WFC3 and Spitzer/IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm data to select photomet- ric samples of galaxies at these high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The data include both the imaging from the original CANDELS sur- vey (HST/ACS F606W, F814W, WFC3 F125W, F160W) and additional imaging from WFC3 F140W (see Footnote 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2022b) then use these data to measure photometric redshifts and redshift probably distribution func- tions, P(z) for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We then matched objects from the catalog from Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2022b) with objects in our MIRI catalogs that are de- tected with S/N > 3 in either the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 or 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1) using a matching radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This sample includes 29 objects, though one object was later identified as foreground star and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Table 5 which lists the ob- served properties of the 28 galaxies in the sample, their ID numbers from Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2022b), their HST/F160W flux densities the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The ta- ble includes the photometric redshifts derived by Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (including the 16th and 84th-percentile range from the 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 1 2 3 5 7 10 wavelength ( m) -------- ACS ------- -- WFC3 -- -- IRAC -- -- MIRI --- Lyman limit H H H H [O III] [O III] [O II] [O II] z=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 z=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 Ly break Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Illustration of galaxy spectra (in relative units of erg s−1 cm−2 Å−1) compared to the broad-band data for the observations in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The bottom panel shows the relative transmission functions for the HST/ACS and WFC3 filters (ACS F606W, F814 and WFC3 F125W, F160W), Spitzer/IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm (Ch1) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm (Ch2), and JWST/MIRI F560W and F770W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The top panel shows model spectra of star- forming galaxies at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 and z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 (which coincidentally match two galaxies with spectroscopic redshifts in this sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Key emission lines and features are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI data probe the shape of the galaxy spectral energy distributions to 8000 Å rest-frame, even for galaxies with z = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' P(z)) used for object selection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The Table also includes the amount of the integrated P(z) contained between ∆z = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 of the stated redshift, for example P(z = zc) = � zc+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 zc−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 P(z′) dz′ (1) in bins with central redshifts of zc = 4, 5, 6, 7, 8, and 9 (see Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These integrated probabilities indi- cate a likelihood that a given galaxy is within the redshift bin to which it is assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For our analysis we rederive the pho- tometric redshifts below (from SED modeling that includes the new 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm MIRI data) but include this here as we use the P(z) in Table 5 as a prior likelihood on the SED fitting (discussed in Section 3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Three of the galaxies in our sample have spectroscopic red- shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This includes a previously known galaxy (ID 6811 in our catalog) with z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='683 from Zitrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2015), and two new redshifts obtained by the CEERS and WERLS collaborations from observations with Keck/DEIMOS and Keck/LRIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The latter two sources are ID 37653 with z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='899 measured by (Stawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2023a, in preparation) and ID 19180 with z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='077 (Stawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2023b, in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In all of these cases the spectroscopic redshifts are consistent with the photometric redshifts in Table 5, and we fix the redshift to the value of the spectroscopic redshift in our analysis of the spectral energy distributions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 3 shows the HST/ACS, HST/WFC3, Spitzer/IRAC, and JWST/MIRI imaging for all the objects in our sample, with the objects ordered by increasing redshift (the full Fig- ure Set of all 28 objects is available online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In all cases the galaxies show prominent “Lyman–breaks” at the location of the redshifted Lyman-limit and/or Lyman-α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In some cases, the flux density appears to be much brighter in a given pass- band compared to the adjacent band (for example, galaxy ID=12514 at z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 shows evidence of enhanced emission at MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm, indicative of strong redshifted Hα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Fig- ure 2 illustrates how the bandpasses are sensitive to different features in the SED of galaxies (using z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 and z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 as examples as these are similar to two of the objects with spec- troscopic redshifts in our sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We will return to these cases below when we explore constraints on the galaxy stel- lar populations by modeling their SEDs (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 1 shows the MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6]−[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] colors for galaxies in our sample, compared to the distribution of MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6]−[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] colors for all objects detected in the MIRI images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The three sources with spectroscopic redshifts are indicated with larger symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The high-redshift sources in our sample have MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6]−[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] colors largely consistent with expectations: most objects have relatively flat ([5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6]−[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] ≈ 0 mag) or blue col- ors ([5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6]−[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] ∼< 0 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This implies that in most cases the MIRI data sample the continuum of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In some cases the MIRI colors suggest very blue colors, [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6]−[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] ∼< −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 to −1 mag, roughly bounded by the bold-dashed line in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The dashed line represents a photoionization model 6 ACS F606W ID=7600 z=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='57 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=13389 z=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='57 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=37653 z=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='899 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=42638 z=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='71 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=19180 z=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='077 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=41545 z=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='19 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=7818 z=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='27 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=12514 z=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='57 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=7364 z=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='52 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=6811 z=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='683 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W ACS F606W ID=26890 z=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='16 ACS F814W WFC3 F125W WFC3 F140W WFC3 F160W IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 m IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 m MIRI F560W MIRI F770W Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Montage of images of a subset of the galaxies used in this study (ordered by increasing redshift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Each row shows images for one galaxy (labeled by galaxy ID and redshift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The images are (left to right), ACS F606W, F814W, WFC3 F125, F160W, IRAC Ch1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm) and Ch2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm), and MIRI F560W and F770W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The images above include the three galaxies with spectroscopic redshifts (ID 37653, 19180, and 6811).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The images are 6′′ ×6′′ centered on the galaxy in each bandpass, as labeled along the top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The complete figure set (28 images) is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Set 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Montage of ACS F606W, F814W, WFC3 F125W, F140W, F160W, IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm, and MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm images for all galaxies in the sample (28 figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 7 with a young, metal-poor stellar population (10 Myr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 Z⊙) driving strong nebular emission (set by an ionization param- eter of logU = −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We will explore evidence for this inter- pretation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' SPECTRAL ENERGY DISTRIBUTION MODELLING We model the spectral energy distributions (SEDs) of the galaxies in our sample using stellar population synthesis models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Our goal is to test how the inferred properties of the stellar populations in the high-redshift galaxies change by including the JWST/MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data, in particular the stellar masses and the SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Previous work (pre-JWST) showed the MIRI data is able to recover these quantities ac- curately (Bisigello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2017), and here we test how they improve the constraints on the stellar population parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is critical for galaxies at higher redshifts, z ∼> 4, where the rest-frame optical features shift to longer wavelengths (rest-frame 4000 Å corresponds to >2 µm), which probes light from longer-lived stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Perhaps more problematic are the effects of nebular emission lines, which can litter the op- tical portion of the SED (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' There are observa- tions that z > 2 galaxies have a higher incidence of “extreme” emission lines with rest-frame EWs up to ≈1000 Å (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Boyett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Pérez-González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022), consistent with inferences made from the >3 µm colors of z > 6 galaxies (Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Roberts- Borsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Hutchison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As the EW scales with redshift as (1+z) this implies these lines have a stronger impact for high- redshift galaxies for bandpasses of fixed wavelength width (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Burgarella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We model each galaxy by fitting HST/ACS and WFC3, Spitzer/IRAC, and JWST/MIRI data with stellar population models using BAGPIPES (Carnall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' BAGPIPES is a Bayesian SED-fitting code that models the multiband photometry (flux densities) with stellar population synthesis models formed over a wide range of user-defined parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The code has flexibility on the type of stellar popula- tion synthesis models, star-formation history, dust attenua- tion, and nebular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' It has the ability to incorporate prior knowledge on parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The code then computes a probability density for model parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', posteriors) given the data by calculating a likelihood weighted by priors on the parameters, and samples the posteriors for the param- eters using the MultiNest nested sampling algorithm (see Feroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Carnall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Table 2 lists the range of parameters considered for the SED fitting in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For all models we use stellar population synthesis models from Bruzual & Charlot (2003) formed with a Chabrier IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The table defines the parameters and their range of parameter values we explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' BAGPIPES also can incorporate priors on these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In most cases we adopt uniform priors, as listed in Table 2, with two exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The first is related to the nebular ioniza- tion parameter, which controls the strength of the nebular emission features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Current evidence from spectroscopy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Oesch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Le Fèvre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2016, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Back- haus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022), including recent JWST spectroscopy (Brinchmann 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Trump et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022), shows that emission lines are common in star-forming galaxies at z ∼> 1 (and the strength appears to increase with increasing redshift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore we fix the ionization parameter to a high, physically plausible value of logU = −2 as this is representative of the values used in previ- ous studies when fitting the SEDs of galaxies (see for exam- ple the discussion in Whitler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We plan to explore how variations in the ionization parameter impact the con- straints on the stellar populations using future data that can include spectroscopy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', from JWST/NIRSpec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The other exception is for galaxies with photometric red- shifts, where we use the photometric redshift posterior, P(z), derived from EAZY as the prior on the redshift (see Chworowsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For galaxies with spectroscopic redshifts, we force the fit to the spectroscopic redshift value listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For the galaxy star-formation histories (SFHs) we test two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' First, we primarily employ “delayed-τ” mod- els, where SFR ∝ (t/τ) × exp(−t/τ) for age, t, and star- formation e-folding timescale, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These models allow SFHs that rise with time (when t/τ ≪ 1) (as is expected for high- redshift galaxies, Finlator et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2011) and for exponentially declining models (when t/τ ≫ 1) and these have the flexibility to broadly reproduce the evolution of galaxies over long time periods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Larson & Tinsley 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Tinsley 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Carnall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' García-Argumánez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Second, following Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2001) we also consider more extreme SFHs that include both the delayed-τ model (above) an early burst of stars that formed at zf = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The reason for this is that a burst of stars formed at the earliest times has the highest mass-to-light ratio possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As such it adds the most amount of stellar mass (at least for commonly assumed IMFs, like the Chabrier one assumed here), with the minimum impact on the observed galaxy SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In contrast, the slowly evolving delayed-τ model provides a fit to the light from the more-recently formed stellar populations that dominate the rest-frame UV and optical light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, the UV/optical light from the stars formed in the maximally old burst can be “lost in the glare” of more recently formed stars (this is also referred to as “outshining”, Conroy 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The choice of zf = 100 is motivated by the fact that current mod- els expect stars to be forming by z = 20−30 Barkana & Loeb 2001, and references in Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The time from z = 100 to z = 20 spans less 200 Myr, during which little stellar evolu- tion occurs for the longer-lived stars that dominate the stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' By using z f = 100 we allow for a “maximally old” stellar population and we constrain any star-formation that may have occurred at the earliest times which could have the 8 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Parameter Settings for BAGPIPES Model Parameter Prior Limits Star-Formation History (1): Delayed-τ, Ψ ∝ (t/τ)exp(−t/τ) e-folding timescale, τ / Gyr Uniform (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='01, 10) age, t / Gyr Uniform (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='01,15) stellar mass, log(M∗/M⊙) Uniform (5, 12) Star-Formation History (2): Burst at zf = 100 and delayed-τ model from (1) burst age, tburst / Gyr Fixed tburst = Age(z) - Age(z f = 100) burst stellar mass, log(M∗/M⊙) Uniform (0, 13) Additional parameters for all models dust attenuation law .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Calzetti (2001) dust attenuation, A(V) / mag Uniform (0, 3) metallicity, Z/Z⊙ Uniform (0,1) ionization parameter, logU Fixed −2 redshift†, z EAZY P(z) (3, 15) †For galaxies with photometric redshifts the redshift prior is the posterior from the photometric redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For galaxies with spectroscopic redshifts the redshift is fixed at the spectroscopic redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' highest possible M/L (for a canonical stellar populations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 By studying the SEDs of galaxies to longer wavelengths we can constrain the amount of light in this population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For ex- ample, Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2001) and Dickinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2003) found that using K-band data, galaxies at z ∼ 3 could hide as much as 75–90% of their stellar mass in early bursts formed at z f = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Indeed, at the risk of foreshadowing, we find that including the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 micron data reduces the amount of possible stellar mass formed in such maximally old bursts by up to an order of magnitude (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Analysis of Galaxy SEDs Figure 4 shows the BAGPIPES SED fits and one- dimenstional (1D) posterior likelihoods for select parameters of the SED fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The Figure shows six galaxies as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The online version of the Paper includes a Figure Set with these plots for the full sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For each galaxy, the plots compare the SED fits with and without the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Tables 6 and 7 provide the medians (50th per- centiles), 16th and 84th percentiles derived from these poste- rior likelihoods for the stellar masses, SFRs, and photometric redshifts for all galaxies in the sample, both with and without using the MIRI data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 3 A redshift of z = 100 is essentially “immediately” in the history of the Universe as it corresponds to an age of only ≃17 Myr after the Big Bang for the assumed cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Stars are expected to form by z ≈ 20 − 30 (Barkana & Loeb 2001 and references in Section 1), and there are reputed candidates from JWST imaging for galaxies at redshifts as high as z ≈ 15 (see Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, zf = 100 seems to be a reasonable upper bound to ensure we capture the earliest time when stars could plausibly form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Using a lower formation redshift, zf < 100, would lower the upper limit on the stellar masses that could form in the bursts as these would be younger, with lower M/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' To test the robustness of the stellar masses and SFRs de- rived from the BAGPIPES fits, we have refit all the galaxies in our sample using several independent SED-fitting codes ( CIGALE, Boquien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' FAST, Kriek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2009, and the codes of Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022a and Pérez-González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Comparing the stellar masses and SFRs, we find they agree in the mean (with bias, µ ≃ 0 dex) and an inter-method scatter of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='23 dex in stellar mass (a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7) and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='27 dex in SFR (a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This scatter is typical in comparisons of SED-fitting results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Mobasher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We therefore interpret the scatter as representative of the systematic uncertainties on the stellar masses and SFRs here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In some cases adding the MIRI data has a small effect on the median values of the stellar mass and SFR, but it does tighten the allowable range of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 4 panels (a) and (b) show galaxy ID 7364 (at z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) and galaxy ID 6811 (with zsp = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='683, Zitrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These have MIRI data that support the interpretation inferred us- ing only the HST and Spitzer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' However, in both of these cases adding the MIRI data tighten the allowed range of mod- els, and thus improve the constraints on the stellar popula- tion parameters (this is true for the sample in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In both cases shown here, the favored range of stellar mass and SFR are improved significantly when MIRI data are included (with improvements in the inter-68%-tile range by more than a factor of two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Below the SED fit for each galaxy, Figure 4 shows the posterior probability densities for the SFR, mass- weighted age (AgeMW), stellar mass, and dust attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Adding the MIRI data typically produce narrower posteriors for SFR and stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is a result of improved con- straints on the dust attenuation (A(V)), and this forces the models to a narrower range of SFR and stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is the case for ID 7364 and 6811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 9 1 2 3 4 6 10 observed wavelength [ m] 22 23 24 25 26 27 AB magnitude log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) SFR/M yr 1 = 52 (+16, 13) z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7) log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3) SFR/M yr 1 = 39 (+24, 16) z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8) ID 7364 with MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] without MIRI 1 2 3 4 6 10 observed wavelength [ m] 23 24 25 26 27 AB magnitude log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) SFR/M yr 1 = 40 (+12, 10) z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='683 log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4) SFR/M yr 1 = 47 (+45, 23) ID 6811 with MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] without MIRI 0 100 SFR/M yr 1 density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 AgeMW/Gyr 9 10 log M /M 0 1 A(V)/mag without MIRI with MIRI (a) 0 200 SFR/M yr 1 density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 AgeMW/Gyr 9 10 log M /M 0 1 A(V)/mag without MIRI with MIRI (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Examples of SED fits for galaxies in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The green data points show the measured flux densities and uncertainties on the HST/WFC3, Spitzer/IRAC, and JWST/MIRI bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The red-shaded regions shows the model fit to all the data points (the shaded region shows the inner-68% range of models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' the solid red line shows the median).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The black-shaded region shows the fit to the data points excluding the MIRI bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The small red and grey points show the median-model photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The inset text gives the median and 68%-tile uncertainties on the stellar masses and SFRs inferred from the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Below each SED plot, the panels show the posteriors (probability density) for the SFR, mass-weighted (MW) age, stellar mass, and dust attenuation for each galaxy (using MIRI and excluding MIRI data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The dashed lines denote the 5, 50, and 95% intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The examples include galaxies where adding the MIRI data yields similar stellar masses and SFRs, but with tighter constraints (panels (a) and (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Other examples show galaxies where adding the MIRI data greatly reduces both the stellar masses and SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Typically this results from contamination in the IRAC data or because of strong emission lines impacting the IRAC data (or both;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' see panels (c) and (d)), or because the stellar continua appear very blue (see panels (e) and (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The complete figure set (28 images) is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Set 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' SED fits and 1D posteriors for SFRs, stellar masses, mass-weighted ages, and dust attenuation for all galaxies in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In other cases adding the MIRI data changes the interpre- tation of the galaxy stellar populations dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Fig- ure 4 panels (c) and (d) show galaxies with ID 19180 (with zsp = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='077) and ID 18441 (at z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In both of these cases, without MIRI data the HST to IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm data implied very red rest-UV–to–optical colors, leading to high dust-attenuation values (A(V) ∼> 1 mag) with high SFRs (∼> 90 − 100 M⊙ yr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The stellar masses in these cases are also elevated primarily because the higher dust attenuation increases the mass-to-light ratio (M/L) of the models, and therefore increases the stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Including the MIRI data changes the favored stellar population models to ones with much bluer rest-UV–to–optical colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As a result the dust attenuation, SFR and stellar mass are decreased, by an order of magnitude in some cases (the decrease in stellar mass is more than that of the SFR, implying the specific SFR declines slightly as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This impacts roughly ≈33% of the sample here (10 of the 28 galaxies based on our visual inspection of the SEDs and 1D posteriors, see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Yet in other cases, the MIRI data forces the constraints on the stellar populations to be bluer than expected based on the HST and Spitzer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 4 panels (e) and (f) show two galaxies that demonstrate these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In both the cases of galaxy ID 12514 (at z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7) and ID 26890 (at z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9) the MIRI data favor very blue UV/optical colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In the case of galaxy 12514, there are indications that the IRAC and MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm data are boosted by the presence of strong emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Having the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm photometry favors a lower stellar continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As a result the SFR and stellar mass are reduced (the presence of redshifted Hα in the MIRI F560W band is apparent even in the galaxy image in Figure 3 which shows the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm flux density is noticeably brighter than the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm flux density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For this reason the SED fit favors a 10 1 2 3 4 6 10 observed wavelength [ m] 22 23 24 25 26 AB magnitude log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) SFR/M yr 1 = 34 (+14, 9) z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='077 log M /M = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) SFR/M yr 1 = 120 (+60, 41) ID 19180 with MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] without MIRI 1 2 3 4 6 10 observed wavelength [ m] 24 25 26 27 28 AB magnitude log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) SFR/M yr 1 = 10 (+4, 3) z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6) log M /M = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3) SFR/M yr 1 = 90 (+67, 45) z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6) ID 18441 with MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] without MIRI 0 200 400 SFR/M yr 1 density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 AgeMW/Gyr 9 10 11 log M /M 0 1 2 A(V)/mag without MIRI with MIRI (c) 0 200 400 SFR/M yr 1 density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='50 AgeMW/Gyr 9 10 11 log M /M 0 2 A(V)/mag without MIRI with MIRI (d) Figure 4 (continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' slightly higher photometric redshift (to accommodate the Hα emission in the F560W bandpass), see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Galaxy ID 26890 is noteworthy in itself because it is the highest redshift galaxy in the sample, and because the HST/WFC3–to–JWST/MIRI colors are H160 − [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 mag, and indicative of stellar populations with very low M/L ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The IRAC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm emission shows indications of enhancement, possibly owing to redshifted Hβ+[O III].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI data reign in the allowable range of stellar population parameters, favoring models with lower SFRs than the con- straints that lack MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, the MIRI data favor bluer rest-UV/optical col- ors compared to the IRAC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In part this may have a physi- cal explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' At the redshifts of the galaxies under consid- eration, the IRAC data contain strong emission lines, includ- ing Hβ+[O III] at 5 < z < 8, [O II]+[Ne III] at 7 < z < 12, and Hα+[N II] at 4 < z < 7 (Labbé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Roberts-Borsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Ends- ley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' At certain redshifts both the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm bands can be both be affected (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Roberts- Borsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Without the SED fits, the models that fit the data may include both models with strong emission lines and/or models with redder colors (indicative of strong dust attenuation) or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The models in Figure 4 show that the presence of strong emission lines in the bands augments the flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This could account for part of the difference between the SED fits with and without MIRI: the MIRI data are evidence in these cases that the stellar populations have bluer colors, and the elevated IRAC flux densities then would require strong emission line EWs to reproduce them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' However, crowding between sources in the IRAC data may be another reason for the increased IRAC flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The lower angular resolution of the IRAC data can cause blending from bright, nearby galaxies, and this can lead to additional uncertainties in the flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As the images in Figure 3 il- lustrate, some galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', ID 19180 and 6811) have bright objects within 3 arcseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The light from these objects makes deblending more difficult and could potentially bias the flux-density measurements (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Laidler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Skelton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For this reason the elevated IRAC flux densities may include systematic mea- surement uncertainties from blended objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In Appendix A we test for the effect of source blending by excluding ob- jects that have any bright neighbor within 3′′ (we show that source blending does not bias our interpretation for the stellar masses nor SFRs, nor in their evolution, that we infer for the galaxy population).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' However this does emphasize the benefit of having data with the enhanced angular resolution available from JWST imaging (for both NIRCam and MIRI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The Impact of MIRI on Redshifts and Implications for Emission Lines Figure 5 compares the redshifts for the galaxies in our sam- ple derived from our BAGPIPES fits for the galaxies exclud- ing the MIRI data (znoMIRI) and when including the MIRI data (zMIRI) as a function of the prior photometric redshift 11 1 2 3 4 6 10 observed wavelength [ m] 24 25 26 27 28 AB magnitude log M /M = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3) SFR/M yr 1 = 5 (+2, 2) z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6) log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4) SFR/M yr 1 = 12 (+13, 7) z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6) ID 12514 with MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] without MIRI 1 2 3 4 6 10 observed wavelength [ m] 24 25 26 27 28 AB magnitude log M /M = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3) SFR/M yr 1 = 7 (+2, 3) z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3) SFR/M yr 1 = 19 (+15, 8) z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) ID 26890 with MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] without MIRI 0 50 SFR/M yr 1 density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 AgeMW/Gyr 8 10 log M /M 0 1 2 A(V)/mag without MIRI with MIRI (e) 0 50 SFR/M yr 1 density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 AgeMW/Gyr 8 10 log M /M 0 1 A(V)/mag without MIRI with MIRI (f) Figure 4 (continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 5 6 7 8 9 redshift, z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 ( znoMIRI zMIRI) / (1 + z) 35896 7818 12514 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Comparison of the redshifts for galaxies in our sample with and without the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The data points show the rela- tive difference between the photometric redshifts (derived from our BAGPIPES fits) for the galaxies excluding MIRI (znoMIRI) and when including MIRI (zMIRI) as a function of the prior photometric red- shift from Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Annotated points indicate ob- jects with |znoMIRI −zMIRI| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These objects indicate shifts in their photometric redshift, where at least in part this is because of nebular emission in one or more bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' from Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For most galaxies there is good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (For completeness we include all 28 galax- ies in this comparison, using the photometric-redshifts even for galaxies with spectrscopic redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=') In several instances we see the median redshift from the posterior shifts appreciably when including the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 5 shows three objects where the shift is greater than ∆z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In all cases the shift in the redshift probability density appears to be related to the effects of one or more emission lines in the IRAC or MIRI passbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 6 illustrates the shift in P(z) from the BAGPIPES fits for these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Two of these galaxies lie at 4 < z < 6 (galaxy IDs 35896 and 7818).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Adding the MIRI data favors having strong nebular emission in both of the IRAC bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This oc- curs at z ∼> 5 when Hβ+[O III] enters the IRAC Ch1 band- pass (at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm) and Hα enters the IRAC Channel 2 band- pass (at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Inspection of the galaxy SEDs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4) shows that the IRAC–to–MIRI col- ors ([3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] − [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] and [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5] − [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6]) are blue for both of these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The BAGPIPES fits that include the MIRI data fa- vor models in which the galaxy has strong nebular emission in the IRAC bands to account for these color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This increases (decreases) the redshift probability density at higher (lower) redshifts for these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The remaining galaxy, ID 12514, lies at z ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 6 shows the redshift probability densities for this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI F560W image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 3 shows evidence of “boosted” flux 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 P(z) ID 35896 with MIRI without MIRI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 P(z) ID 7818 with MIRI without MIRI 6 7 8 9 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 P(z) ID 12514 with MIRI without MIRI Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Comparison of redshift posterior probability densities, P(z), derived from the BAGPIPES fits to three of the galaxies in the sample (as labeled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These three galaxies each have a change of more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 between the median redshift when including the MIRI F560W and F770W data in the fit versus when they are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For the case of galaxy IDs 35896 and 7818, the IRAC−MIRI colors are blue, indicating redshifted emission lines (likely Hβ+[O III] and Hα) inhabit both IRAC bands (while the MIRI data probe the galaxy continua).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In the case of galaxy ID 12514, the MIRI F560W band clearly shows boosted emission, likely from redshifted Hα at z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' compared to the MIRI F770W image, where the MIRI color is [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6]−[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 mag, see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The most likely explanation for this boosted emission appears to be from red- shifted Hα+[N II] at z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 in the MIRI F560W band- pass (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The strength of the flux in the F560W band- pass decreases the probability density for redshifts with z ∼< 7 as these would place Hα at wavelengths shorter than covered by the bandpass (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, nebular emission lines appear to be responsible for the cases where there are larger shifts in the photometric redshift solutions, However, these cases are generally excep- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For most galaxies in the sample the redshift posteriors are consistent, where Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 5 shows the differences are negligi- ble in median redshifts with and without including the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We expect that many (even most) of the galaxies in the sample also exhibit strong emission lines given the pre- ponderance of evidence from other galaxies at these redshifts (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1) and by inspection of the SEDs in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In these cases adding the MIRI data supports the redshifts, either because the emission lines make less of an impact on the redshift likelihoods or because the MIRI data reinforce them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The Impact of MIRI on the Stellar Masses, Dust Attenuation, and SFRs Figure 7 compares the stellar masses and SFRs derived for the galaxies in our sample using the simple delayed-τ models with BAGPIPES for the case where we include the MIRI F560W and F770W data and when we exclude it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In general, including the MIRI data reduces the stellar masses and SFRs for the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We study the offsets in stellar mass and SFR for our galax- ies in two redshift bins, 4 < z < 6 and 6 < z < 10, where the median offsets in stellar mass and SFR are indicated in the Figure as large rectangles, and are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Formally, the offsets in stellar mass are ∆log(M∗/M⊙) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='25 dex for 4 < z < 6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='38 dex for 6 < z < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The inter-68%-tile intervals are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='28 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='44 dex, respectively (estimated us- ing the median absolute deviation, σNMAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For the SFRs the offsets are ∆log(SFR/M⊙ yr−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='15 dex for 4 < z < 6 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Offsets in Stellar Mass and SFRs for 4 < z < 10 galaxies when including the MIRI F560W and F770W data Stellar Mass offsets SFR offsets ⟨∆logM∗⟩ = ⟨ ∆ log SFR ⟩ = logM∗,noMIRI/M∗,MIRI log SFRnoMIRI / SFRMIRI Sample Median Scatter Median Scatter 4 < z < 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='25 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='28 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='15 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='12 dex 6 < z < 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='38 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='44 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='29 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='27 dex NOTE—The quantities with the subscript “noMIRI” denote values derived without the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Quantities with the subscript “MIRI” denote the values derived with the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The scatter is the inter-68-percentile interval derived from the median absolute deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='29 dex for 6 < z < 10, with an inter-68 percentile interval of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='12 dex and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='27 dex, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Because the impact of MIRI is larger on the stellar mass than the SFR, the specific SFRs will be reduced by approximately by ∆logSFR−∆logM∗ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 dex (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The reasons for the offsets are similar to that discussed for the individual galaxies in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Including the MIRI data generally favors stellar populations with bluer rest-UV/optical colors, with lower M/L ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This forces the constraints to lower stellar-mass models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For the SFRs, because the colors are bluer, there is less dust attenuation favored in the models, which lowers the SFRs compared to models with higher dust attenuation (at fixed observed galaxy luminosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI data also remove some of the degen- eracy between models with redder stellar populations versus those with strong emission lines impact select bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These have the combined effect of favoring lower stellar masses and SFRs when including the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' There is significant scatter in the offsets of stellar mass and SFR for individual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In Figure 7 the error bars on the large rectangles show the inter-68-percentile range (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='e,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 13 8 9 10 11 log M [M ] stellar mass with MIRI 8 9 10 11 stellar mass without MIRI log M [M ] 4 6 8 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 log M (dex) 0 1 2 log SFR [M yr 1] SFR with MIRI 0 1 2 SFR without MIRI log SFR [M yr 1] 4 6 8 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 log SFR (dex) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Comparison of stellar masses and SFRs derived from the SED modeling for galaxies including the MIRI F560W and F770W data and without the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The top set of panels show the comparison for the stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The bottom set of panels show the comparison for the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In each row the left panel shows the direct comparison, where the dashed line shows the one-to-one relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The right panel shows the logarithmic difference, defined as ∆logMwith MIRI ∗ −logMwithout MIRI ∗ (similarly for the SFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The symbols are color-coded by redshift (using the right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In the right panel the large rectangular boxes (and error bars) show the median value (and scatter) in two bins of redshift (4 < z < 6 and 6 < z < 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The values for these are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Adding the MIRI data generally lowers the SFRs and stellar masses of these galaxies (though the scatter is significant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The median offset is larger for galaxies at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' the difference between the 16–84 percentiles) for both the 4 < z < 6 and 6 < z < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For some galaxies the offsets are insignificant, with ∆logM∗ ≈ 0 and ∆logSFR ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Two of these galaxies are shown in Figure 4 (IDs 7364 and 6811;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' others are available in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In these cases the MIRI data tighten the existing constraints on the derived stellar masses and SFRs, reducing the uncertainties (by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 dex in stellar mass and by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 dex in SFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In other words, for some cases the MIRI data support the range of stellar popula- tion parameters favored by the fits to the HST/ACS + WFC3 and Spitzer/IRAC data, but the MIRI data improve the accu- racy of the measurements, typically by a factor of order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In other cases the MIRI data dramatically change the inter- pretation of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This was noted above for galaxies 18441 and 19180 (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 and Figure 4), where adding the MIRI data reduce the stellar mass and SFRs substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 7 shows that this is typically the case, where the MIRI data decrease the average stellar mass and SFR for galaxies 14 in our sample, typically by a factor of order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We will ex- plore the implications this has on the evolution of the galaxy stellar-mass density in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The Impact of MIRI on the Inferred Star-Formation History (and the Mass Formed in Bursts) Arguably, one of the most extreme star-formation histo- ries imaginable is the case where a galaxy forms in either one burst at zf → ∞, or (slightly less extreme) in a series of bursts extending back to that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' When a burst forms, the stellar population immediately begins aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' A burst at z f = ∞ has the oldest possible age at any subsequent time, and the smallest amount of light (at a given mass), and there- fore it would have a maximal M/L at any observed epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As the stellar population ages, its colors also become redder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For all these reasons, it is conceivably possible to hide signif- icant amounts of stellar mass formed at earlier times (this is the “outshining” problem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Dickin- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Pforr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Conroy 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' One advantage of focusing on galaxies at high redshifts is that the amount of time for discrete episodes of star- formation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', many individual bursts) is small given the age of the Universe (the Universe has an age of 1 Gyr at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 and 500 Myr at z = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 for the adopted cosmology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is shorter than the lifetimes of stars of spectral type A and later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The short age of the Universe, combined with longer wave- lengths probed by the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data allow us to place tighter constraints on the mass from earlier bursts in high redshift galaxies than has been possible previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We explored the possibility that such an early burst could contribute stellar mass to the galaxies in our sample, and how the JWST/MIRI data can improve the constraints on this pop- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We used the SED fits to the galaxies in our sample using a star-formation history that include both the delayed-τ model (as in Section 3 above) and the burst of stars formed at z f = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The details of the other model parameters are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' To quantify the amount of allowed stellar mass formed in the burst for each galaxy, we use the Bayesian Informa- tion Criterion (BIC, Bailer-Jones 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The BIC provides a criterion for model selection in the case that one intro- duces a model with an additional parameter (in our case we are selecting between between two models, one with a star- formation history that has a delayed-τ model only, and one with both the delayed-τ model and an early burst at zf = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' When comparing the two models, the BIC applies a penalty term for the additional parameter (to determine if the addi- tional parameter improves the fit, or is overfitting the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We use the BIC defined as (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Buat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019), BIC = χ2 0 × klnN, (2) where χ2 0 is the minimum chi-squared from the SED-fitting, k is the number of independently fitted parameters (we use k = 7) and N is the number of data points (N = 8 or 10, depending on if the MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] and [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] data are included or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Our goal is to quantify the upper limit on the stellar mass formed in an early burst that could exist in our galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' To do this, we select models with bursts (zf = 100) that are not excluded by the BIC criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' That is, for a given galaxy we identify all models that satisfy χ2 ≤ BIC, where the BIC is defined in Equation 2, and χ2 is the fit for a given model (this is similar to the approach adopted by Buat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For each galaxy, we select the model with the highest stellar mass in the zf = 100 burst from the subset of models that satisfy the BIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We then take this as upper limit on the stellar mass per- mitted in the burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Comparing this upper limit on the stellar mass formed in bursts to the range of stellar masses from the fits we find that the limiting stellar mass from models that satisfy the BIC criteria corresponds roughly to a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7% up- per limit on the mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', the values we report correspond approximately to a 3σ upper limit on the stellar mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Ta- bles 6 and 7 list the upper limit on the stellar mass in the burst component for all galaxies in the sample for the case that we include and exclude the MIRI data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 8 shows example SED fits for galaxies in our sam- ple, both with and without the bursts, for the both the cases that we include and exclude the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The complete figure set (28 images) is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The examples shown in the Figure include a galaxy where the IRAC data are bright relative to the MIRI data (ID 18441).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In this case, without the MIRI data the allowed stellar mass in the burst can reach logM∗/M⊙ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0, but adding MIRI reduces this by nearly an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For galaxy ID 7364, the MIRI data favor red IRAC–to–MIRI colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Never- theless, because the MIRI data constrain the models at longer wavelengths, they also lower the amount of stellar mass al- lowed in the burst: without the MIRI data the burst can in- clude logM∗/M⊙ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' when MIRI data are included the mass in the burst declines by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 dex (a factor of two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For galaxy ID 6811, the MIRI data show that because they con- strain the SED at longer wavelengths, the amount of stellar mass allowed in the burst is reduced by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 dex (nearly a factor of five).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, the addition of the MIRI F560W and F770W data reduce the amount of stellar mass that can form in early bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 9 shows the change in stellar mass for the case that the star-formation histories include only delayed-τ mod- els (labeled “no burst” in the figure) compared to when an early burst of star formation at zf = 100 is included (labeled “allowed in burst”) in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 9 shows the results for both the case that the MIRI data are excluded (top row) and when the MIRI data are included (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For the fits that lack MIRI data, the amount of stellar mass allowed in the burst is nearly an order of magnitude higher than con- strained in the delayed-τ models: in this case the median differences in the log of the stellar mass of the models with early bursts and those with only delayed-τ models is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 dex at 4 < z < 6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 dex at 6 < z < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For the fits that in- clude the MIRI data, the amount of stellar mass allowed in the bursts is significantly reduced: the median differences in 15 allowed in burst component log M / M =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 allowed in burst component log M / M =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 1 2 3 4 6 10 observed wavelength [ m] 24 25 26 27 28 AB magnitude with burst: log M /M = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3) no burst: log M /M = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3) ID 18441, z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='44 -- without MIRI -- with burst at z = 100 delayed- model only 1 2 3 4 6 10 observed wavelength [ m] 24 25 26 27 28 with burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) no burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='29 -- with MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] -- with burst at z = 100 delayed- model only allowed in burst component log M / M =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 allowed in burst component log M / M =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 1 2 3 4 6 10 observed wavelength [ m] 23 24 25 26 27 AB magnitude with burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3) no burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3) ID 7364, z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='04 -- without MIRI -- with burst at z = 100 delayed- model only 1 2 3 4 6 10 observed wavelength [ m] 23 24 25 26 27 with burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) no burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='90 -- with MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] -- with burst at z = 100 delayed- model only allowed in burst component log M / M =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 allowed in burst component log M / M =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2 1 2 3 4 6 10 observed wavelength [ m] 23 24 25 26 27 AB magnitude with burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4) no burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4) ID 6811, z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='683 -- without MIRI -- with burst at z = 100 delayed- model only 1 2 3 4 6 10 observed wavelength [ m] 23 24 25 26 27 with burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) no burst: log M /M = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2) z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='683 -- with MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6], [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7] -- with burst at z = 100 delayed- model only Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Example fits to SEDs for the galaxies in our sample, comparing the results from fits that include an early burst of star-formation at z f = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Data points and upper limits have the same definitions as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In each panel, the shaded regions show the stellar population fit to the SED using the total model (cyan-shaded = delayed−τ plus the burst) and delayed-τ model only (gray-shaded, these are identical to those in Figure 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' there is almost no difference between the cyan- and gray-shaded models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The tan-shaded region shows the light permitted in the burst component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The labels indicate the amount of stellar mass in each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Each row shows the results for one galaxy, where the Left panel shows the results that exclude the MIRI F560W and F770W data, and the tight panel shows the results including the MIRI F560W and F770W data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The complete figure set (28 images) is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Set 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' SED fits including the MIRI data and excluding the MIRI data for all galaxies comparing the fits with and without bursts at zf = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 16 8 9 10 11 log M [M ] no burst, no MIRI 9 10 11 allowed in burst: log M [M ] 4 6 8 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 log M (dex) 8 9 10 11 log M [M ] no burst, with MIRI 9 10 11 allowed in burst: log M [M ] 4 6 8 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 log M (dex) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Comparison of stellar masses derived for galaxies with and without early bursts (at z f = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The top row shows the results that lack MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The top-left panel shows the stellar masses derived from the delay-τ models only (labeled “no burst”) compared to the models that include the burst (labeled “allowed in burst”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The top-right panel shows the difference between the stellar masses as a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The bottom row shows the same results for the galaxies including the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In the left panels, the dashed lines show the one-to-one relation and the solid lines show the median offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In the right panels, the large rectangles show the medians in two bins of redshift (4 < z < 6 and 6 < z < 9) these are given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Adding the MIRI data reduces the amount of stellar mass allowed in the burst components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 17 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Ratio of the stellar mass allowed in models that include an early burst of star-formation (at z f = 100) to those that include only a delayed−τ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' logM∗(with burst)−logM∗(no burst) with MIRI data no MIRI data Sample Median Scatter Median Scatter 4 < z < 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='59 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='16 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='87 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='23 dex 6 < z < 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='69 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='04 dex 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='11 dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='13 dex NOTE—The values labeled “with burst” denote the upper limit on the stellar mass for models that include bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The values labeled “no burst” denote the stellar masses for models that assume only a delayed-τ star-formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The values “with MIRI” include the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm flux densities from MIRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' this case 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 dex at 4 < z < 6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 dex at 6 < z < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These values are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' On the Colors, Stellar Masses, and Nebular Emission in Early Galaxies One of main findings in this paper is that the MIRI data favor bluer colors in galaxies at 4 < z < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 10 shows this by comparing the relative SED for each galaxy, both in the case of including the MIRI data and without the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Including the MIRI data reduces the derived rest-frame I-band light by approximately ∆m1600 − I ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In- specting Figure 10 this appears to result from many galax- ies favoring bluer SEDs when the MIRI data are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In other words, without the MIRI data, the SED is uncon- strained at longer wavelengths, and this allows for a greater range of SED shape (where the median favors a solution which on average is redder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Adding the MIRI data shifts the likelihood to bluer populations for many galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This has a major impact on the implied M/L, as the blue rest-frame colors implies younger ages, lower dust attenuation, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The fact that adding the MIRI data makes the galaxies bluer largely explains the differences in the derived stellar masses and SFRs observed in Figure 7, where adding the MIRI data lower the stellar masses and SFRs compared what the models favor when the MIRI data are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, our interpretation of the MIRI data is that galax- ies at high redshifts (z > 4) are bluer than inferred from pre- vious studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This adds to other studies that find that galax- ies at high redshifts must have (very) blue colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Studies from the pre-JWST era have argued that galaxies at z > 4 show indications of declining (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', steepening) UV spectral slopes with increasing redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Bouwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Bhatawdekar & Conselice 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These conclusions have been reinforced by early JWST imaging that shows very blue colors among UV-selected galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Nanayakkara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Topping 0 2 with MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 m best (median) SED fits for each galaxy 0 2 m1600Å - m (mag) no MIRI data median rest-frame colors of samples 1000 2000 5000 10000 rest wavelength [Å] 1 0 1 (m1600Å - m ) (mag) redder with MIRI data bluer with MIRI data Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Comparison of median, relative SED for each galaxy, both for the case that MIRI data are used (top panel) and when the MIRI data are excluded (middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The individual lines are the median SED model fit to each galaxy in the sample, shifted to the rest-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The large data points show the median rest-frame magnitude at 1600 Å, 2800 Å, and U, B, V, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The error bars show the scatter in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' All models have been normalized to the 1600 Å magnitude (which accounts for the lack of scatter at that wavelength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The bottom panel shows the difference in color (∆m) between the models with and without the MIRI data (the error bars show the range of the 16th-84th percentiles of the sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The change in the color at the reddest wavelengths probed (about the rest-frame I-band) corresponds to a ∆m ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 mag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We find here that including the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data show strong evidence that the stellar populations are very blue in their rest-frame UV–to–I-band colors, seem- ingly more so than inferred from these previous studies (as in some cases the galaxies in our sample are identical to those in these other studies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, the MIRI data favor lower stellar masses and lower SFRs in distant galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Adding the MIRI data also changes the interpretation of the strength of nebular emission in the galaxies in the sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Many of the galaxies show red colors between the HST/WFC3 and Spitzer/IRAC data (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In the ab- sence of MIRI data, the SED fitting interprets these colors as a combination of higher dust obscuration, older ages, and 18 strong nebular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This has been reported previously for individual galaxies and for stacked samples at z > 6 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Hutchison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Stefanon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The fact that without the MIRI data the mod- els allow for higher dust obscuration and older stellar popu- lations increases the M/L and the stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The higher dust obscuration also leads to higher SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' However, includ- ing the MIRI data changes this interpretation for the galaxy population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Nebular emission lines appear to be the primary explanation for the red HST/WFC3 to Spitzer/IRAC colors (while there are some galaxies where the MIRI data does not change [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', galaxy IDs 6811 and 7364 in Figure 4] this does not change the main result in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This means (1) the nebular emission lines for high redshift galaxies must be strong and (2) the stellar populations are blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Early JWST results show emission lines remain strong in high-redshift galaxies and impact the reddest (4 − 5 µm) bandpasses in NIRCam (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Giménez-Arteaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Topping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Whitler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI data appear necessary to extend the rest- frame wavelength coverage to >7000 Å, past the strongest of the nebular emission features in the rest-frame optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Implications for Early Star-Formation and Stellar Masses in Galaxies The difference between the delayed-τ star-formation his- tory and the model that includes bursts, while simplistic, ar- guably span the gamut of available star-formation histories of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Simulations show that galaxies experience many discrete bursts, but when averaged over long time baselines the evolution is mostly smooth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Diemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Leja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore the smoothly evolv- ing delayed-τ model represents the slowly evolving evolu- tion of galaxy star-formation histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This can reproduce the bluest colors (and lowest M/L) of the stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Of course, galaxies can experience a host of stochastic bursts through changes in gas accretion or events that can sudden changes in the star formation (as a result of mergers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Kartaltepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022, or the sudden onset of strong feed- back from an AGN, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' “Bursts” of star-formation from these events will add stellar mass, but if these occur at z < 100 then will be younger, and will have M/L lower than a model with a burst at zf = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As such, the models with a burst at zf = 100 have the maximum M/L and the oldest possible ages for a stellar population at the ob- served redshift of each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore the models with this burst represent a maximum stellar mass possibly formed in these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Our results show that including MIRI reduces the amount of stellar mass allowed in these models, by an order of mag- nitude in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In itself this is interesting as nearly all galaxies show no direct evidence for such early star- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Comparing the median stellar masses of galax- ies when fit by the delayed-τ models only and those with the delayed-τ models and the early burst at z f = 100 shows they differ by median(∆M∗) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 dex (see the plots in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We find no convincing cases in our sample where the galax- ies require a burst at zf = 100 to better fit their SEDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This would imply that galaxies do not experience early bursts of star-formation (or at least such bursts do not form sufficient mass that we require them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Part of our findings could be impacted by biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' First there is selection bias: all the galaxies studied here were se- lected in HST/WFC3 data, and therefore required some rest- frame UV emission above the HST/WFC3 detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' It will be important to test for objects in, for example, fu- ture JWST/NIRCam-selected populations show evidence for early bursts (especially JWST-selected samples that lack HST counterparts, Glazebrook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Pérez-González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Second, very recent work shows evidence for older stellar populations in the spatially resolved colors of 6 < z < 9 galaxies (Giménez-Arteaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As these issues become better constrained, then the ages of the stellar pop- ulations in distant galaxies could begin to inform us about when galaxies first form stars (see Whitler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Implications for Galaxy Growth The question of how much mass is contained in galaxies is related to the integral of the galaxies’ star-formation his- tories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is important because it contains the integrated record of how rapidly galaxies acquire their baryons, and how efficiently they convert these into stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This has already been discussed as an impossibly early galaxy problem, where galaxies may have acquired too much stellar mass: in typical JWST surveys galaxies should have less stellar mass than a few times 1011 M⊙ (Behroozi & Silk 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Boylan-Kolchin 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The effect of adding the MIRI data already show that stel- lar masses and SFRs derived for galaxies tend to be overesti- mated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The median offsets for the galaxies in our sample are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='15 dex at 4 < z < 6 and rise to ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='3 dex at 6 < z < 9 (Fig- ure 7 and Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Assuming that these (median) offsets ap- ply to previous estimates of galaxy stellar masses, then it im- plies that measurements of the cosmic SFR density (SFRD, which is the average SFR in all galaxies per co-moving vol- ume element) are similarly biased to higher values at these redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 11 shows the impact of these lower SFRs on the cosmic stellar-mass density, ρ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Firstly, the figure shows ρ∗ derived from the integral of the SFRD for two empirical models calibrated against measurements from the literature (Madau & Dickinson 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Finkelstein 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Finkelstein (2016) shows that these models are consistent with a compi- lation of measurements of ρ∗ from the literature at 4 < z < 10 prior to the launch of JWST, (Oesch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Duncan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Grazian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The thick line in Figure 11 labeled “MIRI corrected” shows the empirical model of Finkelstein corrected by the offsets of the SFRs de- rived in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' To derive these corrections we have inter- polated the results from Table 3 assuming median redshifts of z = 5 and 8 for the derived offsets in the two redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Parenthetically, although we use the offsets derived from the SFRs, using those for the stellar masses changes the results 19 4 5 6 7 8 9 10 redshift 4 5 6 7 8 stellar mass density log [M Mpc 3] Finkelstein 2016, original Finkelstein 2016, MIRI corrected Madau & Dickinson 2014 St21 Bh21 Ki20 Sa22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 age of Universe [Gyr] constrained with JWST/MIRI permitted/favored without JWST/MIRI -4 -3 -2 -1 log fraction of (z = 0) Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Evolution of the cosmic stellar mass density, ρ∗, in galaxies from 4 < z < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The lines show pre-JWST constraints from Finkelstein (2016) and Madau & Dickinson (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The data points show recent measurements of ρ∗ at z > 6 from the literature (Kikuchihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2020 [Ki20], Bhatawdekar & Conselice 2021 [Bh21], Stefanon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2021 [St21], Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022a [Sa22]), which largely follow the pre-JWST constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The shaded regions show maximally allowable stellar mass density assuming galaxies experience a burst at z = 100 followed by “normal” star-formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Constraints lacking JWST/MIRI coverage to rest-frame 1 µm allow for a stellar mass density that is up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 dex higher at z = 4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 dex higher at z = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Including MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data lowers the maximum allowed by up to a factor of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' by ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We note, however, that because the MIRI data imply offsets in the SFRs of galaxies, the similarly lower the values of the cosmic SFRD at z > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Secondly, the MIRI data improve the constraints the amount of stellar mass possible in early bursts of star- formation (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is illustrated by the shaded re- gions in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' To derived the area in the shaded swaths, we applied the ratio between the mass permitted in early burst at z f = 100 given to the fiducial value (listed in Table 4), inter- polated in redshift as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, the effects of adding the MIRI data both lower SFR (and stellar masses) and limit the total stellar mass allowed in early bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The combi- nation of these effects reduces the upper bound on the total cosmic stellar mass density allowed by the data by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 dex at z = 4 and by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 dex at z = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As illustrated in Figure 11, this implies that the JWST/MIRI data have constrained the stel- lar mass in galaxies at z = 9 to be less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1% that of the present-day value (ρ∗(z = 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Random Musings The fact that galaxies are bluer than previous constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', they are “bluer than they appear”) has other conse- quences, and likely dovetails with other recent results from JWST NIRCam imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These results in this Paper are likely only the first foray into the properties of distant galaxies using the longer wavelength data available from MIRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Future studies will be able to combine both NIRCam and MIRI imaging, provide JWST–quality data from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8 − −10 µm, which will improve the constraints in this Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Already there are indicates using JWST/NIRCam data only that the number density of luminous galaxies at z > 10 may be much higher than predictions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', Bouwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Donnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Harikane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' One expla- nation for these discoveries could be that the UV–luminosity per unit stellar mass (the UV “efficiency”) may be higher than our models predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This could be a result of changes in the stellar populations (a shift toward bluer/harder ioniz- ing spectra) or a change in the stellar IMF that is weighted toward higher-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' However, a change in the IMF or in the UV efficiency does not change the conclusion that the light from older stars could be lost in the glare of the more recently formed stars, nor would it change the conclusion that there is less light in gen- eral at longer wavelengths from these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Even if the lower-mass cutoff of the IMF is higher, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', > 50M⊙, (Raiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2010) then after ∼10 Myr the mass left in such stars would be effectively zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, any of these effects would further lower the galaxy M/L values, and therefore lower the stellar masses by even more than what we have measured with the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (The only way to add more stellar mass in galaxies at this epoch is if there is a substan- tial population of galaxies at z > 6 that are undetected in HST, which would be an important discovery for JWST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Regard- less, the MIRI data have better constrained the available light in stars at these early epochs and shown that galaxies contain more than three times less “light” at rest-frame 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 − 1 µm than previously known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' CONCLUSIONS In this Paper we have presented results from CEERS on the stellar population parameters for 28 galaxies with red- shifts 4 < z < 9 using new imaging data from JWST/MIRI at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Our galaxy sample was detected in deep data from HST/WFC3 and ACS and has observations from Spitzer/IRAC at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data extend the coverage of the rest-frame spectral energy distribution to nearly 1 micron for galaxies in this redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We use these data to study the improvements in the stellar masses and SFRs of the galaxies at these redshifts when the MIRI data are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Our main results are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Galaxies at 4 < z < 9 have bluer rest-frame UV–I-band colors (m1600 −I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Using the MIRI data we model the SEDs using stellar population synthesis models (with BAGPIPES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' When we compare the average galaxy SED (Figure 10) we find that models that include the MIRI data are (on average) ∆(m1600 − I) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4 mag bluer in their rest-frame colors compared to models that exclude the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Galaxies generally have lower stellar masses and SFRs when the MIRI data are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For the majority of the galaxies (Figure 7) adding the MIRI data reduces the derived stellar masses by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='25 dex at 4 < z < 6 (a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='8) and by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='38 dex at 6 < z < 9 (a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Similarly including the MIRI data reduces the SFRs by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='15 dex at 4 < z < 6 (a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='29 dex at 6 < z < 9 (a factor of 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' There are multiple reasons the stellar masses and SFRs are lowered when we include the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The first reason is that the galaxies are blue, and the fits favor models with lower dust attenuation and mod- els with lower M/L in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The second reason is that in many cases the IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 µm data probe the rest-frame optical, and these show indica- tions of containing light from strong emission lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', redshifted Hβ + [O III], Hα+[N II], [O II], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These boost the flux in these bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In the absence of MIRI data the models can not determine if the red rest-frame UV-optical colors are a result of dust at- tenuation, older stellar populations, or strong emission lines (or all of them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The parameter constraints then give more weight (probably density) to models with higher stellar masses and SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' When the MIRI data are included, then probe more of the stellar continuum at >7000 Å rest-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The model fits that include he MIRI data then show the dominant effect in the rest- frame optical are strong nebular emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This problem will persist for models that use NIRCam data as it also is limited to wavelengths less than 5 µm, but this can be tested with forthcoming datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The amount of stellar mass that could have formed in early bursts is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We estimated the amount of stel- lar mass formed by using a star-formation history that includes an early burst (at zf = 100) in addition to a smoothly evolving component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' A stellar population formed at this early time would fade and redden with time, and it would have the highest M/L at any sub- sequent time and therefore representations an upper limit on the amount of mass that could exist in these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI data improves the constraint on this stellar population by probing longer wavelengths (where the impact from this stellar population is more pronounced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Figure 9 shows that without the MIRI data, the amount of stellar mass in this population can be as much as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='9 dex higher at 4 < z < 6 (a factor of 7) and as much as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 dex higher at 6 < z < 9 (a factor of >10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Including the MIRI data, these drop to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 dex at 4 < z < 6 (a factor of 4) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 dex at 6 < z < 9 (a factor of 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, adding the MIRI reduces the amount of mass in early bursts by a factor of order 2 (compared to when no MIRI data are used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Our analysis of the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm therefore pro- vides evidence that there is less star-formation in dis- tant galaxies (because the SFRs and stellar masses are lowered) than found in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The MIRI data also reduce the limits on the amount of stellar mass possibly formed at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The combination of these results has implications for the evolution of the cosmic stellar-mass density, ρ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We showed (Fig- ure 11) that applying our results to the galaxy popula- tion shows that the amount of stellar mass density in galaxies at z = 9 is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1% of the present day, z = 0, value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This is an order of magnitude lower than implied by previous studies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='e,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' pre-JWST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We wish to thank everyone that brought JWST to fruition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We also thank our other colleagues in the CEERS collabora- tion for their hard work and valuable contributions on this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' CP thanks Marsha and Ralph Schilling for gen- erous support of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Portions of this research were conducted with the advanced computing resources pro- vided by Texas A&M High Performance Research Comput- ing (HPRC, http://hprc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This work benefited from support from the George P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This work acknowledges support from the NASA/ESA/CSA James Webb Space Telescope through the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, In- corporated, under NASA contract NAS5-03127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Support for program No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' JWST-ERS01345 was provided through a grant from the STScI under NASA contract NAS5-03127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Software: AstroPy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2013), BAGPIPES (Carnall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2018), matplotlib (Hunter 2007), NumPy (van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2011), photutils (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2020), PyPHER (Boucaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2016a), SE (Bertin & Arnouts 1996), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2020), Seaborn (Waskom 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' APPENDIX 21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Observed Properties of the Galaxy Sample ID R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (J2000) Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (J2000) F160 E160 F560 E560 F770 E770 zphot z16 z84 zspec P(z = 4) P(z = 5) P(z = 6) P(z = 7) P(z = 8) P(z = 9) (deg) (deg) nJy (nJy) (nJy) (nJy) (nJy) (nJy) 5090 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='04973 52.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='83 NOTE—References for spectroscopic redshifts: ‡ Stawinski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=', in prep;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' † WERLS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' ∗Zitrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 22 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Derived Stellar masses, SFRs, and Redshifts including the MIRI [5.' metadata={'source': 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and 16th and 84th-percentiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' galaxies with no redshift use the spectroscopic redshift in Table 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (5)–(7) stellar mass (50th percentile), and 16th and 84th–percentiles in the delayed-τ model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (8)–(10) SFR median (50th percentile), and 16th and 84th percentiles (all SFRs are averaged over the past 100 Myr) in the delayed-τ model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (11) maximum stellar mass allowed in the “burst” formed at zf = 100, these satisfy the BIC criteria (equation 2 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4) and correspond approximately to a 3σ upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' IMPACT OF CROWDED SOURCES IN IRAC DATA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' One source of potential bias relates to the photometry of our sources in the Spitzer/IRAC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' As illustrated in the images (Figure 3) some objects have bright neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In the case of the IRAC images, the light from the wings of these objects can blend with that for our sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' There is a large body of literature on the subject of performing crowded source photometry (Laidler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Labbé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We have used the catalog from (Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2022b) who used the HST/F160W image as a prior for the locations of sources and neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Source photometry is then carried out using TPHOT (Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2015), which estimates the source flux from objects simultaneously when measuring photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' While this method is theoretically robust, residuals from poorly modeling ePSFs and changes in galaxy morphology with wavelength (the “Morphological K-correction”, Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 2005) can lead to systematic uncertainties in source photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' To test if our results are impacted by blended sources in the IRAC bands, we did the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We first searched around each of the galaxies in our sample and identified galaxies that neighbors in the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm catalog within a radius of r ≤ 3′′ and a magnitude of [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] ≤ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 mag (near the flux limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We selected neighbors in the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm image as the central wavelength is closest to that of IRAC for our dataset (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The IRAC ePSF has a FWHM of ≈2′′, so any source within 3′′ in the MIRI data could therefore have IRAC light blended with our source of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 23 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Derived Stellar masses, SFRs, and Redshifts excluding the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' redshift logM∗/M⊙ (dex) logSFR/M⊙ yr−1 (dex) “Burst” Mass ID z50 z16 z84 logM50 logM16 logM84 log SFR50 log SFR16 log SFR84 logM∗/M⊙ (dex) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 5090 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='45 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='14 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='29 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='31 NOTE—(1) Galaxy ID, (2)–(4) redshift median (50%-tile), and 16th and 84th-percentiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' galaxies with no redshift use the spectro- scopic redshift in Table 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (5)–(7) stellar mass (50th percentile), and 16th and 84th–percentiles in the delayed-τ model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (8)–(10) SFR median (50th percentile), and 16th and 84th percentiles (all SFRs are averaged over the past 100 Myr) in the delayed- τ model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' (11) maximum stellar mass allowed in the “burst” formed at zf = 100, these satisfy the BIC criteria (equation 2 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='4) and correspond approximately to a 3σ upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' From our sample, we identified 11 galaxies that have a neighbor within 3′′ in the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' To estimate their effect on our study we removed these objects from the sample and recomputed the offsets in stellar mass and SFR for the results that include the MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 µm data versus the results that exclude the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These results are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Contrasting this figure with the one for the full sample (Figure 7) shows there is little change in the median offsets in stellar mass and SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The galaxy sample used in this Appendix is obviously smaller, but the median values do not change appreciably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' For the sample that excludes blended objects, the offsets in stellar mass are ∆logM∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='21 dex for 4 < z < 6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='53 for 6 < z < 9 (though the later now includes only three galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The offsets in SFR are ∆logSFR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='13 dex for 4 < z < 6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='42 dex for 6 < z < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' These are within ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 dex of the values reported for the full sample (in Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Similarly, we also investigated how the IRAC data for sources with close neighbors impact our finding that the stellar pop- ulations of the galaxies in our sample are generally “bluer” when the MIRI data are included in the analysis (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 and Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We repeated our analysis of the rest-frame colors in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 with our sample of galaxies that excludes those objects with a neighboring MIRI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6 µm source with [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 mag and within r ≤ 3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We find that in this case the relative rest-frame colors change only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The rest-frame far-UV–I color become bluer by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='015 mag (to have a total rest-frame (blue) color of ∆(m1600 −I) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='42 mag) when the objects with crowded IRAC photometry are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 24 4 6 8 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 log M (dex) excluding IRAC-crowded objects using the full sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4) 4 6 8 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='5 log SFR (dex) excluding IRAC-crowded objects using the full sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' 4) Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Testing the impact of sources with “crowded” IRAC photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' The plots in this figure are similar to those in Figure 7, and compare the stellar masses and SFRs derived from the SED modeling for galaxies including the MIRI F560W and F770W data and without the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In both panels the results show the difference between the mass (SFR) derived without MIRI data and the mass (SFR) deriving including the MIRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' In this figure, we have excluded objects that have a neighbor with r ≤ 3′′ and MIRI [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='6] ≤26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='7 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' This eliminates 11 objects, and allows us to test if crowding in the IRAC data (which has lower angular resolution) impacts object photometry in the IRAC bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' We do not observe any significant offset compared to the results in Figure 7: the median offsets in stellar mass and SFR change by ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore we conclude that blended IRAC photometry does not significantly impact the results here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Therefore, we conclude that our results are not dominated by photometry from sources crowded in the IRAC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content=' Obviously, future studies using JWST/NIRCam will be valuable to testing the IRAC photometry (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfPfba/content/2301.00027v1.pdf'} 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https://www.ankarakongresi.org +E-MAIL: bilgi@ankarakongresi.org +1657 +PREDICTING THE STUDENTS INVOLVEMENTS AND IT’S IMPACTS ON +LEARNING OUTCOMES THROUGH ONLINE EDUCATION DURING COVID-19 + +Muhammad Nadeem +Computer Science Department, University of the Punjab + + +Faisal Bukhari +Data Science Department, University of the Punjab + +Ali Hussain +Computer Science Department, University of the Punjab + +Abstract +Everybody knows very well about the COVID-19 pandemic, lockdown, and its impacts and +effects on every field of life, from childhood to senior citizens, from local to global. The +underlying research study focuses on students' involvement in online classes. This paper +assesses the effect of the COVID-19 pandemic on the students' participation and involvement +during online classes compared to the physical classes, cheating behavior, health effects, and +study styles of the students of diverse degrees and age groups. This research study contributes +to the real problems and challenges that students faced during online classes during the +COVID-19 pandemic. The percentages of the students' responses with different color schemes +shown in Fig. 1, Fig. 2, Fig.3(a), Fig.3(b) and Fig.4 are conveying powerful and meaningful +insight. These figures and the results given in Table I and Table II indicate that most students +are not fully involved during online classes due to technical issues, remote distance, etc. We +applied the Test here because we do not have exact population means. We used ttest_1samp +with default value 0 to compute the variables' statistics and p-value. These values are minimal +in favor of rejecting the null or H0 (hypothesis) and accepting the alternate or H1 +(hypothesis). It further means that students' involvement during online classes is severely +affected. +Keywords: COVID-19, e-Learning, Students Involvements, Cheating Concerns of Students, +Class Participation. + +I. INTRODUCTION +The primary motivation for selecting this topic is that the quality of education is directly +proportional to the involvement of the students during the lecture. Firstly, I found it as a +teacher that many students have left the online lecture physically, but logically they showed +their status as a present. This problem has multiple issues. The respected teacher cannot be +confident about the presence of the students physically during online lectures. Secondly, the +students are facing different issues during online lectures. The impact of these issues is that +they lose interest in learning during online lectures. This research work is a new study focused +mainly on the level of student involvement during online lectures. All the countries attacked +by the villainous COVID-19 virus that has upset each area of life as per economy, from +producers to consumers [1]. During the Covid19 pandemic, the Education sector was also +severely impacted. The forceful impact of this virus sent the students and teachers to study +and teach remotely from face to face system of education. Resultantly, Educational +institutions are searching for another way to teach and evaluate the students [2]. So to keep +every student and teacher safe, all the Educational Institutions closed because of the citywide, +districtwide, and countrywide lockdowns. In such lockup situations, the students and teachers +cannot interact face-to-face [3]. + + + + + + + + +ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII +WEB: https://www.ankarakongresi.org +E-MAIL: bilgi@ankarakongresi.org +1658 +To keep the chain of teaching in COVID-19 virus, the World Bank has been actively trying to +give financial assistance to the underdeveloped or more affected countries. The ultimate goal +of [4] is to provide basic education rights to every student during this viral disease. As far as +online learning is concerned, there is much use of technology. This technology-dependent +way of education becomes a barrier for learners who did not train to use technology [5]. +Similarly, in Pakistan, in 2021, all the educational institutions have closed as the previous +year due to the severity of COVID-19. Pakistani Ministry of Education and Higher Education +Commission (HEC) also provides online and distance learning ways to teach the students. [6]. +The HEC provided the design for online policy guidance notes and guidelines for the +Universities. However, It’s a reality that practical work is not being taught during online +education. This also demotivated the students, and it made an impact on their involvement in +online lectures [7]. In addition to the problems mentioned above and issues of students and +teachers, there are also the problems of admin staff [8]. +Therefore, the teachers are not satisfied with the student’s involvement in online classes +compared to physical classes. +In this connection, to find the answers, this study would work on the following research +objectives: +• To predict why the students are involved is not as much as physical class. +• To find why the students are not interested in attending the full online lecture. +• To discover the issue faced by the students during online lecture. +• To find the impact of taking lectures in class room with the lecture taking online on the +students' learning outcomes. +• To find the family members' realization about their children's online study. +The outcomes of the research would be necessary for the following concerning levels: +• Student +• Teacher +• Parents +• Educational Institution +• Education Ministries +The most crucial stakeholder in the learning process are teachers, and students are aware of +the issues and the factors involved as per the student involvement during an online class. The +parents would also notice the difference in attitude and aptitude to study in the classroom and +at home via online education. The Educational Institution may send reports to the Ministry of +Education and HEC based on the outcomes of the student's involvement during an online +class. In this way, the Ministries can inform the Government to look after the policies to plan +a different mature online education system or to open the educational institution as soon as +possible. + +II. LITERATURE REVIEW +The impacts of COVID-19 on health, society, and education are highlighted in [9]. The +researchers divided their research into four different groups: general demographics, +information about daily online routine, assessment of the learning of online experience and +level of satisfaction of the students, and evaluation of health due to change in lifestyle. +Cheating during the exam is one of the main problems. The research work done by [10] on +cheating shows that an individual's strengths vary according to the achievement settings. +Their findings also concluded that the cheating rate was higher in educational settings than in +work areas and in work sites than in sports venues. Study 1 further suggests that the strengths +of individuals' cheating intentions differ across achievement settings. + + + + + + + + +ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII +WEB: https://www.ankarakongresi.org +E-MAIL: bilgi@ankarakongresi.org +1659 +Intentions to cheat were higher in educational settings than in work settings and higher in +work settings than in sports settings. The outcomes of this research [11] concluded that the +online examination during COVID-19 increased the cheating ratio, which is unrelated to +achievement goals. The studies provided different guidelines to the teachers for setting the +questions and time duration for online exams. The researchers of [12] highlighted the levels of +students' stress, depressive symptoms, loneliness, effects of missing social life, and specific +worries for their undergraduate studies. They also showed extreme crises of the students on +health and research during lockdown due to COVID-19. The authors discussed that they got +212 responses out of 266 from students for the crises suffered. They also recommended +different plans for teachers and academic institution administrators to develop online events +so that they can prepare newcomers very well. The research efforts of [13] discover the +critical problems faced by the students in the present e-learning system. They have also found +the factors influencing online learning during COVID-19. +The authors also discussed the impacts of students' willingness to study alone in an e-learning +environment. In addition, they interviewed 30 students from six Universities and conducted +meetings with 31 e-learning system experts to find the main problems. They also suggested +applicable plans for policymakers, developers, designers, and researchers, enabling them to be +better acquainted with the critical aspects of the e-learning system during the COVID-19 +pandemic. The researchers of [14] have found too much dissatisfaction during the online +study on the COVID-19 situation. The outcomes of this research concluded that the students +of the dental study were dissatisfied with the online teaching during COVID-19. The results +of this research crying that online study is disturbing the student's level of involvement in the +study very severely. The efforts of the analyses highlighted different aspects of students +during the online study in the COVID-19 pandemic worldwide. They discussed and evaluated +severe issues such as technical and economic issues, psychological problems, and students' +fears about the future. It badly affects the study taste of the students and their pace in the +learning process. They also offered different plans and suggestions for the policymakers and +higher authorities to overcome the issues faced by the students and the teachers. The research +study by the authors of [15] observed and evaluated the impact of the perception of e-learning +crashes. They discovered its impact on psychological upset in the students during the COVID- +19 pandemic. They concluded that fear of academic loss had become the main reason for +mental upset during the issues of online study in corona disease. They also suggested +remedies for the policymakers and educational institutions to manage the student's stress +during the online study. The researchers analyzed different types of challenges faced by the +students in Pakistani Universities [16]. The main obstacles highlighted are economic, +technical, lack of skills, family support, etc. They also recommended that the Govt. take a +severe step to overcome the challenges faced by the students. The outcomes of this research +work [17] show that the students do not want to study online. The students expressed their +problems during the survey that they were not prepared and trained for such a learning shift. +They do not have a non-stop electricity facility and well-equipped information technology- +based infrastructure at their homes. + +III. PROBLEM STATEMENT +To find the effect of the COVID-19 pandemic on the involvement of the students during +online classes as compared to the physical classes, cheating behavior, health effects, and study +styles from the students of diverse degrees and age groups. +Hypothesis: +H0 = Student’s involvement during online classes is the same as in physical classes. +H1 = Student’s involvement during online classes is not the same as in physical classes. + + + + + + + +ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII +WEB: https://www.ankarakongresi.org +E-MAIL: bilgi@ankarakongresi.org +1660 +Methodology and Data Collection +The survey methodology used to accomplish this research. Survey is a method for the +collection of the information for the sample of individuals [18]. The findings of the survey +analyzed through statistical analysis. + +• OBJECTIVES OF THE SURVEY +To analyze the levels of the student’s involvement and its impacts on learning outcomes +during online lectures during COVID-19. + +•TARGET POPULATION +Graduate, Undergraduate and Intermediate students of the Universities and Colleges + +•DATA TO BE COLLECTED +A questionnaire developed based on the literature review. Then this questionnaire circulated +online as much as possible to find the maximum responses from the target population due to +the COVID-19 situation. + +•MEASUREMENT `INSTRUMENT' +The measurement instrument of the required survey is a questionnaire. The questions of this +questionnaire were +closed-ended with a Likert scale. The definition of the Likert scale is given below: +1. SA (Strongly Agreed) +2. A (Agreed) +3. U (Undecided), +4. D (Disagreed) +5. SD (Strongly Disagreed) +This questionnaire would be distributed through Google docs to make it available to the +targeted population and to get a maximum number of responses. + +IV. DESIGN OF RESEARCH STUDY +An online survey performed using Google online forms. However, the questionnaire of this +survey consists of the following subsections: +A. Respondents will be requested to answer their following usual demographics: +• Age +• Gender +• Area of residence +B. Getting information routine wise online learning during the shift from face to face study to +online study in colleges/Universities in Pakistan. These information consists of the following: +• Average time given for online study in hours per day +• Quality and the problems of the communication medium +• Actual involvement in virtual lecture same as face to face lecture in physical class +• Level of interruption by the family members during online study period +• Attention and focus level from joining to the end of online class. +• Effects of online learning on Cheating behavior and students involvement to +C. Evaluation of the experience of the student’s level of involvement in virtual class to find +the overall students involvement in online lecture. +D. Evaluation of health during change in learning style from physical class environment +provided by the College/University to the virtual class environment provided by your parents +at home and the effects of virtual class on your involvement of class. + + + + + + + + +ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII +WEB: https://www.ankarakongresi.org +E-MAIL: bilgi@ankarakongresi.org +1661 +The pictorial survey responses are given below: + + +Fig. 1. Getting General Info + + +Fig. 2. Getting General Info + + +Fig. 3(a). Getting Specific Info + + + +Section A: Demographics Info +Chart +100% +80% +60% +40% +20% +0% +6 +1 +4 +4 +5 +8 +5 +9 +3 +2 +3 +3 +3 +4 +4 +4 +5 +5 +5 +Gender +Age +Degree_ Level +Area of ResidenceSection B: Getting General Info Chart +100% +80% +60% +40% +20% +0% +3 +5 +9 +2 +8 +5 +8 +385 +417 +1.49 +3 +4 +8 +3 +6 +1 +1 +T +2 +3 +3 +5 +5 +5 +6 +Time_Spent_SociaMediaLaptopComputerAvail +ISmartPhonesAvail +Class Participation Level +CheatingConcern +StudyLevelifdontExam +I Lack of IT Skills +BetterOnlineLearnSectionC:GettingSpecificInfo(11-17 +Questions) +100% +80% +60% +40% +20% +0% +8 +6 +OnlineAndOffineEqual +Technicallssuelmpact +Economiclssuelmpact +TeacherVoicelssue +Lessrsinteraction +AcademicLossfearinClassParticipation +LackDeficiencyforNonITSt + + + + + +ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII +WEB: https://www.ankarakongresi.org +E-MAIL: bilgi@ankarakongresi.org +1662 +We have created questionnaire. Its soft copy is available at the following link: +https://docs.google.com/forms/d/1zqnXC9EXRXjmNL7VX2FP4hh6OL0NVu- +C_w8QZOFxRsc/edit ) +We have collected 623 responses from different level of degree students. + + +Fig. 3(b). Getting Specific Info + + +Fig. 4. Getting Health Issues info + +V. EXPEIRMENTAL RESULTS +The means and standard deviation of all the variables as per questionnaire are given below: + + + +SectionC:GettingSpecificInfo(1-10Questions) +100% +80% +60% +40% +20% +0% +8 +5 +2 +5 +BetterTimeUtilization +CheatingBehavior +Umwilingness_of_ResponsibilityStudentsHesitancylmpact +TechDificultylmpact +HaveNetAccess +HaveElectricSuppy +InteractionWihTeacher +ClassParticipationChance +AttensionAndFocusDisturbSection D: Getting Health Issues Info +Chart +100% +80% +60% +40% +20% +0% +4 +8 +2 +3 +80 +365 +393 +2 +4 +3 +89 +9 +3 +3 +4 +5 +5 +5 +5 +6 +FitnessOfAttensionLonelinessEffectAnxietyLevelPsycholmpacts + + + + + +ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII +WEB: https://www.ankarakongresi.org +E-MAIL: bilgi@ankarakongresi.org +1663 +TABLE I. +S.# +Variable +Value +Mean of all the variables +SECTION A: DEMOGRAPHICS INFO +1 +Gender +1.400000 +2 +Age +2.028571 +3 +Degree_Level +2.257143 +4 +Area_of_Residence +1.628571 +SECTION B: GETTING GENERAL INFO +1 +Time_Spent_ SociaMedia +2.457143 +2 +LaptopComputerAvail +1.771429 +3 +SmartPhonesAvail +1.571429 +4 +Class_Participation_Level +2.885714 +5 +CheatingConcern +1.942857 +6 +StudyLevelIfdontExam +3.028571 +7 +Lack_of_IT_Skills +2.485714 +8 +BetterOnlineLearn +3.600000 +SECTION C: GETTING SPECIFIC INFO +1 +BetterTimeUtilization +3.342857 +2 +CheatingBehavior +2.285714 +3 +Unwilingness_of_Responsibility +2.114286 +4 +StudentsHesitancyImpact +2.371429 +5 +TechDifficultyImpact +2.200000 +6 +HaveNetAccess +2.514286 +7 +HaveElectricSupply +3.000000 +8 +InteractionWihTeacher +3.085714 +9 +ClassParticipationChance +3.057143 +10 +AttensionAndFocusDisturb +2.342857 +11 +OnlineAndOfflineEqual +3.800000 +12 +TechnicalIssueImpact +1.857143 +13 +EconomicIssueImpact +2.000000 +14 +TeacherVoiceIssue +1.971429 +15 +LessTSInteraction +1.857143 +16 +AcademicLossFearinClassParticipation +2.028571 +16 +AcademicLossFearinClassParticipation +0.970588 +17 +LackDeficiencyforNonITSt +0.747240 +Standard Deviation of all the variables +SECTION A: DEMOGRAPHICS INFO +1 +Gender +0.489898 +2 +Age +0.376883 +3 +Degree_Level +0.552545 +4 +Area_of_Residence +0.483187 + + + + + + + + + +ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII +WEB: https://www.ankarakongresi.org +E-MAIL: bilgi@ankarakongresi.org +1664 +SECTION B: GETTING GENERAL INFO +1 +Time_Spent_ SociaMedia +1.078169 +2 +LaptopComputerAvail +0.897161 +3 +SmartPhonesAvail +0.766652 +4 +Class_Participation_Level +1.259738 +5 +CheatingConcern +1.093954 +6 +StudyLevelIfdontExam +1.502107 +7 +Lack_of_IT_Skills +1.105090 +8 +BetterOnlineLearn +1.515633 +SECTION C: GETTING SPECIFIC INFO +1 +BetterTimeUtilization +1.413059 +2 +CheatingBehavior +1.110249 +3 +Unwilingness_of_Responsibility +1.259738 +4 +StudentsHesitancyImpact +1.332789 +5 +TechDifficultyImpact +0.979796 +6 +HaveNetAccess +1.273273 +7 +HaveElectricSupply +1.309307 +8 +InteractionWihTeacher +1.295518 +9 +ClassParticipationChance +1.286032 +10 +AttensionAndFocusDisturb +1.392692 +11 +OnlineAndOfflineEqual +1.214202 +12 +TechnicalIssueImpact +0.797957 +13 +EconomicIssueImpact +0.956183 +3 +TeacherVoiceIssue +1.027777 +4 +LessTSInteraction +1.045886 + +TABLE II: +TTest Outcomes +Ttest_1sampResult(statistic=array([16.663333 , 31.38507589, +23.81939622, 19.65311057, 13.28871279, +11.51311097, 11.95187108, 13.35711613, 10.35574591, 11.7564528 , +13.11574349, 13.84994208, 13.79421828, 12.0044142 , 9.78640192, +10.375 , 13.09261879, 11.51416659, 13.36038922, 13.88838218, +13.86128572, 9.80912102, 18.24871239, 13.57080199, 12.19631092, +11.18462458, 10.35381536, 12.18694645, 13.15416906, 11.9272551 , +11.34226868, 10.64348064, 11.2720409 ]), pvalue=array([6.29551067e- +18, 1.08636299e-26, 8.60469002e-23, 3.85366635e-20, +5.07753770e-15, 2.80770597e-13, 1.00580609e-13, 4.38220401e-15, +4.72979440e-12, 1.58421613e-13, 7.38588068e-15, 1.53985258e-15, +1.73086219e-15, 8.90861219e-14, 2.02190243e-11, 4.50637008e-12, +7.76732461e-15, 2.80069994e-13, 4.35148773e-15, 1.42079500e-15, +1.50369222e-15, 1.90647656e-11, 3.87615125e-19, 2.77545554e-15, +5.73530395e-14, 6.15085107e-13, 4.75281129e-12, 5.85928814e-14, +6.79392859e-15, 1.06477004e-13, 4.21455437e-13, 2.30654437e-12, +4.98568395e-13])) + + + + + + + + + + +ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII +WEB: https://www.ankarakongresi.org +E-MAIL: bilgi@ankarakongresi.org +1665 +VIII. CONCLUSTION +To evaluate and find the correctness and applicability of the hypothesis as per the problem +statement, we used an online survey approach using Google docs. According to the +percentages of survey responses given in Fig.1 Fig.2, Fig.3(a), Fig.3(b) and Fig. 4, +availability of Laptop/Computer at student homes was 85.3% and smart phones was 87.5%. +Time spent on social media during the online lecture was 71%. The level of Class +participation was 49.6%. The concern of students cheating during the online exam was +70.8%—level of cheating behavior to ignore interest in online due to online exams +encouraged by 60.8% of students. The student's unwillingness was found at 73.2%. Impacts +of technical issues during online classes were 84.4% . The pace of the teacher's voice due to +the Net problem was discovered at 79.5% and Impacts of less interaction of teacher-student +found to be 78%. As per Fig. 4, psychological impacts on learning participation during online +classes were discovered at 73.5% , the stress of loneliness affects students' level of +involvement was 68.5% and the anxiety levels disturb students' level of motivation by 77.9%. +As per the above experiments, the means and standard deviations are given in Table I and +Table II above. Most of the high values of means shows that much percentage of the students +are not fully involved during online lecture. Similarly, most of the values of standard +deviations are far from zero. It shows that data points are far from the mean. We applied the +test here because we don't have actual population means. We used ttest_1samp (Dataset +[:35],0) with a default value of 0 to compute the variables' statistics and p-value. The results +of this test are provided in Table II. These values are minimal, which is in favor of rejecting +the null or H0 hypothesis and accepting the alternate or H1 hypothesis. It further means that +students' involvement during online classes is severely affected. + +ACKNOWLEDGMENT +The authors are very grateful to the management of server room of Faculty of Computing and +Information Technology (FCIT), University of the Punjab to forward our questionnaire to the +students for the responses. We are also thankful to the Students of Undergraduates and +Gradates students of FCIT for the warm participation and sincere responses during the survey +of this research study. + +REFERENCES +[1]. +Fernando, R., “The COVID-19 Pandemic: A call for a reality check.”, +published in Galle Medical Journal, 25 (1). 2020. +[2]. +Myers, A., “After COVID-19: Recalibrating the American educational +system”, Retrieved from https://hub.jhu.edu/2020/04/07/bob-balfanz-education-reform-covid- +19/, accessed 2021. +[3]. +Tam, G., & El-Azar, D. 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Parker, “ The impact of COVID +19 triggered changes to instruction and assessment on university students’ self reported +motivation, engagement and perceptions”, published in Social Psychology of Education +(2021) 24:299–318, Springer. +[12]. Timon ElmerID, Kieran Mepham, Christoph Stadtfeld, “Students under +lockdown: Comparisons of students’ social networks and mental health before and during the +COVID-19 crisis in Switzerland”, published in PLOS ONE, pp: 1-22, 2020. +[13]. Mohammed Amin Almaiah, Ahmad Al-Khasawneh and Ahmad Althunibat, “ +Exploring the critical challenges and factors influencing the E-learning system usage during +COVID-19 pandemic”, published in Journal of Education and Information Technologies, +Springer. +[14]. Huma Sarwar, Hira Akhtar, Meshal Muhammad Naeem, Javeria Ali Khan, +Khadija Waraich, Sumaiya Shabbir, Arshad Hasan and Zohaib Khurshid, “ Self-Reported +Effectiveness of e-Learning Classes during COVID-19 Pandemic: A Nation-Wide Survey of +Pakistani Undergraduate Dentistry Students”, published in European Journal of Dentistry, +2020;14(suppl S1):S34–S43. +[15]. Aqsa Arshad, Madiha Afzal, * Dr. Muhammad Sabboor Hussain, “ Sudden +Switch to Post-COVID-19 Online Classes and Cognitive Transformation of ESL Learners: +Critical Analysis of Discourse of Fear”, published in Research Journal of Social Sciences & +Economics Review, Vol. 1, Issue 3, 2020, PP: 188-199. +[16]. Najmul Hasan, Yukun Bao, “ Impact of “e-Learning crack-up” perception on +psychological distress among college students during COVID-19 pandemic: A mediating role +of “fear of academic year loss”, published in Children and Youth Services Review 118 (2020) +PP: 1-10. +[17]. Muhammad Anwar, Anwar Khan, Khalid Sultan, “The barriers and challenges +faced by students in online education during covid-19 pandemic in Pakistan”, published in +Gomal University Journal of Research, Volume 36, Issue 1, JUNE, 2020. PP. 52-62. +[18]. Brochure, what is a survey? Bill Kalsbeek, 1995 publications officer, ASA +section on survey research methods, 1995. + diff --git a/2tAyT4oBgHgl3EQfPvai/content/tmp_files/load_file.txt b/2tAyT4oBgHgl3EQfPvai/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75689b04066d921a3e392bfecbba0a408f9ba067 --- /dev/null +++ b/2tAyT4oBgHgl3EQfPvai/content/tmp_files/load_file.txt @@ -0,0 +1,407 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf,len=406 +page_content='ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org 1657 PREDICTING THE STUDENTS INVOLVEMENTS AND IT’S IMPACTS ON LEARNING OUTCOMES THROUGH ONLINE EDUCATION DURING COVID-19 Muhammad Nadeem Computer Science Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' University of the Punjab Faisal Bukhari Data Science Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' University of the Punjab Ali Hussain Computer Science Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' University of the Punjab Abstract Everybody knows very well about the COVID-19 pandemic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' lockdown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' and its impacts and effects on every field of life,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' from childhood to senior citizens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' from local to global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" The underlying research study focuses on students' involvement in online classes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" This paper assesses the effect of the COVID-19 pandemic on the students' participation and involvement during online classes compared to the physical classes, cheating behavior, health effects, and study styles of the students of diverse degrees and age groups." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' This research study contributes to the real problems and challenges that students faced during online classes during the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" The percentages of the students' responses with different color schemes shown in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='3(a), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='4 are conveying powerful and meaningful insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' These figures and the results given in Table I and Table II indicate that most students are not fully involved during online classes due to technical issues, remote distance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' We applied the Test here because we do not have exact population means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" We used ttest_1samp with default value 0 to compute the variables' statistics and p-value." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' These values are minimal in favor of rejecting the null or H0 (hypothesis) and accepting the alternate or H1 (hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" It further means that students' involvement during online classes is severely affected." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Keywords: COVID-19, e-Learning, Students Involvements, Cheating Concerns of Students, Class Participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' INTRODUCTION The primary motivation for selecting this topic is that the quality of education is directly proportional to the involvement of the students during the lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Firstly, I found it as a teacher that many students have left the online lecture physically, but logically they showed their status as a present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' This problem has multiple issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The respected teacher cannot be confident about the presence of the students physically during online lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Secondly, the students are facing different issues during online lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The impact of these issues is that they lose interest in learning during online lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' This research work is a new study focused mainly on the level of student involvement during online lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' All the countries attacked by the villainous COVID-19 virus that has upset each area of life as per economy, from producers to consumers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' During the Covid19 pandemic, the Education sector was also severely impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The forceful impact of this virus sent the students and teachers to study and teach remotely from face to face system of education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Resultantly, Educational institutions are searching for another way to teach and evaluate the students [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' So to keep every student and teacher safe, all the Educational Institutions closed because of the citywide, districtwide, and countrywide lockdowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' In such lockup situations, the students and teachers cannot interact face-to-face [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org 1658 To keep the chain of teaching in COVID-19 virus, the World Bank has been actively trying to give financial assistance to the underdeveloped or more affected countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The ultimate goal of [4] is to provide basic education rights to every student during this viral disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' As far as online learning is concerned, there is much use of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' This technology-dependent way of education becomes a barrier for learners who did not train to use technology [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Similarly, in Pakistan, in 2021, all the educational institutions have closed as the previous year due to the severity of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Pakistani Ministry of Education and Higher Education Commission (HEC) also provides online and distance learning ways to teach the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The HEC provided the design for online policy guidance notes and guidelines for the Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' However, It’s a reality that practical work is not being taught during online education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' This also demotivated the students, and it made an impact on their involvement in online lectures [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' In addition to the problems mentioned above and issues of students and teachers, there are also the problems of admin staff [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Therefore, the teachers are not satisfied with the student’s involvement in online classes compared to physical classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' In this connection, to find the answers, this study would work on the following research objectives: • To predict why the students are involved is not as much as physical class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' • To find why the students are not interested in attending the full online lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' • To discover the issue faced by the students during online lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" • To find the impact of taking lectures in class room with the lecture taking online on the students' learning outcomes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" • To find the family members' realization about their children's online study." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The outcomes of the research would be necessary for the following concerning levels: • Student • Teacher • Parents • Educational Institution • Education Ministries The most crucial stakeholder in the learning process are teachers, and students are aware of the issues and the factors involved as per the student involvement during an online class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The parents would also notice the difference in attitude and aptitude to study in the classroom and at home via online education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" The Educational Institution may send reports to the Ministry of Education and HEC based on the outcomes of the student's involvement during an online class." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' In this way, the Ministries can inform the Government to look after the policies to plan a different mature online education system or to open the educational institution as soon as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' LITERATURE REVIEW The impacts of COVID-19 on health, society, and education are highlighted in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The researchers divided their research into four different groups: general demographics, information about daily online routine, assessment of the learning of online experience and level of satisfaction of the students, and evaluation of health due to change in lifestyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Cheating during the exam is one of the main problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" The research work done by [10] on cheating shows that an individual's strengths vary according to the achievement settings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Their findings also concluded that the cheating rate was higher in educational settings than in work areas and in work sites than in sports venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" Study 1 further suggests that the strengths of individuals' cheating intentions differ across achievement settings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org 1659 Intentions to cheat were higher in educational settings than in work settings and higher in work settings than in sports settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The outcomes of this research [11] concluded that the online examination during COVID-19 increased the cheating ratio, which is unrelated to achievement goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The studies provided different guidelines to the teachers for setting the questions and time duration for online exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" The researchers of [12] highlighted the levels of students' stress, depressive symptoms, loneliness, effects of missing social life, and specific worries for their undergraduate studies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' They also showed extreme crises of the students on health and research during lockdown due to COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The authors discussed that they got 212 responses out of 266 from students for the crises suffered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' They also recommended different plans for teachers and academic institution administrators to develop online events so that they can prepare newcomers very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The research efforts of [13] discover the critical problems faced by the students in the present e-learning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' They have also found the factors influencing online learning during COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" The authors also discussed the impacts of students' willingness to study alone in an e-learning environment." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' In addition, they interviewed 30 students from six Universities and conducted meetings with 31 e-learning system experts to find the main problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' They also suggested applicable plans for policymakers, developers, designers, and researchers, enabling them to be better acquainted with the critical aspects of the e-learning system during the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The researchers of [14] have found too much dissatisfaction during the online study on the COVID-19 situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The outcomes of this research concluded that the students of the dental study were dissatisfied with the online teaching during COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" The results of this research crying that online study is disturbing the student's level of involvement in the study very severely." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The efforts of the analyses highlighted different aspects of students during the online study in the COVID-19 pandemic worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" They discussed and evaluated severe issues such as technical and economic issues, psychological problems, and students' fears about the future." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' It badly affects the study taste of the students and their pace in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' They also offered different plans and suggestions for the policymakers and higher authorities to overcome the issues faced by the students and the teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The research study by the authors of [15] observed and evaluated the impact of the perception of e-learning crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' They discovered its impact on psychological upset in the students during the COVID- 19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' They concluded that fear of academic loss had become the main reason for mental upset during the issues of online study in corona disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" They also suggested remedies for the policymakers and educational institutions to manage the student's stress during the online study." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The researchers analyzed different types of challenges faced by the students in Pakistani Universities [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The main obstacles highlighted are economic, technical, lack of skills, family support, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' They also recommended that the Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' take a severe step to overcome the challenges faced by the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The outcomes of this research work [17] show that the students do not want to study online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The students expressed their problems during the survey that they were not prepared and trained for such a learning shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' They do not have a non-stop electricity facility and well-equipped information technology- based infrastructure at their homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' PROBLEM STATEMENT To find the effect of the COVID-19 pandemic on the involvement of the students during online classes as compared to the physical classes, cheating behavior, health effects, and study styles from the students of diverse degrees and age groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Hypothesis: H0 = Student’s involvement during online classes is the same as in physical classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' H1 = Student’s involvement during online classes is not the same as in physical classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org 1660 Methodology and Data Collection The survey methodology used to accomplish this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Survey is a method for the collection of the information for the sample of individuals [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The findings of the survey analyzed through statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' OBJECTIVES OF THE SURVEY To analyze the levels of the student’s involvement and its impacts on learning outcomes during online lectures during COVID 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' TARGET POPULATION Graduate, Undergraduate and Intermediate students of the Universities and Colleges DATA TO BE COLLECTED A questionnaire developed based on the literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Then this questionnaire circulated online as much as possible to find the maximum responses from the target population due to the COVID 19 situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" MEASUREMENT `INSTRUMENT' The measurement instrument of the required survey is a questionnaire." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The questions of this questionnaire were closed ended with a Likert scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The definition of the Likert scale is given below: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' SA (Strongly Agreed) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' A (Agreed) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' U (Undecided), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' D (Disagreed) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' SD (Strongly Disagreed) This questionnaire would be distributed through Google docs to make it available to the targeted population and to get a maximum number of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' DESIGN OF RESEARCH STUDY An online survey performed using Google online forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' However, the questionnaire of this survey consists of the following subsections: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Respondents will be requested to answer their following usual demographics: • Age • Gender • Area of residence B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Getting information routine wise online learning during the shift from face to face study to online study in colleges/Universities in Pakistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' These information consists of the following: • Average time given for online study in hours per day • Quality and the problems of the communication medium • Actual involvement in virtual lecture same as face to face lecture in physical class • Level of interruption by the family members during online study period • Attention and focus level from joining to the end of online class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' • Effects of online learning on Cheating behavior and students involvement to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Evaluation of the experience of the student’s level of involvement in virtual class to find the overall students involvement in online lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Evaluation of health during change in learning style from physical class environment provided by the College/University to the virtual class environment provided by your parents at home and the effects of virtual class on your involvement of class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org 1661 The pictorial survey responses are given below: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Getting General Info Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Getting General Info Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Getting Specific Info Section A: Demographics Info Chart 100% 80% 60% 40% 20% 0% 6 1 4 4 5 8 5 9 3 2 3 3 3 4 4 4 5 5 5 Gender Age Degree_ Level Area of ResidenceSection B: Getting General Info Chart 100% 80% 60% 40% 20% 0% 3 5 9 2 8 5 8 385 417 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='49 3 4 8 3 6 1 1 T 2 3 3 5 5 5 6 Time_Spent_SociaMediaLaptopComputerAvail ISmartPhonesAvail Class Participation Level CheatingConcern StudyLevelifdontExam I Lack of IT Skills BetterOnlineLearnSectionC:GettingSpecificInfo(11-17 Questions) 100% 80% 60% 40% 20% 0% 8 6 OnlineAndOffineEqual Technicallssuelmpact Economiclssuelmpact TeacherVoicelssue Lessrsinteraction AcademicLossfearinClassParticipation LackDeficiencyforNonITSt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org 1662 We have created questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Its soft copy is available at the following link: https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='com/forms/d/1zqnXC9EXRXjmNL7VX2FP4hh6OL0NVu- C_w8QZOFxRsc/edit ) We have collected 623 responses from different level of degree students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Getting Specific Info Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Getting Health Issues info V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' EXPEIRMENTAL RESULTS The means and standard deviation of all the variables as per questionnaire are given below: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='SectionC:GettingSpecificInfo(1-10Questions) 100% 80% 60% 40% 20% 0% 8 5 2 5 BetterTimeUtilization CheatingBehavior Umwilingness_of_ResponsibilityStudentsHesitancylmpact TechDificultylmpact HaveNetAccess HaveElectricSuppy InteractionWihTeacher ClassParticipationChance AttensionAndFocusDisturbSection D: Getting Health Issues Info Chart 100% 80% 60% 40% 20% 0% 4 8 2 3 80 365 393 2 4 3 89 9 3 3 4 5 5 5 5 6 FitnessOfAttensionLonelinessEffectAnxietyLevelPsycholmpacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org 1663 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='# Variable Value Mean of all the variables SECTION A: DEMOGRAPHICS INFO 1 Gender 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='400000 2 Age 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='028571 3 Degree_Level 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='257143 4 Area_of_Residence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='628571 SECTION B: GETTING GENERAL INFO 1 Time_Spent_ SociaMedia 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='457143 2 LaptopComputerAvail 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='771429 3 SmartPhonesAvail 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='571429 4 Class_Participation_Level 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='885714 5 CheatingConcern 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='942857 6 StudyLevelIfdontExam 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='028571 7 Lack_of_IT_Skills 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='485714 8 BetterOnlineLearn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='600000 SECTION C: GETTING SPECIFIC INFO 1 BetterTimeUtilization 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='342857 2 CheatingBehavior 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='285714 3 Unwilingness_of_Responsibility 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='114286 4 StudentsHesitancyImpact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='371429 5 TechDifficultyImpact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='200000 6 HaveNetAccess 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='514286 7 HaveElectricSupply 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='000000 8 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='21455437e-13, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='30654437e-12, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='98568395e-13])) ANKARA INTERNATIONAL CONGRESS ON SCIENTIFIC RESEARCH-VII WEB: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org E-MAIL: bilgi@ankarakongresi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='org 1665 VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' CONCLUSTION To evaluate and find the correctness and applicability of the hypothesis as per the problem statement, we used an online survey approach using Google docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' According to the percentages of survey responses given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='3(a), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 4, availability of Laptop/Computer at student homes was 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='3% and smart phones was 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Time spent on social media during the online lecture was 71%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The level of Class participation was 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The concern of students cheating during the online exam was 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='8%—level of cheating behavior to ignore interest in online due to online exams encouraged by 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='8% of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" The student's unwillingness was found at 73." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Impacts of technical issues during online classes were 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='4% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" The pace of the teacher's voice due to the Net problem was discovered at 79." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='5% and Impacts of less interaction of teacher-student found to be 78%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' As per Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 4, psychological impacts on learning participation during online classes were discovered at 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content="5% , the stress of loneliness affects students' level of involvement was 68." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content="5% and the anxiety levels disturb students' level of motivation by 77." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' As per the above experiments, the means and standard deviations are given in Table I and Table II above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Most of the high values of means shows that much percentage of the students are not fully involved during online lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Similarly, most of the values of standard deviations are far from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' It shows that data points are far from the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" We applied the test here because we don't have actual population means." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" We used ttest_1samp (Dataset [:35],0) with a default value of 0 to compute the variables' statistics and p-value." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' The results of this test are provided in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' These values are minimal, which is in favor of rejecting the null or H0 hypothesis and accepting the alternate or H1 hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=" It further means that students' involvement during online classes is severely affected." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors are very grateful to the management of server room of Faculty of Computing and Information Technology (FCIT), University of the Punjab to forward our questionnaire to the students for the responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' We are also thankful to the Students of Undergraduates and Gradates students of FCIT for the warm participation and sincere responses during the survey of this research study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' REFERENCES [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Fernando, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=', “The COVID-19 Pandemic: A call for a reality check.”, published in Galle Medical Journal, 25 (1).' 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Classes and Cognitive Transformation of ESL Learners: Critical Analysis of Discourse of Fear”, published in Research Journal of Social Sciences & Economics Review, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 1, Issue 3, 2020, PP: 188-199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Najmul Hasan, Yukun Bao, “ Impact of “e-Learning crack-up” perception on psychological distress among college students during COVID-19 pandemic: A mediating role of “fear of academic year loss”, published in Children and Youth Services Review 118 (2020) PP: 1-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Muhammad Anwar, Anwar Khan, Khalid Sultan, “The barriers and challenges faced by students in online education during covid-19 pandemic in Pakistan”, published in Gomal University Journal of Research, Volume 36, Issue 1, JUNE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' PP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' 52-62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Brochure, what is a survey?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} +page_content=' Bill Kalsbeek, 1995 publications officer, ASA section on survey research methods, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfPvai/content/2301.00031v1.pdf'} diff --git a/3dE3T4oBgHgl3EQfPwmd/content/tmp_files/2301.04406v1.pdf.txt b/3dE3T4oBgHgl3EQfPwmd/content/tmp_files/2301.04406v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..573079b743ced9c7bae94183226c52c491930dc5 --- /dev/null +++ b/3dE3T4oBgHgl3EQfPwmd/content/tmp_files/2301.04406v1.pdf.txt @@ -0,0 +1,794 @@ +arXiv:2301.04406v1 [cs.DS] 11 Jan 2023 +A Note on Property Testing of the Binary Rank +Nader H. Bshouty +Dept. of Computer Science +Technion, Haifa, Israel. +January 12, 2023 +Abstract +Let M be a n × m (0, 1)-matrix. We define the s-binary rank, brs(M), of M to be the +minimal integer d such that there are d monochromatic rectangles that cover all the 1-entries in +the matrix, and each 1-entry is covered by at most s rectangles. When s = 1, this is the binary +rank, br(M), known from the literature. +Let R(M) and C(M) be the set of rows and columns of M, respectively. We use the result of +Sgall [8] to prove that if M has s-binary rank at most d, then |R(M)| · |C(M)| ≤ +� d +≤s +� +2d where +� d +≤s +� += �s +i=0 +�d +i +� +. This bound is tight; that is, there exists a matrix M ′ of s-binary rank d such +that |R(M ′)| · |C(M ′)| = +� d +≤s +� +2d. +Using this result, we give a new one-sided adaptive and non-adaptive testers for (0, 1)- +matrices of s-binary rank at most d (and exactly d) that makes ˜O +�� d +≤s +� +2d/ǫ +� +and ˜O +�� d +≤s +� +2d/ǫ2� +queries, respectively. +For a fixed s, this improves the query complexity of the tester of Parnas et al. in [7] by a +factor of ˜Θ(2d). +1 +Introduction +Let M be a n × m (0, 1)-matrix. We define the s-binary rank, brs(M), of M to be the minimal +integer d such that there are d sets (rectangles) Ik×Jk where Ik ⊆ [n] := {1, . . . , n}, Jk ⊂ [m], k ∈ [d] +such that1 M[i, j] = 1 for all (i, j) ∈ Ik × Jk, k ∈ [d] (monochromatic rectangles), and for every +(i, j) ∈ [n] × [m] where M[i, j] = 1, there are at least one and at most s integers t ∈ [d] such +that (i, j) ∈ It × Jt (each 1-entry in M is covered by at least one and at most s monochromatic +rectangles). When s = 1, br1(M), is the binary rank, br(M), and when s = ∞, br∞(M) is the +Boolean rank. Both are known from the literature. See, for example, [4]. +The binary rank can also be defined as follows. The binary rank of a n × m (0, 1)-matrix M +is equal to the minimal d, where there are n × d (0, 1)-matrix N and d × m (0, 1)-matrix L such +that M = NL. It is also equal to the minimal number of bipartite cliques needed to partition all +the edges of a bipartite graph whose adjacent matrix is M. The s-binary rank of M is the minimal +number of bipartite cliques needed to cover all edges of a bipartite graph whose adjacent matrix +is M, where each edge is covered by at most s bipartite cliques. In [2], it was shown that it is +NP-hard to approximating the binary rank to within a factor of n1−δ for any given δ. +1For M, the (i, j) entry of the matrix is denoted by M[i, j]. +1 + +A property-testing algorithm (tester) of the s-binary rank [7] is given as input 0 < ǫ < 1, integers +d, n, m, and query access to the entries of a n×m (0, 1)-matrix M. If M has s-binary rank at most +d (resp. equal d), then the tester accepts with probability at least 2/3. If M is ǫ-far from having +s-binary rank at most d (resp. equal d), i.e., more than ǫ-fraction of the entries of M should be +modified to get a matrix with s-binary rank at most d (resp. equal to d), then the tester rejects +with probability at least 2/3. If the tester accepts matrices having s-binary rank at most d (resp. +equal to d) with probability 1, then we call it a one-sided error tester. In adaptive testing, the +queries can depend on the answers to the previous queries, whereas in non-adaptive testing, all the +queries are fixed in advance by the tester. The goal is to construct a tester that makes a minimal +number of queries. +The testability of s-binary rank at most d of (0, 1)-matrices was studied in [6, 7]. In [6], Nakar +and Ron gave a non-adaptive one-sided error tester for s = 1, that makes ˜O(24d/ǫ4). In [7], Parnas +et al. gave a non-adaptive and adaptive one-sided error tester for s = 1 that makes O(22d/ǫ2) and +O(22d/ǫ) queries, respectively. The results in [7] also hold for s-binary rank at most d. In this +paper, for s-binary at most d and equal to d, we prove +Theorem 1. There exists an adaptive one-sided error tester for s-binary rank of n × m (0, 1)- +matrices that makes ˜O +�� d +≤s +� +2d/ǫ +� +queries. +Theorem 2. There exists a non-adaptive one-sided error tester for s-binary rank of n × m (0, 1)- +matrices that makes ˜O +�� d +≤s +� +2d/ǫ2� +queries. +For fixed s, this improves the query complexity of Parnas et al. in [7] by a factor of ˜O(2d). +1.1 +Our Approach +The tester of Parnas et al. [7] uses the fact that if M′ is a k × k sub-matrix of M and M′ is of +s-binary rank at most d, then +1. M′ has at most 2d distinct rows and at most 2d distinct columns. +2. If M is ǫ-far from having s-binary rank at most d, then extending M′ by one more uniformly +at random row and column of M, gives a (k + 1) × (k + 1) sub-matrix M′′ of M that, with +probability at least Ω(ǫ), satisfies: the number of distinct rows in M′′ is greater by one than +the number of distinct rows in M′, or, the number of distinct columns in M′′ is greater by +one than the number of distinct columns in M′. +So, their adaptive tester runs O(2d/ǫ) iterations. At each iteration, it extends M′ by uniformly at +random one row and one column. Let M′′ be the resulting sub-matrix. If the s-binary rank of M′′ +is greater than d, the tester rejects. If the number of distinct rows or columns in M′′ is greater +than the number in M′, then it continues to the next iteration with M′ ← M′′. Otherwise, it +continues to the next iteration with M′. If, after O(2d/ǫ) iterations, M′ has s-binary rank d, the +tester accepts. +If the s-binary rank of M is d, then every sub-matrix has a s-binary rank d, and the tester +accepts. If M is ǫ-far from having s-binary rank at most d, then: since, at each iteration, with +probability at least Ω(ǫ), the number of distinct rows or columns of M′ is increased by one, and +since matrices of s-binary rank d has at most 2d distinct rows and at most 2d distinct columns, +with high probability, we get M′ with s-binary rank greater than d and the tester rejects. The +2 + +query complexity of the tester is O(22d/ǫ), which is the number of entries of the matrix M′, O(22d), +times the number of trials O(1/ǫ) for extending M′ by one row and one column. +We now give our approach. Call a sub-matrix M′ of M perfect if it has distinct rows and distinct +columns. Our adaptive tester uses the fact that if M′ is a perfect k×k′ sub-matrix of M of s-binary +rank d, then +1. kk′ ≤ +� d +≤s +� +2d. +2. If M is ǫ-far from having s-binary rank at most d, then at least one of the following occurs +(a) With probability at least Ω(ǫ), extending M′ by one uniformly at random column of M, +gives a perfect k × (k′ + 1) sub-matrix M′′ of M. +(b) With probability at least Ω(ǫ), extending M′ by one uniformly at random row of M, +gives a perfect (k + 1) × k′ sub-matrix M′′ of M. +(c) With probability at least Ω(ǫ), extending M′ by one uniformly at random column and +one uniformly at random row of M, gives a perfect2 (k + 1) × (k′ + 1) sub-matrix M′′ of +M. +Item 1 follows from Sgall result in [8] (See Section 3), and item 2 is Claim 10 in [7]. Now, the +tester strategy is as follows. If k ≤ k′, the tester first tries to extend M′ with a new column. If +it succeeds, it moves to the next iteration. Otherwise, it tries to extend M′ with a new row. If it +succeeds, it moves to the next iteration. Otherwise, it tries to extend M′ with a new row and a +new column. If it succeeds, it moves to the next iteration. If it fails, it accepts. If k′ < k, it starts +with the row, then the column, and then both. +Using this strategy, we show that the query complexity will be, at most, the order of the size +kk′ ≤ +� d +≤s +� +2d of M′ times the number of trials, ˜O(1/ǫ), to find the new row, column, or both. This +achieves the query complexity in Theorem 1. +For the non-adaptive tester, the tester, uniformly at random, chooses t = ˜O +�� d +≤s +� +2d/ǫ2� +rows +r1, . . . , rt ∈ [n] and t columns c1, . . . , ct ∈ [m] and queries all M[ri, cj] for all i · j ≤ t and puts them +in a table. Then it runs the above non-adaptive tester. When the non-adaptive tester asks for +uniformly at random row or column, it provides the next element ri or cj, respectively. The queries +are then answered from the table. We show that the adaptive algorithm does not need to make +queries that are not in the table before it halts. This achieves the query complexity in Theorem 2. +1.2 +Other Rank Problems +The real rank of a n × m-matrix M over any field F is the minimal d, such that there is a n × d +matrix N over F and a d × m matrix L over F such that M = NL. The testability of the real +rank was studied in [1, ?, 5]. In [1], Balcan et al. gave a non-adaptive tester for the real rank that +makes ˜O(d2/ǫ) queries. They also show that this query complexity is optimal. +The Boolean rank (∞-binary rank) was studied in [6, 7]. Parnas et al. in [7] gave a non-adaptive +tester for the Boolean rank that makes ˜O(d4/ǫ4) queries3. +2It may happen that events (a) and (b) do not occur and (c) does +3The query complexity in [7] is ˜O(d4/ǫ6). +We’ve noticed that Lemma 3 in [7] is also true when we replace +(ǫ2/64)n2 with (ǫ/4)n2. To prove that, in the proof of Lemma 3, replace Modification rules 1 and 2 with the following +modification: Modify to 0 all beneficial entries. This gives the result stated here,[3]. +3 + +2 +Definitions and Preliminary Results +Let M be a n × m (0, 1)-matrix. We denote by R(M) and C(M) the set of rows and columns +of M, respectively. The number of distinct rows and columns of M are denoted by r(M) = |R(M)| +and, c(M) = |C(M)|, respectively. The binary rank of a n × m-matrix M, br(M), is equal to the +minimal d, where there is a n × d (0, 1)-matrix N and a d × m (0, 1)-matrix L such that M = NL. +We define the s-binary rank, brs(M), of M to be the minimal integer d such that there are d +sets (rectangles) Ik × Jk where Ik ⊆ [n] := {1, . . . , n}, Jk ⊂ [m], k ∈ [d] such that M[i, j] = 1 for all +(i, j) ∈ Ik ×Jk, k ∈ [d] (monochromatic rectangles) and for every (i, j) ∈ [n]×[m] where M[i, j] = 1 +there are at least one and at most s integers t ∈ [d] such that (i, j) ∈ It × Jt (each 1-entry in M is +covered by at least one and at most s monochromatic rectangles). +We now prove. +Lemma 1. Let M be a n × m (0, 1)-matrix. The s-binary rank of M, brs(M), is equal to the +minimal integer d, where there is a n × d (0, 1)-matrix N and a d × m (0, 1)-matrix L such that: +For P = NL, +1. For every (i, j) ∈ [n] × [m], M[i, j] = 0 if and only if P[i, j] = 0. +2. For every (i, j) ∈ [n] × [m], P[i, j] ≤ s. +Proof. If M is of s-binary rank d, then there are rectangles {Ik ×Jk}k∈[d], Ik ⊆ [n], Jk ⊂ [m], k ∈ [d] +such that M[i, j] = 1 for all (i, j) ∈ Ik ×Jk, k ∈ [d] and for every (i, j) ∈ [n]×[m] where M[i, j] = 1 +there are at least one and at most s integers t ∈ [d] such that (i, j) ∈ It × Jt. Define row vectors +a(k) ∈ {0, 1}n and b(k) ∈ {0, 1}m where a(k) +i += 1 iff (if and only if) i ∈ Ik, and b(k) +j += 1 iff j ∈ Jk. +Then define4 P = a(1)′b(1)+· · ·+a(d)′b(d). It is easy to see that (a(k)′b(k))[i, j] = 1 iff (i, j) ∈ Ik ×Jk. +Therefore, P[i, j] = 0 iff M[i, j] = 0 and P[i, j] ≤ s for all (i, j) ∈ [n]×[m]. Define the n×d matrix +N = +� +a(1)′| · · · |a(d)′� +and the d × m matrix L = +� +b(1)′| · · · |b(d)′�′ +. +It is again easy to see that +P = NL. +The other direction can be easily seen by tracing backward in the above proof. +We now prove the following, +Lemma 2. Let M be a n × m matrix. Let N and L be n × d (0, 1)-matrix and d × m (0, 1)-matrix, +respectively, such that P = NL. Then r(P) ≤ r(N) and c(P) ≤ c(L). +Proof. We prove the result for r. +The proof for c is similar. +Let r1, . . . , rn be the rows of N +and p1, . . . , pn be the rows of P. +Then pi = riL. +Therefore, if ri = rj, then pi = pj. +Thus, +r(P) ≤ r(N). +Let M be a n × m matrix. For x ∈ X ⊆ [n], y ∈ Y ⊆ [m], we denote by M[X, Y ] the |X| × |Y | +sub-matrix of M, (M[x′, y′])x′∈X,y′∈Y . Denote by M[X, y] the column vector (M[x′, y])x′∈X and by +M[x, Y ] the row vector (M(x, y′))y′∈Y . +For x ∈ [n] (resp. y ∈ [m]) we say that M[X, y] is a new column (resp. M[x, Y ] is a new row) +to M[X, Y ] if it is not equal to any of the columns (resp. rows) of M[X, Y ]. +4Here x′ is the transpose of x. +4 + +Lemma 3. Let M be a n × m matrix, x ∈ [n], X ⊆ [n], y ∈ [m], and Y ⊆ [m]. Suppose M[x, Y ] is +not a new row to M[X, Y ], and M[X, y] is not a new column to M[X, Y ]. Then M[x, Y ∪ {y}] is +not a new row to M[X, Y ∪{y}] if and only if M[X ∪{x}, y] is not a new column to M[X ∪{x}, Y ]. +Proof. If M[x, Y ∪ {y}] is not a new row to M[X, Y ∪ {y}], then there is x′ ∈ X such that +M[x, Y ∪ {y}] = M[x′, Y ∪ {y}]. Since M[X, y] is not a new column to M[X, Y ], there is y′ ∈ Y +such that M[X, y] = M[X, y′]. Since M[x, Y ∪ {y}] = M[x′, Y ∪ {y}], we have M[x′, y′] = M[x, y′] +and M[x, y] = M[x′, y]. +Since M[X, y] = M[X, y′], we have M[x′, y] = M[x′, y′]. +Therefore, +M[x, y] = M[x, y′] and M[X ∪ {x}, y] = M[X ∪ {x}, y′]. Thus, M[X ∪ {x}, y] is not a new column +to M[X ∪ {x}, Y ]. +Similarly, the other direction follows. +3 +Matrices of s-Binary Rank d +In this section, we prove the following two Lemmas. +Lemma 4. For any n × m (0, 1)-matrix M of s-binary rank at most d, we have +r(M) · c(M) ≤ +� d +≤ s +� +2d. +Lemma 5. There is a (0, 1)-matrix M′ of s-binary rank d that satisfies r(M′) · c(M′) = +� d +≤s +� +2d. +To prove Lemma 4, we use the following Sgall’s lemma. +Lemma 6. [8]. Let A, B ⊆ 2[d] be such that for every A ∈ A and B ∈ B, |A ∩ B| ≤ s. Then +|A| · |B| ≤ +� d +≤s +� +2d. +We now prove Lemma 4. +Proof. Since the s-binary rank of M is at most d, by Lemma 1, there is a n × d (0, 1)-matrix N +and a d × m (0, 1)-matrix L such that, for P = NL +1. For every (i, j) ∈ [n] × [m], M[i, j] = 0 if and only if P[i, j] = 0. +2. For every (i, j) ∈ [n] × [m], P[i, j] ≤ s. +Obviously, r(M) ≤ r(P) and c(M) ≤ c(P). +Consider A = {A1, . . . , An} ⊆ 2[d] and B = +{B1, . . . , Bm} ⊆ 2[d], where Ai = {j|Ni,j = 1} and Bk = {j|Lj,k = 1}. +Since the entries of +P = NL are at most s, for every i ∈ [n] and k ∈ [m], |Ai ∩ Bk| ≤ s. +By Lemma 2 and 6, +r(M) · c(M) ≤ r(P) · c(P) ≤ r(N) · c(L) = |A| · |B| ≤ +� d +≤ s +� +2d. +We now prove Lemma 5 +5 + +Proof. Let N be a 2d × d (0, 1)-matrix where its rows contain all the vectors in {0, 1}d. Let L be a +d × +� d +≤s +� +matrix where its columns contain all the vectors in {0, 1}d of weight at most s. Obviously, +P = NL is 2d × +� d +≤s +� +with entries that are less than or equal to s. Define a 2d × +� d +≤s +� +(0, 1)-matrix +M′ where M′[i, j] = 0 if and only if P[i, j] = 0. Then, by Lemma 1, M′ is of s-binary rank at +most d. We now show that r(M′) · c(M′) = +� d +≤s +� +2d. +Since the identity d × d matrix Id is a sub-matrix of L, we have that NId = N is (0, 1)-matrix +and a sub-matrix of P and therefore of M′. Therefore, r(M′) ≥ r(N) = 2d. Since Id is a sub- +matrix of N, by the same argument, c(M′) ≥ c(L) = +� d +≤s +� +. Therefore r(M′) · c(M′) ≥ +� d +≤s +� +2d. +Thus, r(M′) · c(M′) = +� d +≤s +� +2d. +We now show that M′ has s-binary rank d. Suppose the contrary, i.e., M′ has binary rank +d′ < d. Then there are 2d × d′ (0, 1)-matrix N and d′ × +� d +≤s +� +(0, 1)-matrix L such that P = NL +and M′[i, j] = 0 iff P[i, j] = 0. Now by Lemma 2, r(M′) ≤ r(P) ≤ r(N) ≤ 2d′ < 2d, which gives a +contradiction. +4 +Testing The s-Binary Rank +In this section, we present the adaptive and non-adaptive testing algorithms for s-binary rank at +most d. We first give the adaptive algorithm and prove Theorem 1. +4.1 +The Adaptive Tester +In this section, we prove Theorem 1. +Consider the tester Adaptive-Test-Rank in Figure 1. The tester, at every iteration of the +main While-loop (step 2) has a set X of rows of M and a set Y of columns of M. If |X| ≥ |Y | +(step 5), the tester first tries to extend M[X, Y ] with a new column (steps 6-8). If it succeeds, it +moves to the next iteration. Otherwise, it tries to extend M[X, Y ] with a new row (steps 9-12). If +it succeeds, it moves to the next iteration. Otherwise, it tries to extend M[X, Y ] with a new row +and a new column (steps 21-26). If it succeeds, it moves to the next iteration. If it fails, it accepts +(step 27). If |X| < |Y | (step 13), it starts with the row of M[X, Y ] (steps 14-16), then the column +(steps 18-20), and then both (steps 21-26). If it fails, it accepts (step 27). +If |X| · |Y | > +� d +≤s +� +2d (step 2 and then step 28) or the s-binary rank of M[X, Y ] is greater than +d (step 3), then it rejects. +We first prove +Lemma 7. Let t = 9d/ǫ. Tester Adaptive-Test-Rank makes at most 2 +� d +≤s +� +2dt = ˜O +�� d +≤s +� +2d� +/ǫ +queries. +Proof. We prove by induction that at every iteration of the main While-loop (step 2), the tester +knows the entries of M[X, Y ], and the total number of queries, qX,Y , is at most 2|X||Y |t. Since +the While-loop condition is |X||Y | ≤ +� d +≤s +� +2d, the result follows. +At the beginning of the algorithm, no queries are made, and |X| = |Y | = 1. Then 2|X||Y |t = +2t > 0 = qX,Y . +Suppose, at the kth iteration, the tester knows the entries of M[X, Y ] and +qX,Y ≤ 2|X||Y |t. We prove the result for the (k + 1)th iteration. +We have the following cases (at the (k + 1)th iteration) +Case I. |X| ≥ |Y | (step 5) and, for some x, M[x, Y ] is a new row to M[X, Y ] (step 8). +6 + +Adaptive-Test-Rank(d, s, M, n, m, ǫ) +Input: Oracle that accesses the entries of n × m (0, 1)-matrix M. +Output: Either “Accept” or “Reject” +1. X ← {1}; Y ← {1}; t = 9d/ǫ. +2. While |X| · |Y | ≤ +� d +≤s +� +2d do +3. +If the s-binary rank of M[X, Y ] is greater than d, then Reject. +4. +Finish ← False; X′ ← Ø; Y ′ ← Ø. /∗ X′ and Y ′ are multi-sets. +5. +If |X| ≥ |Y | then +6. +While (NOT Finish) AND |X′| < t +7. +Draw uniformly at random x ∈ [n]\X; X′ ← X′ ∪ {x}; +8. +If M[x, Y ] is a new row to M[X, Y ] then X ← X ∪ {x}; Finish ← True. +9. +If (NOT Finish) then +10. +While (NOT Finish) AND |Y ′| < t +11. +Draw uniformly at random y ∈ [m]\Y ; Y ′ ← Y ′ ∪ {y}. +12. +If M[X, y] is new column to M[X, Y ] then Y ← Y ∪ {y}; Finish ← True. +13. +Else (|X| < |Y |) +14. +While (NOT Finish) AND |Y ′| < t +15. +Draw uniformly at random y ∈ [m]\Y ; Y ′ ← Y ′ ∪ {y}; +16. +If M[X, y] is a new column to M[X, Y ] then Y ← Y ∪ {y}; Finish ← True. +17. +If (NOT Finish) then +18. +While (NOT Finish) AND |X′| < t +19. +Draw uniformly at random x ∈ [n]\X; X′ ← X′ ∪ {x} +20. +If M[x, Y ] is a new row to M[X, Y ] then X ← X ∪ {x}; Finish ← True. +21. +While (NOT Finish) AND X′ ̸= Ø do +22. +Draw uniformly at random x ∈ X′ and y ∈ Y ′ +23. +If M[x, Y ∪ {y}] is a new row to M[X, Y ∪ {y}] OR, equivalently, +24. +M[X ∪ {x}, y] is a new column to M[X ∪ {x}, Y ] +25. +then X ← X ∪ {x}; Y ← Y ∪ {y}; Finish ← True. +26. +else X′ ← X′\{x}; Y ′ ← Y ′\{y}. +27. +If (NOT Finish) then Accept +28.Reject +Figure 1: An adaptive tester for s-binary rank at most d. +In that case, Finish becomes true, and no other sub-while-loop is executed. Therefore, the +number of queries made at this iteration is at most |Y |t (to find all M[x, Y ]), and one element x is +added to X. Then, the tester knows all the entries of M[X ∪ {x}, Y ] and +qX∪{x},Y = qX,Y + |Y |t ≤ 2|X||Y |t + |Y |t ≤ 2|X ∪ {x}| · |Y |t, +and the result follows. +Case II. |X| ≥ |Y | (step 5), for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ] (step 8), and +for some y, M[X, y] is a new column to M[X, Y ] (step 12). +7 + +In that case, Finish becomes true, and no other sub-while-loop is executed after the second +sub-while-loop (step 10). +Therefore, in this case, the number of queries made at this iteration is at most |Y |t + |X|t. +|X|t queries in the first sub-while-loop (to find M[x, Y ] for all x ∈ X′), and at most |Y |t queries +in the second sub-while-loop (to find M[X, y′] for all y′ ∈ Y ′). Then one element y is added to Y . +Therefore, the tester knows the entries of M[X, Y ∪ {y}] and, since |Y | ≤ |X|, +qX,Y ∪{y} = qX,Y + |X|t + |Y |t ≤ 2|X||Y |t + 2|X|t = 2|X| · |Y ∪ {y}|t, +and the result follows. +Case III. |X| ≥ |Y |, for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ], for all y′ ∈ Y ′, M[X, y′] +is not a new column to M[X, Y ], and for some x ∈ X′, y ∈ Y ′, M[x, Y ∪ {y}] is a new row to +M[X, Y ∪ {y}] (step 23). +In this case, |X′| = |Y ′| = t, the number of queries is |X|t + |Y |t + t. Exactly |X|t queries in +the first sub-while-loop, |Y |t queries in the second sub-while-loop, and at most5 t queries in the +sub-while-loop in step 21. Then one element x is added to X, and one element y is added to Y . +Then the tester knows the entries of M[X ∪ {x}, Y ∪ {y}] and +qX∪{x},Y ∪{y} = qX,Y + |X|t + |Y |t + t ≤ 2|X| · |Y |t + |X|t + |Y |t + t ≤ 2|X ∪ {x}| · |Y ∪ {y}|t. +Case IV. |X| ≥ |Y |, for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ], for all y′ ∈ Y ′, M[X, y′] +is not a new column to M[X, Y ], and for all the drawn pairs x ∈ X′, y ∈ Y ′, M[x, Y ∪ {y}] is not +a new row to M[X, Y ∪ {y}] (step 23). +In this case, Finish will have value False, and the tester accepts in step 27. +The analysis of the case when |X| < |Y | is similar to the above analysis. +We now prove the completeness of the tester. +Lemma 8. If M is a n × m (0, 1)-matrix of s-binary rank at most d, then the tester Adaptive- +Test-Rank accepts with probability 1. +Proof. The tester rejects if and only if one of the following occurs, +1. M[X, Y ] has s-binary rank greater than d. +2. |X| · |Y | > +� d +≤s +� +2d. +If M[X, Y ] has s-binary rank greater than d, then M has s-binary rank greater than d. This is +because, if M = NL, then M[X, Y ] = N[X, [d]] · L[[d], Y ]. So item 1 cannot occur. +Before we show that item 2 cannot occur, we prove the following: +Claim 1. The rows (resp. columns) of M[X, Y ] are distinct. +Proof. The steps in the tester where we add rows or columns are steps 8, 12 16, 20, and 23. In +steps 8, 12 16, 20 it is clear that a row (resp. column) is added only if it is a new row (resp. +column) to M[X, Y ]. Consider step 23 and suppose, w.l.o.g |X| ≥ |Y |. This step is executed only +when Finish = False. This happens when |X′| = |Y ′| = t, for every x ∈ X′, M[x, Y ] is not a +new row to M[X, Y ], and for every y ∈ Y ′, M[X, y] is not a new column to M[X, Y ]. Then x +5This is because, for x ∈ X′, y ∈ Y ′, the tester already knows M[x, Y ] and M[X, y] from the first and second +sub-while-loop and only needs to query M[x, y]. +8 + +and y are added to X and Y , respectively, if M[x, Y ∪ {y}] is a new row to M[X, Y ∪ {y}]. Then, +by Lemma 3, M[X ∪ {x}, y] is a new column to M[X ∪ {x}, Y ]. So, the rows (and columns) in +M[X ∪ {x}, Y ∪ {y}] are distinct. This implies the result. +Suppose, to the contrary, |X| · |Y | > +� d +≤s +� +2d. Since M′ = M[X, Y ] satisfies r(M′)c(M′) = +|X| · |Y | > +� d +≤s +� +2d, by Lemma 4, the s-binary rank of M′, and therefore of M, is greater than d. A +contradiction. +We now prove the soundness of the tester. +We first prove the following. +Claim 2. Let M be a n×m (0, 1)-matrix, X ⊆ [n], and Y ⊆ [m]. Suppose there are two functions, +′ : [n] → X and ′′ : [m] → Y , such that +1. For every x ∈ [n], M[x, Y ] = M[x′, Y ]. +2. For every y ∈ [m], M[X, y] = M[X, y′′]. +3. For every x ∈ [n] and y ∈ [m], M[x, y] = M[x′, y′′]. +Then M has at most |X| distinct rows and |Y | distinct columns, and its s-binary rank is the s-binary +rank of M[X, Y ]. +Proof. Let x ∈ [n]\X. For every y, M[x, y] = M[x′, y′′] = M[x′, y]. Therefore, row x in M is equal +to row x′. Similarly, column y in M is equal to column y′′. +Since adding equal columns and rows to a matrix does not change the s-binary rank6, we have +brs(M[X, Y ]) = brs(M[X, [m]]) = brs(M). +The following Claim is proved in [7] (Claim 10). Here, we give the proof for completeness. +Claim 3. Let M be a (0, 1)-matrix that is ǫ-far from having s-binary rank at most d. Let X ⊆ [n] +and Y ⊆ [m], such that brs(M[X, Y ]) ≤ d, the columns of M[X, Y ] are distinct, and the rows of +M[X, Y ] are distinct. Then one of the following must hold: +1. The number of rows x ∈ [n] where M[x, Y ] is a new row to M[X, Y ] is at least nǫ/3. +2. The number of columns y ∈ [m] where M[X, y] is a new column to M[X, Y ] is at least mǫ/3. +3. The number pairs (x, y), x ̸∈ X, y ̸∈ Y , where, M[x, Y ] = M[x′, Y ] for some x′ ∈ X, +M[X, y] = M[X, y′′] for some y′′ ∈ Y , and M[x, y] ̸= M[x′, y′′], is at least mnǫ/3. +Proof. Assume, to the contrary, that none of the above statements holds. Change every row x in +M where M[x, Y ] is a new row to M[X, Y ] to a zero row. Let X′ be the set of such rows. Change +every column y in M where M[X, y] is a new row to M[X, Y ] to a zero column. Let Y ′ be the +set of such columns. For every other entry (x, y), x ̸∈ X, y ̸∈ Y that is not changed to zero and +M[x, y] ̸= M[x′, y′′], change M[x, y] to M[x′, y′′]. Let M′ be the matrix obtained from the above +changes. +The number of entries (x, y) where M[x, y] ̸= M′[x, y] is less than (nǫ/3)m + (mǫ/3)n + +mnǫ/3 = ǫmn. Therefore, M′ is ǫ-close to M. By claim 3, brs(M′) = brs(M[[n]\X′, [m]\Y ′]) = +brs(M[X, Y ]) ≤ d. A contradiction. +6If we add a column to a matrix that is equal to column y, then the rectangles that cover column y can be extended +to cover the added column. +9 + +We now prove the completeness of the tester. +Lemma 9. If M is ǫ-far from having s-binary rank d, then with probability at least 2/3, Adaptive- +Test-Rank rejects. +Proof. Consider the while-loop in step 2 at some iteration i. If brs(M[X, Y ]) > d, then the tester +rejects in step 3. We will now show that if brs(M[X, Y ]) ≤ d, then, with probability at most 3e−2d, +the tester accepts at iteration i. +To this end, let brs(M[X, Y ]) ≤ d. Then, by Claim 3, one of the following holds. +1. The number of rows x ∈ [n] where M[x, Y ] is a new row to M[X, Y ] is at least nǫ/3. +2. The number of columns y ∈ [m] where M[X, y] is a new column to M[X, Y ] is at least mǫ/3. +3. The number pairs (x, y), x ̸∈ X, y ̸∈ Y , where, M[x, Y ] = M[x′, Y ] for some x′ ∈ X, +M[X, y] = M[X, y′′] for some y′′ ∈ Y , and M[x, y] ̸= M[x′, y′′], is at least mnǫ/3. +Now at the ith iteration, suppose w.l.o.g, |X| ≥ |Y | (the other case |Y | < |X| is similar). If item 1 +occurs, then with probability at least p = 1 − (1 − ǫ/3)t ≥ 1 − e−2d, the tester finds a new row +to M[X, Y ] and does not accept at iteration i. If item 2 occurs, then if it does not find a new +row to M[X, Y ], with probability at least p, the tester finds a new column to M[X, Y ] and does +not accept. If item 3 occurs, and it does not find a new row or column to M[X, Y ], then with +probability at least p, it finds such a pair and does not accept. Therefore, with probability at most +3(1 − p) ≤ 3e−2d, the tester accepts at iteration i. +Since the while-loop runs at most |X| + |Y | ≤ 2|X||Y | ≤ 2 +� d +≤s +� +2d ≤ 22d+1 iterations, with +probability at most 3e−2d22d+1 ≤ 1/3, the tester accepts in while-loop. Therefore, with proba- +bility at least 2/3, the tester does not accept in the while-loop. Thus, it either rejects because +brs(M[X, Y ]) > d or rejects in step 28. +4.2 +The Non-Adaptive Tester +In this section, we prove Theorem 2. +First, consider Adaptive-Test-Rank in Figure 1. Consider steps 7,11,15, and 19, where it +draws a new column or row. We prove. +Lemma 10. Let t = 9d/ǫ. At each iteration of Adaptive-Test-Rank, the total number of uni- +formly at random rows x ∈ [n] drawn is at most (|X| + min(|X|, |Y | − 1))t, and the number of +uniformly at random rows y ∈ [m] drawn is at most (|Y | + min(|X|, |Y |))t. +Proof. We prove by induction that at every iteration of the main While-loop (step 2), the total +number of random rows drawn by the tester, nX,Y , is at most (|X| + min(|X|, |Y | − 1))t, and the +total number of random columns drawn, mX,Y , is at most (|Y | + min(|X|, |Y |))t. +At the beginning, |X| = |Y | = 1, and the number of columns and rows is 1. In that case,7, +nX,Y = 1 ≤ t and mX,Y = 1 ≤ 2t. Suppose, at the kth iteration, the induction statement is true. +We prove the result for the (k + 1)th iteration. +At the (k + 1)th iteration, we have the following cases. +Case I. |X| ≥ |Y | (step 5) and, for some x, M[x, Y ] is a new row to M[X, Y ] (step 8). +7We assume that the first column/row drawn is column/row one +10 + +Non-Adaptive-Test-Rank(d, s, M, n, m, ǫ) +Input: Oracle that accesses the entries of (0, 1)-matrix M. +Output: Either “Accept” or “Reject”. +1. +T ← +324·d2( d +≤s)2d +ǫ2 +. +2. +Dray uniformly at random x(1), . . . , x(T) ∈ [n]. +3. +Dray uniformly at random y(1), . . . , y(T) ∈ [m]. +4. +For every i ∈ [T] and j ∈ [T] such that i · j ≤ T +5. +D[i, j] ← Query M[x(i), y(j)] +6. +u = 1; w = 1. +7. +Run Adaptive-Test-Rank(d, s, M, n, m, ǫ) +When the tester asks for a uniform at random x - return x(u); u ← u + 1 +When the tester asks for a uniform at random y - return y(w); w ← w + 1 +When the tester makes the Query M[x(i), y(j)] - return D[i, j] +Figure 2: A non-adaptive tester for s-binary rank at most d. +In that case, Finish becomes true, and no other sub-while-loop is executed. Therefore, the +number of rows drawn at this iteration is at most t, and one element x is added to X. No columns +are drawn. Then, +nX∪{x},Y ≤ nX,Y + t ≤ (|X| + min(|X|, |Y | − 1) + 1)t ≤ (|X ∪ {x}| + min(|X ∪ {x}|, |Y | − 1))t, +and +mX∪{x},Y = mX,Y ≤ (|Y | + min(|X|, |Y |))t ≤ (|Y | + min(|X ∪ {x}|, |Y |))t. +Thus, the result follows for this case. +Case II. |X| ≥ |Y | (step 5), for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ] (step 8), and +for some y, M[X, y] is a new column to M[X, Y ] (step 12). +In that case, Finish becomes true, and no other sub-while-loop is executed after the second +sub-while-loop (step 10). +Therefore, in this case, the number of rows drawn at this iteration is t, one element y is added +to Y , and the number of columns drawn is at most t. Then +nX,Y ∪{y} = nX,Y + t +≤ +(|X| + min(|X|, |Y | − 1) + 1)t += +(|X| + |Y |)t = (|X| + min(|X|, |Y ∪ {y}| − 1))t, +and +mX,Y ∪{y} ≤ mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t ≤ (|Y ∪ {y}| + min(|X|, |Y ∪ {y}|))t. +Thus, the result follows for this case. +Case III. |X| < |Y | (step 13), and for some y, M[X, y] is a new column to M[X, Y ] (step 16). +11 + +In that case, Finish becomes true, and no other sub-while-loop is executed. Therefore, the +number of columns drawn at this iteration is at most t, and one element y is added to Y . No rows +are drawn. Then, +nX,Y ∪{y} = nX,Y ≤ (|X| + min(|X|, |Y | − 1))t ≤ (|X| + min(|X|, |Y ∪ {y}| − 1))t, +and +mX,Y ∪{y} ≤ mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t = (|Y ∪ {y}| + min(|X|, |Y ∪ {y}|))t. +Thus, the result follows for this case. +Case IV. |X| < |Y | (step 13), for all y′ ∈ Y ′, M[X, y′] is not a new row to M[X, Y ], and for some +x, M[x, Y ] is a new column to M[X, Y ] (step 20). In that case, Finish becomes true, and no other +sub-while-loop is executed after the fourth sub-while-loop (step 18). +In this case, the number of rows drawn at this iteration is t, one element x is added to X, and +the number of columns drawn is at most t. Then +nX∪{x},Y = nX,Y + t +≤ +(|X| + min(|X|, |Y | − 1) + 1)t +≤ +(|X ∪ {x}| + min(|X ∪ {x}|, |Y | − 1))t +mX∪{x},Y ≤ mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t = (|Y | + min(|X ∪ {x}|, |Y |))t. +Thus, the result follows for this case. +Case V. For all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ], for all y′ ∈ Y ′, M[X, y′] is not a +new column to M[X, Y ], and for some x ∈ X′, y ∈ Y ′, M[x, Y ∪{y}] is a new row to M[X, Y ∪{y}] +(step 23). +In this case, the number of rows drawn at this iteration is t, the number of columns drawn is t, +one element x is added to X, and one element y is added to Y . Then +nX∪{x},Y ∪{y} = nX,Y + t +≤ +(|X| + min(|X|, |Y | − 1) + 1)t +≤ +(|X ∪ {x}| + min(|X ∪ {x}|, |Y ∪ {y}| − 1))t. +mX∪{x},Y ∪{y} = mX,Y + t +≤ +(|Y | + min(|X|, |Y |) + 1)t +≤ +(|Y ∪ {y}| + min(|X ∪ {x}|, |Y ∪ {y}|))t. +We are now ready to prove Theorem 2. +Proof. By Lemma 10, the total number of rows and columns drawn in Adaptive-Test-Rank +up to iteration t is at most n′ := 9(|X| + min(|X|, |Y | − 1))d/ǫ ≤ 18|X|d/ǫ and m′ := 9(|Y | + +min(|X|, |Y |)d/ǫ ≤ 18|Y |d/ǫ, respectively. We also have |X| · |Y | ≤ +� d +≤s +� +2d. So +n′ · m′ ≤ 324|X||Y |d2/ǫ2 ≤ T := +324 · d2� d +≤s +� +2d +ǫ2 +. +Consider the tester Non-Adaptive-Test-Rank in Figure 2. The tester draws T rows x(1), . . . , +x(T) ∈ [n], and columns y(1), . . . , y(T) ∈ [m] and queries all M[x(i), y(j)] where ij ≤ T and puts the +12 + +result in the table D. Then it runs Adaptive-Test-Random using the above-drawn rows and +columns. We now show that all the queries that Adaptive-Test-Random makes can be fetched +from the table D. +At any iteration, the number of rows drawn is at most n′, and the number of rows drawn is at +most m′. Therefore, the tester needs to know (in the worst case) all the entries M[x(i), y(j)] where +i ≤ n′ and j ≤ m′. Since ij ≤ n′m′ ≤ T, the result follows. +The number of queries that the tester makes is +T +� +i=1 +T +i = O(T ln T) = ˜O +�� d +≤s +� +2d +ǫ2 +� +. +5 +Testing the Exact s-Binary Rank +We first prove the following. +Lemma 11. Let M and M′ be n × m (0, 1)-matrices that differ in one row (or column). Then +|brs(M) − brs(M′)| ≤ 1. +Proof. Suppose brs(M) = d and M′ differ from M in row k. Let N and L be n × d (0, 1)-matrix +and d × m (0, 1)-matrix, respectively, such that P = NL, for every (i, j) ∈ [n] × [m], P[i, j] ≤ s, +and P[i, j] = 0 if and only if M[i, j] = 0. Add to N a column (as a (d + 1)th column) that all its +entries are zero except the k-th entry, which equals 1. Then change N[k, j] to zero for all j ∈ [d]. +Let N ′ be the resulting matrix. Add to L another row (as a (d + 1)th row) equal to the k-th row +of M′. Let L′ be the resulting matrix. Let P ′ = N ′L′. It is easy to see that P ′[i, j] = P[i, j] for all +i ̸= k and j, and the kth row of P ′ is equal to the kth row of M′. Then, for every (i, j) ∈ [n] × [m], +P ′[i, j] ≤ s, and P ′[i, j] = 0 if and only if M′[i, j] = 0. Therefore, brs(M′) ≤ d + 1 = brs(M) + 1. +In the same way, brs(M) ≤ brs(M′) + 1. +Lemma 12. Let η = d2/(nm). Let M be n × m (0, 1)-matrix. If M is ǫ-close to having s-binary +rank at most d, then M is (ǫ + η)-close to having s-binary rank d. +Proof. We will show that for every n × m (0, 1)-matrix H of s-binary rank at most d − 1, there is a +n × m (0, 1)-matrix G of s-binary rank d that is η-close to H. Therefore, if M is ǫ-close to having +s-binary rank at most d, then it is (ǫ + η)-close to having s-binary rank d. +Define the n×m (0, 1)-matrices Gk, k ∈ [d]∪{0}, where G0 = H and for k ≥ 1, Gk[i, j] = H[i, j] +if j > k or i > d, and Gk[[d], [k]] = Id[[d], [k]] where Id is the d × d identity matrix. +Since +Gd[[d], [d]] = Id, we have brs(Gd) ≥ d. It is clear that for every k ∈ [d] ∪ {0}, Gk is (d2/nm)-close +to H. If brs(Gd) = d, then take G = Gd, and we are done. Otherwise, suppose brs(Gd) > d. +Now consider a sequence H = G0, G1, G2, . . . , Gd. By Lemma 11, we have brs(Gi−1) − 1 ≤ +brs(Gi) ≤ brs(Gi−1) + 1. Now since brs(G0) = brs(H) ≤ d − 1 and brs(Gd) > d, by the discrete +intermediate value theorem, there must be k ∈ [d] such that brs(Gk) = d. Then take G = Gk, and +we are done. +Now, the tester for testing the s-binary rank d runs as follows. +If mn < 2d2/ǫ, then find +all the entries of M with mn < 2d2/ǫ queries. If brs(M) = d, then accept. Otherwise, reject. +13 + +If mn ≥ 2d2/ǫ, then run Adaptive-Test-Rank(d, s, M, n, m, ǫ/2) (for the non-adaptive, we run +Non-Adaptive-Test-Rank(d, s, M, n, m, ǫ/2)) and output its answer. +We now show the correctness of this algorithm. If M is of s-binary rank d, then it is of s-binary +rank at most d, and the tester accepts. +Now, suppose f is ǫ-far from having s-binary rank d. If mn < 2d2/ǫ, the tester rejects. If +mn ≥ 2d2/ǫ, then, by Lemma 12, f is (ǫ − η)-far from having s-binary rank at most d, where +η = d2/(nm). Since η = d2/(nm) ≤ ǫ/2, the function f is (ǫ/2)-far from having s-binary rank at +most d, and therefore the tester, with probability at least 2/3, rejects. +References +[1] Maria-Florina Balcan, Yi Li, David P. Woodruff, and Hongyang Zhang. Testing matrix rank, +optimally. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algo- +rithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 727–746, 2019. +doi:10.1137/1.9781611975482.46. +[2] Parinya Chalermsook, Sandy Heydrich, Eugenia Holm, and Andreas Karrenbauer. Nearly tight +approximability results for minimum biclique cover and partition. In Andreas S. Schulz and +Dorothea Wagner, editors, Algorithms - ESA 2014 - 22th Annual European Symposium, Wro- +claw, Poland, September 8-10, 2014. Proceedings, volume 8737 of Lecture Notes in Computer +Science, pages 235–246. Springer, 2014. doi:10.1007/978-3-662-44777-2\_20. +[3] Dana Ron. Private Communication. +[4] David A. Gregory, Norman J. Pullman, Kathryn F. Jones, and J. Richard Lundgren. Biclique +coverings of regular bigraphs and minimum semiring ranks of regular matrices. J. Comb. Theory, +Ser. B, 51(1):73–89, 1991. doi:10.1016/0095-8956(91)90006-6. +[5] Yi Li, Zhengyu Wang, and David P. Woodruff. +Improved testing of low rank matrices. +In The 20th ACM SIGKDD International Conference on Knowledge Discovery and Data +Mining, KDD ’14, New York, NY, USA - August 24 - 27, 2014, pages 691–700, 2014. +doi:10.1145/2623330.2623736. +[6] Yonatan Nakar and Dana Ron. +On the testability of graph partition properties. +In +Eric Blais, Klaus Jansen, Jos´e D. P. Rolim, and David Steurer, editors, Approxima- +tion, Randomization, and Combinatorial Optimization. Algorithms and Techniques, AP- +PROX/RANDOM 2018, +August 20-22, +2018 +- Princeton, +NJ, USA, volume +116 +of +LIPIcs, +pages +53:1–53:13. +Schloss +Dagstuhl +- +Leibniz-Zentrum +f¨ur +Informatik, +2018. +doi:10.4230/LIPIcs.APPROX-RANDOM.2018.53. +[7] Michal Parnas, Dana Ron, and Adi Shraibman. Property testing of the boolean and binary +rank. Theory Comput. Syst., 65(8):1193–1210, 2021. doi:10.1007/s00224-021-10047-8. +[8] Jir´ı Sgall. Bounds on pairs of families with restricted intersections. Comb., 19(4):555–566, 1999. +doi:10.1007/s004939970007. +14 + diff --git a/3dE3T4oBgHgl3EQfPwmd/content/tmp_files/load_file.txt b/3dE3T4oBgHgl3EQfPwmd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28ffb9090e7ae826f9234db3ccedba5a0812338a --- /dev/null +++ b/3dE3T4oBgHgl3EQfPwmd/content/tmp_files/load_file.txt @@ -0,0 +1,624 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf,len=623 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='04406v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='DS] 11 Jan 2023 A Note on Property Testing of the Binary Rank Nader H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Bshouty Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' of Computer Science Technion, Haifa, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' January 12, 2023 Abstract Let M be a n × m (0, 1)-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We define the s-binary rank, brs(M), of M to be the minimal integer d such that there are d monochromatic rectangles that cover all the 1-entries in the matrix, and each 1-entry is covered by at most s rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' When s = 1, this is the binary rank, br(M), known from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let R(M) and C(M) be the set of rows and columns of M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We use the result of Sgall [8] to prove that if M has s-binary rank at most d, then |R(M)| · |C(M)| ≤ � d ≤s � 2d where � d ≤s � = �s i=0 �d i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' This bound is tight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' that is, there exists a matrix M ′ of s-binary rank d such that |R(M ′)| · |C(M ′)| = � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Using this result, we give a new one-sided adaptive and non-adaptive testers for (0, 1)- matrices of s-binary rank at most d (and exactly d) that makes ˜O �� d ≤s � 2d/ǫ � and ˜O �� d ≤s � 2d/ǫ2� queries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For a fixed s, this improves the query complexity of the tester of Parnas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' in [7] by a factor of ˜Θ(2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 1 Introduction Let M be a n × m (0, 1)-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We define the s-binary rank, brs(M), of M to be the minimal integer d such that there are d sets (rectangles) Ik×Jk where Ik ⊆ [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , n}, Jk ⊂ [m], k ∈ [d] such that1 M[i, j] = 1 for all (i, j) ∈ Ik × Jk, k ∈ [d] (monochromatic rectangles), and for every (i, j) ∈ [n] × [m] where M[i, j] = 1, there are at least one and at most s integers t ∈ [d] such that (i, j) ∈ It × Jt (each 1-entry in M is covered by at least one and at most s monochromatic rectangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' When s = 1, br1(M), is the binary rank, br(M), and when s = ∞, br∞(M) is the Boolean rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Both are known from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' See, for example, [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The binary rank can also be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The binary rank of a n × m (0, 1)-matrix M is equal to the minimal d, where there are n × d (0, 1)-matrix N and d × m (0, 1)-matrix L such that M = NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' It is also equal to the minimal number of bipartite cliques needed to partition all the edges of a bipartite graph whose adjacent matrix is M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The s-binary rank of M is the minimal number of bipartite cliques needed to cover all edges of a bipartite graph whose adjacent matrix is M, where each edge is covered by at most s bipartite cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In [2], it was shown that it is NP-hard to approximating the binary rank to within a factor of n1−δ for any given δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 1For M, the (i, j) entry of the matrix is denoted by M[i, j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 1 A property-testing algorithm (tester) of the s-binary rank [7] is given as input 0 < ǫ < 1, integers d, n, m, and query access to the entries of a n×m (0, 1)-matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M has s-binary rank at most d (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' equal d), then the tester accepts with probability at least 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M is ǫ-far from having s-binary rank at most d (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' equal d), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=', more than ǫ-fraction of the entries of M should be modified to get a matrix with s-binary rank at most d (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' equal to d), then the tester rejects with probability at least 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If the tester accepts matrices having s-binary rank at most d (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' equal to d) with probability 1, then we call it a one-sided error tester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In adaptive testing, the queries can depend on the answers to the previous queries, whereas in non-adaptive testing, all the queries are fixed in advance by the tester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The goal is to construct a tester that makes a minimal number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The testability of s-binary rank at most d of (0, 1)-matrices was studied in [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In [6], Nakar and Ron gave a non-adaptive one-sided error tester for s = 1, that makes ˜O(24d/ǫ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In [7], Parnas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' gave a non-adaptive and adaptive one-sided error tester for s = 1 that makes O(22d/ǫ2) and O(22d/ǫ) queries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The results in [7] also hold for s-binary rank at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In this paper, for s-binary at most d and equal to d, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' There exists an adaptive one-sided error tester for s-binary rank of n × m (0, 1)- matrices that makes ˜O �� d ≤s � 2d/ǫ � queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' There exists a non-adaptive one-sided error tester for s-binary rank of n × m (0, 1)- matrices that makes ˜O �� d ≤s � 2d/ǫ2� queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For fixed s, this improves the query complexity of Parnas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' in [7] by a factor of ˜O(2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='1 Our Approach The tester of Parnas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' [7] uses the fact that if M′ is a k × k sub-matrix of M and M′ is of s-binary rank at most d, then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' M′ has at most 2d distinct rows and at most 2d distinct columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M is ǫ-far from having s-binary rank at most d, then extending M′ by one more uniformly at random row and column of M, gives a (k + 1) × (k + 1) sub-matrix M′′ of M that, with probability at least Ω(ǫ), satisfies: the number of distinct rows in M′′ is greater by one than the number of distinct rows in M′, or, the number of distinct columns in M′′ is greater by one than the number of distinct columns in M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' So, their adaptive tester runs O(2d/ǫ) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' At each iteration, it extends M′ by uniformly at random one row and one column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M′′ be the resulting sub-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If the s-binary rank of M′′ is greater than d, the tester rejects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If the number of distinct rows or columns in M′′ is greater than the number in M′, then it continues to the next iteration with M′ ← M′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Otherwise, it continues to the next iteration with M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If, after O(2d/ǫ) iterations, M′ has s-binary rank d, the tester accepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If the s-binary rank of M is d, then every sub-matrix has a s-binary rank d, and the tester accepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M is ǫ-far from having s-binary rank at most d, then: since, at each iteration, with probability at least Ω(ǫ), the number of distinct rows or columns of M′ is increased by one, and since matrices of s-binary rank d has at most 2d distinct rows and at most 2d distinct columns, with high probability, we get M′ with s-binary rank greater than d and the tester rejects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The 2 query complexity of the tester is O(22d/ǫ), which is the number of entries of the matrix M′, O(22d), times the number of trials O(1/ǫ) for extending M′ by one row and one column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now give our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Call a sub-matrix M′ of M perfect if it has distinct rows and distinct columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Our adaptive tester uses the fact that if M′ is a perfect k×k′ sub-matrix of M of s-binary rank d, then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' kk′ ≤ � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M is ǫ-far from having s-binary rank at most d, then at least one of the following occurs (a) With probability at least Ω(ǫ), extending M′ by one uniformly at random column of M, gives a perfect k × (k′ + 1) sub-matrix M′′ of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' (b) With probability at least Ω(ǫ), extending M′ by one uniformly at random row of M, gives a perfect (k + 1) × k′ sub-matrix M′′ of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' (c) With probability at least Ω(ǫ), extending M′ by one uniformly at random column and one uniformly at random row of M, gives a perfect2 (k + 1) × (k′ + 1) sub-matrix M′′ of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Item 1 follows from Sgall result in [8] (See Section 3), and item 2 is Claim 10 in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Now, the tester strategy is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If k ≤ k′, the tester first tries to extend M′ with a new column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If it succeeds, it moves to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Otherwise, it tries to extend M′ with a new row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If it succeeds, it moves to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Otherwise, it tries to extend M′ with a new row and a new column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If it succeeds, it moves to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If it fails, it accepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If k′ < k, it starts with the row, then the column, and then both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Using this strategy, we show that the query complexity will be, at most, the order of the size kk′ ≤ � d ≤s � 2d of M′ times the number of trials, ˜O(1/ǫ), to find the new row, column, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' This achieves the query complexity in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For the non-adaptive tester, the tester, uniformly at random, chooses t = ˜O �� d ≤s � 2d/ǫ2� rows r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , rt ∈ [n] and t columns c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , ct ∈ [m] and queries all M[ri, cj] for all i · j ≤ t and puts them in a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then it runs the above non-adaptive tester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' When the non-adaptive tester asks for uniformly at random row or column, it provides the next element ri or cj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The queries are then answered from the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We show that the adaptive algorithm does not need to make queries that are not in the table before it halts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' This achieves the query complexity in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='2 Other Rank Problems The real rank of a n × m-matrix M over any field F is the minimal d, such that there is a n × d matrix N over F and a d × m matrix L over F such that M = NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The testability of the real rank was studied in [1, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=', 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In [1], Balcan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' gave a non-adaptive tester for the real rank that makes ˜O(d2/ǫ) queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' They also show that this query complexity is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The Boolean rank (∞-binary rank) was studied in [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Parnas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' in [7] gave a non-adaptive tester for the Boolean rank that makes ˜O(d4/ǫ4) queries3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2It may happen that events (a) and (b) do not occur and (c) does 3The query complexity in [7] is ˜O(d4/ǫ6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We’ve noticed that Lemma 3 in [7] is also true when we replace (ǫ2/64)n2 with (ǫ/4)n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' To prove that, in the proof of Lemma 3, replace Modification rules 1 and 2 with the following modification: Modify to 0 all beneficial entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' This gives the result stated here,[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 3 2 Definitions and Preliminary Results Let M be a n × m (0, 1)-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We denote by R(M) and C(M) the set of rows and columns of M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The number of distinct rows and columns of M are denoted by r(M) = |R(M)| and, c(M) = |C(M)|, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The binary rank of a n × m-matrix M, br(M), is equal to the minimal d, where there is a n × d (0, 1)-matrix N and a d × m (0, 1)-matrix L such that M = NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We define the s-binary rank, brs(M), of M to be the minimal integer d such that there are d sets (rectangles) Ik × Jk where Ik ⊆ [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , n}, Jk ⊂ [m], k ∈ [d] such that M[i, j] = 1 for all (i, j) ∈ Ik ×Jk, k ∈ [d] (monochromatic rectangles) and for every (i, j) ∈ [n]×[m] where M[i, j] = 1 there are at least one and at most s integers t ∈ [d] such that (i, j) ∈ It × Jt (each 1-entry in M is covered by at least one and at most s monochromatic rectangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M be a n × m (0, 1)-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The s-binary rank of M, brs(M), is equal to the minimal integer d, where there is a n × d (0, 1)-matrix N and a d × m (0, 1)-matrix L such that: For P = NL, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every (i, j) ∈ [n] × [m], M[i, j] = 0 if and only if P[i, j] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every (i, j) ∈ [n] × [m], P[i, j] ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M is of s-binary rank d, then there are rectangles {Ik ×Jk}k∈[d], Ik ⊆ [n], Jk ⊂ [m], k ∈ [d] such that M[i, j] = 1 for all (i, j) ∈ Ik ×Jk, k ∈ [d] and for every (i, j) ∈ [n]×[m] where M[i, j] = 1 there are at least one and at most s integers t ∈ [d] such that (i, j) ∈ It × Jt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Define row vectors a(k) ∈ {0, 1}n and b(k) ∈ {0, 1}m where a(k) i = 1 iff (if and only if) i ∈ Ik, and b(k) j = 1 iff j ∈ Jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then define4 P = a(1)′b(1)+· · ·+a(d)′b(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' It is easy to see that (a(k)′b(k))[i, j] = 1 iff (i, j) ∈ Ik ×Jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, P[i, j] = 0 iff M[i, j] = 0 and P[i, j] ≤ s for all (i, j) ∈ [n]×[m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Define the n×d matrix N = � a(1)′| · · · |a(d)′� and the d × m matrix L = � b(1)′| · · · |b(d)′�′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' It is again easy to see that P = NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The other direction can be easily seen by tracing backward in the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now prove the following, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M be a n × m matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let N and L be n × d (0, 1)-matrix and d × m (0, 1)-matrix, respectively, such that P = NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then r(P) ≤ r(N) and c(P) ≤ c(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We prove the result for r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The proof for c is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , rn be the rows of N and p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , pn be the rows of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then pi = riL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, if ri = rj, then pi = pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Thus, r(P) ≤ r(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M be a n × m matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For x ∈ X ⊆ [n], y ∈ Y ⊆ [m], we denote by M[X, Y ] the |X| × |Y | sub-matrix of M, (M[x′, y′])x′∈X,y′∈Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Denote by M[X, y] the column vector (M[x′, y])x′∈X and by M[x, Y ] the row vector (M(x, y′))y′∈Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For x ∈ [n] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' y ∈ [m]) we say that M[X, y] is a new column (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' M[x, Y ] is a new row) to M[X, Y ] if it is not equal to any of the columns (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' rows) of M[X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 4Here x′ is the transpose of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 4 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M be a n × m matrix, x ∈ [n], X ⊆ [n], y ∈ [m], and Y ⊆ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Suppose M[x, Y ] is not a new row to M[X, Y ], and M[X, y] is not a new column to M[X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then M[x, Y ∪ {y}] is not a new row to M[X, Y ∪{y}] if and only if M[X ∪{x}, y] is not a new column to M[X ∪{x}, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M[x, Y ∪ {y}] is not a new row to M[X, Y ∪ {y}], then there is x′ ∈ X such that M[x, Y ∪ {y}] = M[x′, Y ∪ {y}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since M[X, y] is not a new column to M[X, Y ], there is y′ ∈ Y such that M[X, y] = M[X, y′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since M[x, Y ∪ {y}] = M[x′, Y ∪ {y}], we have M[x′, y′] = M[x, y′] and M[x, y] = M[x′, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since M[X, y] = M[X, y′], we have M[x′, y] = M[x′, y′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, M[x, y] = M[x, y′] and M[X ∪ {x}, y] = M[X ∪ {x}, y′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Thus, M[X ∪ {x}, y] is not a new column to M[X ∪ {x}, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Similarly, the other direction follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 3 Matrices of s-Binary Rank d In this section, we prove the following two Lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For any n × m (0, 1)-matrix M of s-binary rank at most d, we have r(M) · c(M) ≤ � d ≤ s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' There is a (0, 1)-matrix M′ of s-binary rank d that satisfies r(M′) · c(M′) = � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' To prove Lemma 4, we use the following Sgall’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let A, B ⊆ 2[d] be such that for every A ∈ A and B ∈ B, |A ∩ B| ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then |A| · |B| ≤ � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since the s-binary rank of M is at most d, by Lemma 1, there is a n × d (0, 1)-matrix N and a d × m (0, 1)-matrix L such that, for P = NL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every (i, j) ∈ [n] × [m], M[i, j] = 0 if and only if P[i, j] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every (i, j) ∈ [n] × [m], P[i, j] ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Obviously, r(M) ≤ r(P) and c(M) ≤ c(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Consider A = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , An} ⊆ 2[d] and B = {B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , Bm} ⊆ 2[d], where Ai = {j|Ni,j = 1} and Bk = {j|Lj,k = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since the entries of P = NL are at most s, for every i ∈ [n] and k ∈ [m], |Ai ∩ Bk| ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' By Lemma 2 and 6, r(M) · c(M) ≤ r(P) · c(P) ≤ r(N) · c(L) = |A| · |B| ≤ � d ≤ s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now prove Lemma 5 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let N be a 2d × d (0, 1)-matrix where its rows contain all the vectors in {0, 1}d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let L be a d × � d ≤s � matrix where its columns contain all the vectors in {0, 1}d of weight at most s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Obviously, P = NL is 2d × � d ≤s � with entries that are less than or equal to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Define a 2d × � d ≤s � (0, 1)-matrix M′ where M′[i, j] = 0 if and only if P[i, j] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then, by Lemma 1, M′ is of s-binary rank at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now show that r(M′) · c(M′) = � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since the identity d × d matrix Id is a sub-matrix of L, we have that NId = N is (0, 1)-matrix and a sub-matrix of P and therefore of M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, r(M′) ≥ r(N) = 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since Id is a sub- matrix of N, by the same argument, c(M′) ≥ c(L) = � d ≤s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore r(M′) · c(M′) ≥ � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Thus, r(M′) · c(M′) = � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now show that M′ has s-binary rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Suppose the contrary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=', M′ has binary rank d′ < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then there are 2d × d′ (0, 1)-matrix N and d′ × � d ≤s � (0, 1)-matrix L such that P = NL and M′[i, j] = 0 iff P[i, j] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Now by Lemma 2, r(M′) ≤ r(P) ≤ r(N) ≤ 2d′ < 2d, which gives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 4 Testing The s-Binary Rank In this section, we present the adaptive and non-adaptive testing algorithms for s-binary rank at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We first give the adaptive algorithm and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='1 The Adaptive Tester In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Consider the tester Adaptive-Test-Rank in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The tester, at every iteration of the main While-loop (step 2) has a set X of rows of M and a set Y of columns of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If |X| ≥ |Y | (step 5), the tester first tries to extend M[X, Y ] with a new column (steps 6-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If it succeeds, it moves to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Otherwise, it tries to extend M[X, Y ] with a new row (steps 9-12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If it succeeds, it moves to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Otherwise, it tries to extend M[X, Y ] with a new row and a new column (steps 21-26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If it succeeds, it moves to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If it fails, it accepts (step 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If |X| < |Y | (step 13), it starts with the row of M[X, Y ] (steps 14-16), then the column (steps 18-20), and then both (steps 21-26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If it fails, it accepts (step 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If |X| · |Y | > � d ≤s � 2d (step 2 and then step 28) or the s-binary rank of M[X, Y ] is greater than d (step 3), then it rejects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We first prove Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let t = 9d/ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Tester Adaptive-Test-Rank makes at most 2 � d ≤s � 2dt = ˜O �� d ≤s � 2d� /ǫ queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We prove by induction that at every iteration of the main While-loop (step 2), the tester knows the entries of M[X, Y ], and the total number of queries, qX,Y , is at most 2|X||Y |t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since the While-loop condition is |X||Y | ≤ � d ≤s � 2d, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' At the beginning of the algorithm, no queries are made, and |X| = |Y | = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then 2|X||Y |t = 2t > 0 = qX,Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Suppose, at the kth iteration, the tester knows the entries of M[X, Y ] and qX,Y ≤ 2|X||Y |t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We prove the result for the (k + 1)th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We have the following cases (at the (k + 1)th iteration) Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X| ≥ |Y | (step 5) and, for some x, M[x, Y ] is a new row to M[X, Y ] (step 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 6 Adaptive-Test-Rank(d, s, M, n, m, ǫ) Input: Oracle that accesses the entries of n × m (0, 1)-matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Output: Either “Accept” or “Reject” 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' X ← {1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Y ← {1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' t = 9d/ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' While |X| · |Y | ≤ � d ≤s � 2d do 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If the s-binary rank of M[X, Y ] is greater than d, then Reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Finish ← False;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' X′ ← Ø;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Y ′ ← Ø.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' /∗ X′ and Y ′ are multi-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If |X| ≥ |Y | then 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' While (NOT Finish) AND |X′| < t 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Draw uniformly at random x ∈ [n]\\X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' X′ ← X′ ∪ {x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M[x, Y ] is a new row to M[X, Y ] then X ← X ∪ {x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Finish ← True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If (NOT Finish) then 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' While (NOT Finish) AND |Y ′| < t 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Draw uniformly at random y ∈ [m]\\Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Y ′ ← Y ′ ∪ {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M[X, y] is new column to M[X, Y ] then Y ← Y ∪ {y};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Finish ← True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Else (|X| < |Y |) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' While (NOT Finish) AND |Y ′| < t 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Draw uniformly at random y ∈ [m]\\Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Y ′ ← Y ′ ∪ {y};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M[X, y] is a new column to M[X, Y ] then Y ← Y ∪ {y};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Finish ← True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If (NOT Finish) then 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' While (NOT Finish) AND |X′| < t 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Draw uniformly at random x ∈ [n]\\X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' X′ ← X′ ∪ {x} 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M[x, Y ] is a new row to M[X, Y ] then X ← X ∪ {x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Finish ← True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' While (NOT Finish) AND X′ ̸= Ø do 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Draw uniformly at random x ∈ X′ and y ∈ Y ′ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M[x, Y ∪ {y}] is a new row to M[X, Y ∪ {y}] OR, equivalently, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' M[X ∪ {x}, y] is a new column to M[X ∪ {x}, Y ] 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' then X ← X ∪ {x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Y ← Y ∪ {y};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Finish ← True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' else X′ ← X′\\{x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Y ′ ← Y ′\\{y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If (NOT Finish) then Accept 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='Reject Figure 1: An adaptive tester for s-binary rank at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In that case, Finish becomes true, and no other sub-while-loop is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, the number of queries made at this iteration is at most |Y |t (to find all M[x, Y ]), and one element x is added to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then, the tester knows all the entries of M[X ∪ {x}, Y ] and qX∪{x},Y = qX,Y + |Y |t ≤ 2|X||Y |t + |Y |t ≤ 2|X ∪ {x}| · |Y |t, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X| ≥ |Y | (step 5), for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ] (step 8), and for some y, M[X, y] is a new column to M[X, Y ] (step 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 7 In that case, Finish becomes true, and no other sub-while-loop is executed after the second sub-while-loop (step 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, in this case, the number of queries made at this iteration is at most |Y |t + |X|t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X|t queries in the first sub-while-loop (to find M[x, Y ] for all x ∈ X′), and at most |Y |t queries in the second sub-while-loop (to find M[X, y′] for all y′ ∈ Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then one element y is added to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, the tester knows the entries of M[X, Y ∪ {y}] and, since |Y | ≤ |X|, qX,Y ∪{y} = qX,Y + |X|t + |Y |t ≤ 2|X||Y |t + 2|X|t = 2|X| · |Y ∪ {y}|t, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Case III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X| ≥ |Y |, for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ], for all y′ ∈ Y ′, M[X, y′] is not a new column to M[X, Y ], and for some x ∈ X′, y ∈ Y ′, M[x, Y ∪ {y}] is a new row to M[X, Y ∪ {y}] (step 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In this case, |X′| = |Y ′| = t, the number of queries is |X|t + |Y |t + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Exactly |X|t queries in the first sub-while-loop, |Y |t queries in the second sub-while-loop, and at most5 t queries in the sub-while-loop in step 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then one element x is added to X, and one element y is added to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then the tester knows the entries of M[X ∪ {x}, Y ∪ {y}] and qX∪{x},Y ∪{y} = qX,Y + |X|t + |Y |t + t ≤ 2|X| · |Y |t + |X|t + |Y |t + t ≤ 2|X ∪ {x}| · |Y ∪ {y}|t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Case IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X| ≥ |Y |, for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ], for all y′ ∈ Y ′, M[X, y′] is not a new column to M[X, Y ], and for all the drawn pairs x ∈ X′, y ∈ Y ′, M[x, Y ∪ {y}] is not a new row to M[X, Y ∪ {y}] (step 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In this case, Finish will have value False, and the tester accepts in step 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The analysis of the case when |X| < |Y | is similar to the above analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now prove the completeness of the tester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M is a n × m (0, 1)-matrix of s-binary rank at most d, then the tester Adaptive- Test-Rank accepts with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The tester rejects if and only if one of the following occurs, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' M[X, Y ] has s-binary rank greater than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X| · |Y | > � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M[X, Y ] has s-binary rank greater than d, then M has s-binary rank greater than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' This is because, if M = NL, then M[X, Y ] = N[X, [d]] · L[[d], Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' So item 1 cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Before we show that item 2 cannot occur, we prove the following: Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The rows (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' columns) of M[X, Y ] are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The steps in the tester where we add rows or columns are steps 8, 12 16, 20, and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In steps 8, 12 16, 20 it is clear that a row (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' column) is added only if it is a new row (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' column) to M[X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Consider step 23 and suppose, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='g |X| ≥ |Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' This step is executed only when Finish = False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' This happens when |X′| = |Y ′| = t, for every x ∈ X′, M[x, Y ] is not a new row to M[X, Y ], and for every y ∈ Y ′, M[X, y] is not a new column to M[X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then x 5This is because, for x ∈ X′, y ∈ Y ′, the tester already knows M[x, Y ] and M[X, y] from the first and second sub-while-loop and only needs to query M[x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 8 and y are added to X and Y , respectively, if M[x, Y ∪ {y}] is a new row to M[X, Y ∪ {y}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then, by Lemma 3, M[X ∪ {x}, y] is a new column to M[X ∪ {x}, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' So, the rows (and columns) in M[X ∪ {x}, Y ∪ {y}] are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' This implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Suppose, to the contrary, |X| · |Y | > � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since M′ = M[X, Y ] satisfies r(M′)c(M′) = |X| · |Y | > � d ≤s � 2d, by Lemma 4, the s-binary rank of M′, and therefore of M, is greater than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now prove the soundness of the tester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We first prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M be a n×m (0, 1)-matrix, X ⊆ [n], and Y ⊆ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Suppose there are two functions, ′ : [n] → X and ′′ : [m] → Y , such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every x ∈ [n], M[x, Y ] = M[x′, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every y ∈ [m], M[X, y] = M[X, y′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every x ∈ [n] and y ∈ [m], M[x, y] = M[x′, y′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then M has at most |X| distinct rows and |Y | distinct columns, and its s-binary rank is the s-binary rank of M[X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let x ∈ [n]\\X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every y, M[x, y] = M[x′, y′′] = M[x′, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, row x in M is equal to row x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Similarly, column y in M is equal to column y′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since adding equal columns and rows to a matrix does not change the s-binary rank6, we have brs(M[X, Y ]) = brs(M[X, [m]]) = brs(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The following Claim is proved in [7] (Claim 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Here, we give the proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M be a (0, 1)-matrix that is ǫ-far from having s-binary rank at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let X ⊆ [n] and Y ⊆ [m], such that brs(M[X, Y ]) ≤ d, the columns of M[X, Y ] are distinct, and the rows of M[X, Y ] are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then one of the following must hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The number of rows x ∈ [n] where M[x, Y ] is a new row to M[X, Y ] is at least nǫ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The number of columns y ∈ [m] where M[X, y] is a new column to M[X, Y ] is at least mǫ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The number pairs (x, y), x ̸∈ X, y ̸∈ Y , where, M[x, Y ] = M[x′, Y ] for some x′ ∈ X, M[X, y] = M[X, y′′] for some y′′ ∈ Y , and M[x, y] ̸= M[x′, y′′], is at least mnǫ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Assume, to the contrary, that none of the above statements holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Change every row x in M where M[x, Y ] is a new row to M[X, Y ] to a zero row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let X′ be the set of such rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Change every column y in M where M[X, y] is a new row to M[X, Y ] to a zero column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let Y ′ be the set of such columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every other entry (x, y), x ̸∈ X, y ̸∈ Y that is not changed to zero and M[x, y] ̸= M[x′, y′′], change M[x, y] to M[x′, y′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M′ be the matrix obtained from the above changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The number of entries (x, y) where M[x, y] ̸= M′[x, y] is less than (nǫ/3)m + (mǫ/3)n + mnǫ/3 = ǫmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, M′ is ǫ-close to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' By claim 3, brs(M′) = brs(M[[n]\\X′, [m]\\Y ′]) = brs(M[X, Y ]) ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 6If we add a column to a matrix that is equal to column y, then the rectangles that cover column y can be extended to cover the added column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 9 We now prove the completeness of the tester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M is ǫ-far from having s-binary rank d, then with probability at least 2/3, Adaptive- Test-Rank rejects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Consider the while-loop in step 2 at some iteration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If brs(M[X, Y ]) > d, then the tester rejects in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We will now show that if brs(M[X, Y ]) ≤ d, then, with probability at most 3e−2d, the tester accepts at iteration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' To this end, let brs(M[X, Y ]) ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then, by Claim 3, one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The number of rows x ∈ [n] where M[x, Y ] is a new row to M[X, Y ] is at least nǫ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The number of columns y ∈ [m] where M[X, y] is a new column to M[X, Y ] is at least mǫ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The number pairs (x, y), x ̸∈ X, y ̸∈ Y , where, M[x, Y ] = M[x′, Y ] for some x′ ∈ X, M[X, y] = M[X, y′′] for some y′′ ∈ Y , and M[x, y] ̸= M[x′, y′′], is at least mnǫ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Now at the ith iteration, suppose w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='g, |X| ≥ |Y | (the other case |Y | < |X| is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If item 1 occurs, then with probability at least p = 1 − (1 − ǫ/3)t ≥ 1 − e−2d, the tester finds a new row to M[X, Y ] and does not accept at iteration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If item 2 occurs, then if it does not find a new row to M[X, Y ], with probability at least p, the tester finds a new column to M[X, Y ] and does not accept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If item 3 occurs, and it does not find a new row or column to M[X, Y ], then with probability at least p, it finds such a pair and does not accept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, with probability at most 3(1 − p) ≤ 3e−2d, the tester accepts at iteration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since the while-loop runs at most |X| + |Y | ≤ 2|X||Y | ≤ 2 � d ≤s � 2d ≤ 22d+1 iterations, with probability at most 3e−2d22d+1 ≤ 1/3, the tester accepts in while-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, with proba- bility at least 2/3, the tester does not accept in the while-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Thus, it either rejects because brs(M[X, Y ]) > d or rejects in step 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content='2 The Non-Adaptive Tester In this section, we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' First, consider Adaptive-Test-Rank in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Consider steps 7,11,15, and 19, where it draws a new column or row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let t = 9d/ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' At each iteration of Adaptive-Test-Rank, the total number of uni- formly at random rows x ∈ [n] drawn is at most (|X| + min(|X|, |Y | − 1))t, and the number of uniformly at random rows y ∈ [m] drawn is at most (|Y | + min(|X|, |Y |))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We prove by induction that at every iteration of the main While-loop (step 2), the total number of random rows drawn by the tester, nX,Y , is at most (|X| + min(|X|, |Y | − 1))t, and the total number of random columns drawn, mX,Y , is at most (|Y | + min(|X|, |Y |))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' At the beginning, |X| = |Y | = 1, and the number of columns and rows is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In that case,7, nX,Y = 1 ≤ t and mX,Y = 1 ≤ 2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Suppose, at the kth iteration, the induction statement is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We prove the result for the (k + 1)th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' At the (k + 1)th iteration, we have the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X| ≥ |Y | (step 5) and, for some x, M[x, Y ] is a new row to M[X, Y ] (step 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 7We assume that the first column/row drawn is column/row one 10 Non-Adaptive-Test-Rank(d, s, M, n, m, ǫ) Input: Oracle that accesses the entries of (0, 1)-matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Output: Either “Accept” or “Reject”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' T ← 324·d2( d ≤s)2d ǫ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Dray uniformly at random x(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , x(T) ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Dray uniformly at random y(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , y(T) ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For every i ∈ [T] and j ∈ [T] such that i · j ≤ T 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' D[i, j] ← Query M[x(i), y(j)] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' u = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Run Adaptive-Test-Rank(d, s, M, n, m, ǫ) When the tester asks for a uniform at random x - return x(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' u ← u + 1 When the tester asks for a uniform at random y - return y(w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' w ← w + 1 When the tester makes the Query M[x(i), y(j)] - return D[i, j] Figure 2: A non-adaptive tester for s-binary rank at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In that case, Finish becomes true, and no other sub-while-loop is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, the number of rows drawn at this iteration is at most t, and one element x is added to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' No columns are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then, nX∪{x},Y ≤ nX,Y + t ≤ (|X| + min(|X|, |Y | − 1) + 1)t ≤ (|X ∪ {x}| + min(|X ∪ {x}|, |Y | − 1))t, and mX∪{x},Y = mX,Y ≤ (|Y | + min(|X|, |Y |))t ≤ (|Y | + min(|X ∪ {x}|, |Y |))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Thus, the result follows for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X| ≥ |Y | (step 5), for all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ] (step 8), and for some y, M[X, y] is a new column to M[X, Y ] (step 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In that case, Finish becomes true, and no other sub-while-loop is executed after the second sub-while-loop (step 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, in this case, the number of rows drawn at this iteration is t, one element y is added to Y , and the number of columns drawn is at most t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then nX,Y ∪{y} = nX,Y + t ≤ (|X| + min(|X|, |Y | − 1) + 1)t = (|X| + |Y |)t = (|X| + min(|X|, |Y ∪ {y}| − 1))t, and mX,Y ∪{y} ≤ mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t ≤ (|Y ∪ {y}| + min(|X|, |Y ∪ {y}|))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Thus, the result follows for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Case III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X| < |Y | (step 13), and for some y, M[X, y] is a new column to M[X, Y ] (step 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 11 In that case, Finish becomes true, and no other sub-while-loop is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, the number of columns drawn at this iteration is at most t, and one element y is added to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' No rows are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then, nX,Y ∪{y} = nX,Y ≤ (|X| + min(|X|, |Y | − 1))t ≤ (|X| + min(|X|, |Y ∪ {y}| − 1))t, and mX,Y ∪{y} ≤ mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t = (|Y ∪ {y}| + min(|X|, |Y ∪ {y}|))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Thus, the result follows for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Case IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' |X| < |Y | (step 13), for all y′ ∈ Y ′, M[X, y′] is not a new row to M[X, Y ], and for some x, M[x, Y ] is a new column to M[X, Y ] (step 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In that case, Finish becomes true, and no other sub-while-loop is executed after the fourth sub-while-loop (step 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In this case, the number of rows drawn at this iteration is t, one element x is added to X, and the number of columns drawn is at most t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then nX∪{x},Y = nX,Y + t ≤ (|X| + min(|X|, |Y | − 1) + 1)t ≤ (|X ∪ {x}| + min(|X ∪ {x}|, |Y | − 1))t mX∪{x},Y ≤ mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t = (|Y | + min(|X ∪ {x}|, |Y |))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Thus, the result follows for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Case V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' For all x′ ∈ X′, M[x′, Y ] is not a new row to M[X, Y ], for all y′ ∈ Y ′, M[X, y′] is not a new column to M[X, Y ], and for some x ∈ X′, y ∈ Y ′, M[x, Y ∪{y}] is a new row to M[X, Y ∪{y}] (step 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In this case, the number of rows drawn at this iteration is t, the number of columns drawn is t, one element x is added to X, and one element y is added to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then nX∪{x},Y ∪{y} = nX,Y + t ≤ (|X| + min(|X|, |Y | − 1) + 1)t ≤ (|X ∪ {x}| + min(|X ∪ {x}|, |Y ∪ {y}| − 1))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' mX∪{x},Y ∪{y} = mX,Y + t ≤ (|Y | + min(|X|, |Y |) + 1)t ≤ (|Y ∪ {y}| + min(|X ∪ {x}|, |Y ∪ {y}|))t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We are now ready to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' By Lemma 10, the total number of rows and columns drawn in Adaptive-Test-Rank up to iteration t is at most n′ := 9(|X| + min(|X|, |Y | − 1))d/ǫ ≤ 18|X|d/ǫ and m′ := 9(|Y | + min(|X|, |Y |)d/ǫ ≤ 18|Y |d/ǫ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We also have |X| · |Y | ≤ � d ≤s � 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' So n′ · m′ ≤ 324|X||Y |d2/ǫ2 ≤ T := 324 · d2� d ≤s � 2d ǫ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Consider the tester Non-Adaptive-Test-Rank in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The tester draws T rows x(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , x(T) ∈ [n], and columns y(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , y(T) ∈ [m] and queries all M[x(i), y(j)] where ij ≤ T and puts the 12 result in the table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then it runs Adaptive-Test-Random using the above-drawn rows and columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now show that all the queries that Adaptive-Test-Random makes can be fetched from the table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' At any iteration, the number of rows drawn is at most n′, and the number of rows drawn is at most m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, the tester needs to know (in the worst case) all the entries M[x(i), y(j)] where i ≤ n′ and j ≤ m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since ij ≤ n′m′ ≤ T, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' The number of queries that the tester makes is T � i=1 T i = O(T ln T) = ˜O �� d ≤s � 2d ǫ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 5 Testing the Exact s-Binary Rank We first prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M and M′ be n × m (0, 1)-matrices that differ in one row (or column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then |brs(M) − brs(M′)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Suppose brs(M) = d and M′ differ from M in row k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let N and L be n × d (0, 1)-matrix and d × m (0, 1)-matrix, respectively, such that P = NL, for every (i, j) ∈ [n] × [m], P[i, j] ≤ s, and P[i, j] = 0 if and only if M[i, j] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Add to N a column (as a (d + 1)th column) that all its entries are zero except the k-th entry, which equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then change N[k, j] to zero for all j ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let N ′ be the resulting matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Add to L another row (as a (d + 1)th row) equal to the k-th row of M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let L′ be the resulting matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let P ′ = N ′L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' It is easy to see that P ′[i, j] = P[i, j] for all i ̸= k and j, and the kth row of P ′ is equal to the kth row of M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then, for every (i, j) ∈ [n] × [m], P ′[i, j] ≤ s, and P ′[i, j] = 0 if and only if M′[i, j] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, brs(M′) ≤ d + 1 = brs(M) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In the same way, brs(M) ≤ brs(M′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let η = d2/(nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Let M be n × m (0, 1)-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M is ǫ-close to having s-binary rank at most d, then M is (ǫ + η)-close to having s-binary rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We will show that for every n × m (0, 1)-matrix H of s-binary rank at most d − 1, there is a n × m (0, 1)-matrix G of s-binary rank d that is η-close to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Therefore, if M is ǫ-close to having s-binary rank at most d, then it is (ǫ + η)-close to having s-binary rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Define the n×m (0, 1)-matrices Gk, k ∈ [d]∪{0}, where G0 = H and for k ≥ 1, Gk[i, j] = H[i, j] if j > k or i > d, and Gk[[d], [k]] = Id[[d], [k]] where Id is the d × d identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since Gd[[d], [d]] = Id, we have brs(Gd) ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' It is clear that for every k ∈ [d] ∪ {0}, Gk is (d2/nm)-close to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If brs(Gd) = d, then take G = Gd, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Otherwise, suppose brs(Gd) > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Now consider a sequence H = G0, G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' , Gd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' By Lemma 11, we have brs(Gi−1) − 1 ≤ brs(Gi) ≤ brs(Gi−1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Now since brs(G0) = brs(H) ≤ d − 1 and brs(Gd) > d, by the discrete intermediate value theorem, there must be k ∈ [d] such that brs(Gk) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Then take G = Gk, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Now, the tester for testing the s-binary rank d runs as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If mn < 2d2/ǫ, then find all the entries of M with mn < 2d2/ǫ queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If brs(M) = d, then accept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Otherwise, reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' 13 If mn ≥ 2d2/ǫ, then run Adaptive-Test-Rank(d, s, M, n, m, ǫ/2) (for the non-adaptive, we run Non-Adaptive-Test-Rank(d, s, M, n, m, ǫ/2)) and output its answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' We now show the correctness of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If M is of s-binary rank d, then it is of s-binary rank at most d, and the tester accepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Now, suppose f is ǫ-far from having s-binary rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If mn < 2d2/ǫ, the tester rejects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' If mn ≥ 2d2/ǫ, then, by Lemma 12, f is (ǫ − η)-far from having s-binary rank at most d, where η = d2/(nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Since η = d2/(nm) ≤ ǫ/2, the function f is (ǫ/2)-far from having s-binary rank at most d, and therefore the tester, with probability at least 2/3, rejects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' References [1] Maria-Florina Balcan, Yi Li, David P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Woodruff, and Hongyang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' Testing matrix rank, optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algo- rithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 727–746, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfPwmd/content/2301.04406v1.pdf'} +page_content=' doi:10.' metadata={'source': 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b/49AyT4oBgHgl3EQfcPf_/content/tmp_files/2301.00281v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..98d78d217c9a7ffd1d4da69c8fec73dbc72cb049 --- /dev/null +++ b/49AyT4oBgHgl3EQfcPf_/content/tmp_files/2301.00281v1.pdf.txt @@ -0,0 +1,305 @@ +arXiv:2301.00281v1 [cs.LG] 31 Dec 2022 +arXiv® 2022 (cs.LG) 1-7 +Submitted 12/22; Published 12/22 +Lightmorphic Signatures Analysis Toolkit +Dumitru Damian +dumitrudamian@yahoo.com +Information and Communication Engineering +Research and development consultant +Timis,oara, RO +Abstract +In this paper we discuss the theory used in the design of an open source lightmorphic sig- +natures analysis toolkit (LSAT). In addition to providing a core functionality, the software +package enables specific optimizations with its modular and customizable design. +To promote its usage and inspire future contributions, LSAT is publicly available. By +using a self-supervised neural network and augmented machine learning algorithms, LSAT +provides an easy-to-use interface with ample documentation. +The experiments demonstrate that LSAT improves the otherwise tedious and error- +prone tasks of translating lightmorphic associated data into usable spectrograms, enhanced +with parameter tuning and performance analysis. +With the provided mathematical functions, LSAT validates the nonlinearity encoun- +tered in the data conversion process while ensuring suitability of the forecasting algorithms. +Keywords: +lightmorphic, machine learning, spectrogram, graph chord, neural network +1. Introduction +It is common knowledge, in the machine learning domain, to use differential values, since +they provide a simple way to model the data. However, such algorithms may not fit the +lightmorphic signature properly, leading to a reduced quality of the obtained results. Train- +ing a neural network to predict the lightmorphic signature can significantly increase the data +quality. This is the task that LSAT tries to accomplish. +As such we define the lightmorphic metric learning (LML) as a branch of machine +learning algorithms, set out with the purpose of learning lightmorphic signatures from +multiple datasets trough usage of vibrating graph chords. +In the pursuing sections we describe the main features of the toolkit, explain the general +mathematical concepts and finally detail the plans regarding future functionalities. +2. General mathematical concepts +In this section we expand the mathematical concepts and link them with the reasoning +encountered in the implemented code. +We define the lightmorphic signature as a function of: light intensity (I) that varies +according to seasons and local weather conditions, trajectory distribution characteristics +©2022 Dumitru Damian. +License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/. +Typeset in LaTex using the JMLR LaTeX style file https://jmlr.org/author-info.html. + +Damian +(D), and specific adjustments (T): +fL⊙ = +I +� +1 +D +� +1 +T +� +1 +Γtζtdt +(1) +where: +• Γt – trajectory tensor +• ζt – point in time specificity +Storage of these trajectory specific lightmorphic signatures is done in a database (Θ). +The segments containing isochronous surfaces with similarities are stored in another database +(Φ) that serves as a baseline for training the neural network implementation. +The isochronous surfaces that constitute the lightmorphic signature are interlinked +trough the definition and usage of graph chords (δ(t)). Observing their vibrational am- +plitude allows the prediction of alternative lightmorphic signatures and, at the same time, +correction of the already known values. +Since the primary light source considered is the Earth’s Sun, specific spacetime metrics +(ex. gµν, ηµν, h+, h×, Gµν) have to be used in order to describe the encountered anisotropies. +These are implemented as a function of distant astrophysical forces that stretch and com- +press the fabric of spacetime. +According to special relativity, spacetime is seen as a four dimensional manifold de- +scribed by a flat Minkowski metric defined in Cartesian coordinates (t, x, y, z, c = 1) +as: +ηµν = + + + + +−1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 + + + + , +(2) +When considering the geometry of curved space, we have made use of the metric gµν, +that replaces the flat Minkowski metric ηµν. This substitution was done considering that +the geometry of curved space will eventually reduce to the flat spacetime of special relativity +at a sufficiently small scale. +The interaction between curvature of spacetime and the mass distribution was modeled +following (Blackburn (2010)) work, as: +Gµν = kTµν +(3) +where Gµν is defined as the Einstein curvature tensor, Tµν is the stress-energy tensor and +represents the mass-energy distribution, while k describes the Einstein constant of gravita- +tion defined as: +k = 8πG +c4 +(4) +where c is the speed of light in a vacuum. +2 + +Lightmorphic Signatures Analysis Toolkit +At the same time, in order to improve the results quality, the Einstein tensor was also +considered under the form: +Gµν = Rµν − 1 +2gµνR, +(5) +where Rµν is the Riemann tensor for the local spacetime, and R is the Ricci scalar. +Since there is not one general solution for the complex Einstein equations, but a large +variety of possible solutions that apply to particular circumstances, we’ve considered a weak- +field approximation, where the nonlinear Einstein equations where approximated towards +linearity. +For example, a very small perturbation specific to a gravitational wave, will impact the +flat spacetime and it is defined as hµν(x) and it’s value will be |hµν| << 1. +Thus, the Einstein equation becomes: +gµν(x) = ηµν + hµν(x). +(6) +or by simply considering the induced strain variations: +□hµν(x) = 0, +(7) +By further pursuing such linearization, we can represent in the TT gauge, a propagating +wave, under the following form: +hTT +µν = + + + + +0 +0 +0 +0 +0 +h+ +h× +0 +0 +h× +−h+ +0 +0 +0 +0 +0 + + + + , +(8) +where the constant amplitudes (h+, h×) represent the two gravitational wave polariza- +tions, the plus- and cross-polarization. +We represent the distance between two neighboring points as defined by (Berit (2013)) +for a flat spacetime, trough the following expression: +ds2 = −c2dt2 + dx2 + dy2 + dz2 = −c2dt2 + [1 + h+(t)]dx2 + [1 − h+(t)]dy2 +That allows us to model in the TT gauge, the gravitational wave stretching along the x +axis and compression along the y axis with the specific factor of: +� +1 ± h+(t) ≃ 1 + 1 +2h+(t) +Having modeled the photon’s traveling path in outer space, in order to simplify the +inherent path inhomogeneities, we separated the domains into outer space domain, atmo- +spheric domain and Earth specific domains (lithosphere, hydrosphere, biosphere, noises, +etc). +We further define the phase of an electromagnetic wave of frequency ω0 as φ. Following +Driggers (2015)’s work, we consider that the starting light phase is at 0 and it travels at +the speed of light c. After a distance L it will have a phase δφspace that can be expressed as +a distance integral over the spacetime metric, +δφspace = ω0 +c +� L +0 +gdx, +(9) +3 + +Damian +with g(t) = η+h(t), where η is the Minkowski metric and h(t) is the dimensionless spacetime +strain. +Summing the light phase shift δφatm and the δφEarth which is derived from the noise +sources like seismic or electromagnetic interferences, leads to the dataset of trajectory spe- +cific lightmorphic signatures: +ΦΓIDT = +N +� +j=1 +Γj +IDT +(10) +The signature parameter estimation is performed considering a prior distribution p(Φ|L⊙) +that is updated upon receiving the new data d to give a posterior distribution p(Φ|d, L⊙) +p(Φ|d, L⊙) = p(Φ|L⊙)p(d|Φ, L⊙) +p(d|L⊙) +(11) +While observing the distribution of multiple light segments within the dataset ΦΓIDT , +it will be possible to estimate the probability for trajectory specific lightmorphic evolution: +pΦ = f(ρk · pΦk) +(12) +where pΦk is the database’s k-th segment specific probability, ρk is the prediction weight +for the k-th segment. +3. Software package design +The distribution matrices specific to the isochronous segmentation surfaces, which define +the lightmorphic signature model, form the LSAT core. +As such we’ve used a design principle that ensures simplicity for the whole package, +while making the source codes easy to read and maintain. As the toolkit is written in a +modular way, new functionalities can be easily plugged in. This makes the LSAT not only +a lightmorphic signature machine learning tool but also an experimental platform. +LSAT comes with plenty of documentation for all the interface functionalities and related +data structures. The README file describes the installation process and interface usage. +For developers who use the toolkit in their applications, the API documentation can provide +additional information related to functionality calls. +4. Practical Usage +In the examples, we provide sample values for the lightmorphic signature updates, as a +function of δφatm derived by the neural network from the values of a large dataset of at- +mospheric meteorological data for 317 cities in Romania, with hundreds of thousands data +points. +Automatic learning is supported trough API calls to the domain specific data +providers. +Beyond this simple way of running the lightmorphic signatures analysis toolkit, there are +several enhancement options for advanced usage. As example, one may activate additional +functionalities that consider input parameters like complex space weather forecasting, dif- +ferent electromagnetic wave disturbances or lithosphere, hydrosphere and biosphere specific +localized data. +4 + +Lightmorphic Signatures Analysis Toolkit +5. Conclusion and Future Work +With the lightmorphic signatures analysis toolkit we provided an open source SW package +that is simple and easy-to-use. +Experiments and analysis conclude that the modular design and customization support +are performing excellent in practice and can provide the base for additional research on +lightmorphic signatures. +The toolkit is constantly being improved by new research results and user feed-back +with the ultimate goal of having an automated toolkit to use in maintaining and updating +a large database of high-quality light signatures. +Future work will focus on probability estimates, additional functionalities that mitigate +the large uncertainties in the available observational input data which arise from the complex +interaction processes. In addition, the inclusion of artificial intelligence (AI) options will +be considered while building a national/international network for lightmorphic signature +analysis. +5 + +Damian +6. 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Romano and Neil J. Cornish. Detection methods for stochastic gravitational- +wave backgrounds: a unified treatment. Living Rev. Rel., 20(1):2, 2017. doi: 10.1007/ +s41114-017-0004-1. +Michael P. Ross. Precision Mechanical Rotation Sensors for Terrestrial Gravitational Wave +Observatories. PhD thesis, University of Washington, 2020. +6 + +Lightmorphic Signatures Analysis Toolkit +Darkhan Tuyenbayev. Extending the scientific reach of Advanced LIGO by compensating +for temporal variations in the calibration of the detectors. PhD thesis, The University of +Texas at San Antonio, 2017. +Madeline Wade. Gravitational-Wave Science with the Laser Interferometer Gravitational- +Wave Observatory. PhD thesis, University of Wisconsin–Milwaukee, 2015. +7 + diff --git a/49AyT4oBgHgl3EQfcPf_/content/tmp_files/load_file.txt b/49AyT4oBgHgl3EQfcPf_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d20bf6dc17941da175d942ce88a5c3ad8944bbbb --- /dev/null +++ b/49AyT4oBgHgl3EQfcPf_/content/tmp_files/load_file.txt @@ -0,0 +1,160 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf,len=159 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content='00281v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content='LG] 31 Dec 2022 arXiv® 2022 (cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content='LG) 1-7 Submitted 12/22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Published 12/22 Lightmorphic Signatures Analysis Toolkit Dumitru Damian dumitrudamian@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content='com Information and Communication Engineering Research and development consultant Timis,oara, RO Abstract In this paper we discuss the theory used in the design of an open source lightmorphic sig- natures analysis toolkit (LSAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' In addition to providing a core functionality, the software package enables specific optimizations with its modular and customizable design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' To promote its usage and inspire future contributions, LSAT is publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' By using a self-supervised neural network and augmented machine learning algorithms, LSAT provides an easy-to-use interface with ample documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' The experiments demonstrate that LSAT improves the otherwise tedious and error- prone tasks of translating lightmorphic associated data into usable spectrograms, enhanced with parameter tuning and performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' With the provided mathematical functions, LSAT validates the nonlinearity encoun- tered in the data conversion process while ensuring suitability of the forecasting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Keywords: lightmorphic, machine learning, spectrogram, graph chord, neural network 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Introduction It is common knowledge, in the machine learning domain, to use differential values, since they provide a simple way to model the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' However, such algorithms may not fit the lightmorphic signature properly, leading to a reduced quality of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Train- ing a neural network to predict the lightmorphic signature can significantly increase the data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' This is the task that LSAT tries to accomplish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' As such we define the lightmorphic metric learning (LML) as a branch of machine learning algorithms, set out with the purpose of learning lightmorphic signatures from multiple datasets trough usage of vibrating graph chords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' In the pursuing sections we describe the main features of the toolkit, explain the general mathematical concepts and finally detail the plans regarding future functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' General mathematical concepts In this section we expand the mathematical concepts and link them with the reasoning encountered in the implemented code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' We define the lightmorphic signature as a function of: light intensity (I) that varies according to seasons and local weather conditions, trajectory distribution characteristics ©2022 Dumitru Damian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' License: CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content='0, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content='0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Typeset in LaTex using the JMLR LaTeX style file https://jmlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content='org/author-info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Damian (D), and specific adjustments (T): fL⊙ = I � 1 D � 1 T � 1 Γtζtdt (1) where: Γt – trajectory tensor ζt – point in time specificity Storage of these trajectory specific lightmorphic signatures is done in a database (Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' The segments containing isochronous surfaces with similarities are stored in another database (Φ) that serves as a baseline for training the neural network implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' The isochronous surfaces that constitute the lightmorphic signature are interlinked trough the definition and usage of graph chords (δ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Observing their vibrational am- plitude allows the prediction of alternative lightmorphic signatures and, at the same time, correction of the already known values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Since the primary light source considered is the Earth’s Sun, specific spacetime metrics (ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' gµν, ηµν, h+, h×, Gµν) have to be used in order to describe the encountered anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' These are implemented as a function of distant astrophysical forces that stretch and com- press the fabric of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' According to special relativity, spacetime is seen as a four dimensional manifold de- scribed by a flat Minkowski metric defined in Cartesian coordinates (t, x, y, z, c = 1) as: ηµν = \uf8eb \uf8ec \uf8ec \uf8ed −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , (2) When considering the geometry of curved space, we have made use of the metric gµν, that replaces the flat Minkowski metric ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' This substitution was done considering that the geometry of curved space will eventually reduce to the flat spacetime of special relativity at a sufficiently small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' The interaction between curvature of spacetime and the mass distribution was modeled following (Blackburn (2010)) work, as: Gµν = kTµν (3) where Gµν is defined as the Einstein curvature tensor, Tµν is the stress-energy tensor and represents the mass-energy distribution, while k describes the Einstein constant of gravita- tion defined as: k = 8πG c4 (4) where c is the speed of light in a vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' 2 Lightmorphic Signatures Analysis Toolkit At the same time, in order to improve the results quality, the Einstein tensor was also considered under the form: Gµν = Rµν − 1 2gµνR, (5) where Rµν is the Riemann tensor for the local spacetime, and R is the Ricci scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Since there is not one general solution for the complex Einstein equations, but a large variety of possible solutions that apply to particular circumstances, we’ve considered a weak- field approximation, where the nonlinear Einstein equations where approximated towards linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' For example, a very small perturbation specific to a gravitational wave, will impact the flat spacetime and it is defined as hµν(x) and it’s value will be |hµν| << 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Thus, the Einstein equation becomes: gµν(x) = ηµν + hµν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' (6) or by simply considering the induced strain variations: □hµν(x) = 0, (7) By further pursuing such linearization, we can represent in the TT gauge, a propagating wave, under the following form: hTT µν = \uf8eb \uf8ec \uf8ec \uf8ed 0 0 0 0 0 h+ h× 0 0 h× −h+ 0 0 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , (8) where the constant amplitudes (h+, h×) represent the two gravitational wave polariza- tions, the plus- and cross-polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' We represent the distance between two neighboring points as defined by (Berit (2013)) for a flat spacetime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' trough the following expression: ds2 = −c2dt2 + dx2 + dy2 + dz2 = −c2dt2 + [1 + h+(t)]dx2 + [1 − h+(t)]dy2 That allows us to model in the TT gauge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' the gravitational wave stretching along the x axis and compression along the y axis with the specific factor of: � 1 ± h+(t) ≃ 1 + 1 2h+(t) Having modeled the photon’s traveling path in outer space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' in order to simplify the inherent path inhomogeneities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' we separated the domains into outer space domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' atmo- spheric domain and Earth specific domains (lithosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' hydrosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' biosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' noises,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' We further define the phase of an electromagnetic wave of frequency ω0 as φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Following Driggers (2015)’s work, we consider that the starting light phase is at 0 and it travels at the speed of light c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' After a distance L it will have a phase δφspace that can be expressed as a distance integral over the spacetime metric, δφspace = ω0 c � L 0 gdx, (9) 3 Damian with g(t) = η+h(t), where η is the Minkowski metric and h(t) is the dimensionless spacetime strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Summing the light phase shift δφatm and the δφEarth which is derived from the noise sources like seismic or electromagnetic interferences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' leads to the dataset of trajectory spe- cific lightmorphic signatures: ΦΓIDT = N � j=1 Γj IDT (10) The signature parameter estimation is performed considering a prior distribution p(Φ|L⊙) that is updated upon receiving the new data d to give a posterior distribution p(Φ|d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' L⊙) p(Φ|d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' L⊙) = p(Φ|L⊙)p(d|Φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' L⊙) p(d|L⊙) (11) While observing the distribution of multiple light segments within the dataset ΦΓIDT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' it will be possible to estimate the probability for trajectory specific lightmorphic evolution: pΦ = f(ρk · pΦk) (12) where pΦk is the database’s k-th segment specific probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' ρk is the prediction weight for the k-th segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Software package design The distribution matrices specific to the isochronous segmentation surfaces, which define the lightmorphic signature model, form the LSAT core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' As such we’ve used a design principle that ensures simplicity for the whole package, while making the source codes easy to read and maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' As the toolkit is written in a modular way, new functionalities can be easily plugged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' This makes the LSAT not only a lightmorphic signature machine learning tool but also an experimental platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' LSAT comes with plenty of documentation for all the interface functionalities and related data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' The README file describes the installation process and interface usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' For developers who use the toolkit in their applications, the API documentation can provide additional information related to functionality calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Practical Usage In the examples, we provide sample values for the lightmorphic signature updates, as a function of δφatm derived by the neural network from the values of a large dataset of at- mospheric meteorological data for 317 cities in Romania, with hundreds of thousands data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Automatic learning is supported trough API calls to the domain specific data providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Beyond this simple way of running the lightmorphic signatures analysis toolkit, there are several enhancement options for advanced usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' As example, one may activate additional functionalities that consider input parameters like complex space weather forecasting, dif- ferent electromagnetic wave disturbances or lithosphere, hydrosphere and biosphere specific localized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' 4 Lightmorphic Signatures Analysis Toolkit 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Conclusion and Future Work With the lightmorphic signatures analysis toolkit we provided an open source SW package that is simple and easy-to-use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Experiments and analysis conclude that the modular design and customization support are performing excellent in practice and can provide the base for additional research on lightmorphic signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' The toolkit is constantly being improved by new research results and user feed-back with the ultimate goal of having an automated toolkit to use in maintaining and updating a large database of high-quality light signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Future work will focus on probability estimates, additional functionalities that mitigate the large uncertainties in the available observational input data which arise from the complex interaction processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' In addition, the inclusion of artificial intelligence (AI) options will be considered while building a national/international network for lightmorphic signature analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' 5 Damian 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' References References Rana Adhikari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Sensitivity and noise analysis of 4 km laser interferometric gravitational wave antennae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' PhD thesis, Massachusetts Institute of Technology, 2004.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' PhD thesis, University of Washington, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' 6 Lightmorphic Signatures Analysis Toolkit Darkhan Tuyenbayev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Extending the scientific reach of Advanced LIGO by compensating for temporal variations in the calibration of the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' PhD thesis, The University of Texas at San Antonio, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Madeline Wade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' Gravitational-Wave Science with the Laser Interferometer Gravitational- Wave Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' PhD thesis, University of Wisconsin–Milwaukee, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfcPf_/content/2301.00281v1.pdf'} diff --git a/4NFAT4oBgHgl3EQfExwU/content/tmp_files/2301.08423v1.pdf.txt b/4NFAT4oBgHgl3EQfExwU/content/tmp_files/2301.08423v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..24f6635b6e88c768a64748cc809a2a9954b235aa --- /dev/null +++ b/4NFAT4oBgHgl3EQfExwU/content/tmp_files/2301.08423v1.pdf.txt @@ -0,0 +1,6495 @@ +Under consideration for publication in J. Fluid Mech. +1 +Adjoint-based variational optimal mixed models for +large-eddy simulation of turbulence +Zelong Yuan, Yunpeng Wang, Xiaoning Wang and Jianchun Wang† +1Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, +Shenzhen 518055, People’s Republic of China +2Guangdong–Hong Kong–Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering +Applications, Southern University of Science and Technology, Shenzhen 518055, People’s Republic of +China +(Received xx; revised xx; accepted xx) +An adjoint-based variational optimal mixed model (VOMM) is proposed for subgrid-scale (SGS) +closure in large-eddy simulation (LES) of turbulence. The stabilized adjoint LES equations are +formulated by introducing a minimal regularization to address the numerical instabilities of +the long-term gradient evaluations in chaotic turbulent flows. The VOMM model parameters +are optimized by minimizing the discrepancy of energy dissipation spectra between LES +calculations and a priori knowledge of direct numerical simulation (DNS) using the gradient- +based optimization. The a posteriori performance of the VOMM model is comprehensively +examined in LES of three turbulent flows, including the forced homogeneous isotropic turbulence, +decaying homogenous isotropic turbulence, and temporally evolving turbulent mixing layer. The +VOMM model outperforms the dynamic Smagorinsky model (DSM), dynamic mixed model +(DMM) and approximate deconvolution model (ADM) in predictions of various turbulence +statistics, including the velocity spectrum, structure functions, statistics of velocity increments +and vorticity, temporal evolutions of the turbulent kinetic energy, dissipation rate, momentum +thickness and Reynolds stress, as well as the instantaneous vortex structures at different grid +resolutions and times. In addition, the VOMM model only takes up 30% time of the DMM model +for all flow scenarios. These results demonstrate that the proposed VOMM model improves the +numerical stability of LES and has high a posteriori accuracy and computational efficiency by +incorporating the a priori information of turbulence statistics, highlighting that the VOMM model +has a great potential to develop advanced SGS models in the LES of turbulence. +Key words: subgrid-scale model, variational optimal models, adjoint-based optimization, large- +eddy simulation, incompressible turbulence +1. Introduction +Large-eddy simulation (LES) has become an effective tool for the investigation of turbulent +flows, and has been widely applied to many industrial problems including the aeroacoustics, +combustions, meteorological physics, interfacial mixing, etc (Sagaut 2006; Garnier et al. 2009). +The dominant large-scale motions of turbulence are directly resolved by the LES, leaving the +effects of residual subgrid scales (SGS) on the resolved large scales modeled by the SGS models +(Lesieur & Metais 1996; Meneveau & Katz 2000). In contrast, direct numerical simulation (DNS) +of turbulence requires a sufficiently high mesh resolution to fully resolve all flow scales down +† Email address for correspondence: wangjc@sustech.edu.cn +arXiv:2301.08423v1 [physics.flu-dyn] 20 Jan 2023 + +2 +to the size of the Kolmogorov eddies, whose computational cost is prohibitively expensive at a +high Reynolds number (Pope 2000). Therefore, LES is much more computationally efficient than +the DNS by significantly reducing the degrees of freedom of turbulence, meanwhile accurately +reconstructing large-scale flow structures (Pope 2000; Sagaut 2006; Durbin 2018). +The modeling of the unclosed SGS stress is crucial for the accuracy of predictions in LES. SGS +models can be generally categorized into functional models, structural models and mixed models +(Sagaut 2006; Garnier et al. 2009). The functional SGS models utilize the explicit dissipative +terms to correctly reconstruct the forward kinetic energy cascade from large scales to small +scales (Rozema et al. 2015; Abkar et al. 2016). The Smagorinsky model is one of the most +popular functional SGS models and is favored for its substantial numerical stability and excellent +robustness of LES calculations (Smagorinsky 1963; Lilly 1967). However, the functional SGS +models generally exhibit excessive dissipation and fail to predict the sophisticated small-scale +flow structures. In contrast, the structural SGS models recover the unclosed SGS stress with +high a priori accuracy by exactly truncating the Taylor series expansions or the assumption of +scale similarity. These structural models include the approximate deconvolution method (Stolz +& Adams 1999; Stolz et al. 2001), scale-similarity model (Bardina et al. 1980; Liu et al. 1994), +velocity gradient model (Clark et al. 1979), etc. The structural SGS models can accurately capture +the spatial distribution of SGS energy flux and backscatter of the kinetic energy, but suffer from +the numerical instability without sufficient SGS dissipation in the a posteriori studies of LES. +The mixed models consist of the structural models and functional eddy-viscosity models +to balance the numerical stability and accuracy of LES and compensate their inherent model +deficiencies. The Clark model combines the velocity gradient model with the Smagorinsky eddy +viscosity (Clark et al. 1979). Erlebacher et al. (1992) proposed a mixed model which consists of +the scale-similarity model and the dissipative Smagorinsky term. In the early stage, the SGS model +parameters were either theoretically derived from the isotropic turbulent flows (Lilly 1967) or +estimated by the a priori analysis of DNS and experimental observations (Deardorff 1970; Clark +et al. 1979), yielding poor predictions in the a posteriori LES (Lesieur & Metais 1996; Meneveau +& Katz 2000). A pioneering dynamical procedure with the Germano identity was developed +to determine the Smagorinsky coefficient adaptively by the least-squares algorithm (Germano +et al. 1991; Lilly 1992). Subsequently, the dynamic versions of mixed models were successively +proposed, including the one-parameter (Zang et al. 1992) and two-parameter dynamical mixed +models (Liu et al. 1994; Shi et al. 2008), the dynamic Clark model (Vreman et al. 1994) and +dynamic ADM model (Habisreutinger et al. 2007), etc. The coefficients of a general multi- +parameter dynamic mixed model (DMM) can be conveniently determined by the Germano- +identity-based dynamic approach (Sagaut et al. 2000). However, extensive previous studies +have shown that these DMM models are excessively dissipative in the transitional regions, but +underestimate the SGS dissipation in situations of coarse mesh resolutions and grid anisotropy +(Meneveau & Katz 2000; Moser et al. 2021). In addition, the dissipative Smagorinsky part in the +DMM models is usually dominant over the structural part, leading to little advantage in the high +a priori accuracy of structural models. The basis tensors of the DMM model, comprising the +functional eddy-viscosity and the accurate structural part, give a complete representation of the +SGS stress and SGS energy flux (SGS dissipation), which is essential for the SGS modeling of +LES. Yuan et al. (2022) preliminarily explored a scale-similarity dynamic procedure (SSD) with +a dynamic nonlinear algebraic model, yielding more accurate predictions of various turbulence +statistics and instantaneous vortex structures for both a priori and a posteriori analyses of LES than +the Germano-identity-based dynamic (GID) approach in the homogeneous isotropic turbulence. +However, the SSD procedure still suffers from the numerical instability at coarse-grid-resolution +cases, where the spatial discretization error dominates the SGS modeling error. It might be +challenging to develop a general dynamic framework for the model coefficient determination +at various grid resolutions applicable to different types of turbulence problems. These results + +3 +demonstrate that the adjustment of SGS model parameters can effectively improve the accuracy +of SGS modeling and enhance the predictions of LES. +Besides, additional artificial viscous or penalized regularization terms have been also in- +troduced to enhance the a posteriori stability of structural models. A secondary filtering +regularization technique was proposed by Stolz et al. (2001) and Adams et al. (2004) to +maintain the numerical stability of ADM models. Vollant et al. (2016) efficiently regularized +the velocity gradient model by dynamically clipping the SGS backscatter. A spectral-vanishing- +viscosity method (Tadmor 1989) was proposed to effectively suppress the Gibbs oscillations at +high wavenumbers (Cerutti et al. 2000) and has been successfully applied to the prediction of +turbulent channel flows (Karamanos & Karniadakis 2000). Xie et al. (2020a) used a hyperviscosity +term to address the stability issue of the spatial-artificial-neural-network models. The effective +hyperviscosity term was further applied to other data-driven SGS models (Yuan et al. 2020; Wang +et al. 2021). Yuan et al. (2021b) developed a small-scale eddy-viscosity model to enhance the +a posteriori stability of dynamic iterative approximate deconvolution models, without affecting +the accurate predictions of large-scale flow structures. A kinetic-energy-flux constrained SGS +model proposed by Yu et al. (2022) regularizes the DSM model by the correct kinetic energy flux +approximated by the tensor-diffusivity model and accurately predicts the transition to turbulence +of a compressible flat-plate boundary layer. It is noteworthy that additional numerical parameters +would be introduced for most regularization techniques, which are sensitive to the grid resolution +of LES, requiring multiple tedious testings for different turbulence scenarios. To our knowledge, +there might not be a unified adaptive regularization framework proposed for the stability of +structural SGS models that can be universally applied to various types of turbulence with +different grid resolutions of LES calculations. The dependence of SGS model parameters on +grid resolutions of LES might be effectively addressed by incorporating the a priori knowledge +of DNS or experimental observations. +In recent years, many data-driven closure approaches (Tracey et al. 2015; Ling et al. 2016a; +Xiao et al. 2016; Maulik & San 2017; Wang et al. 2018; Zhou et al. 2019; Yang et al. 2019; Park +& Choi 2021; Guan et al. 2022) have been extensively developed to improve the modeling of +unclosed terms in turbulence, as more high-fidelity DNS or experimental data become available +(Kutz 2017; Duraisamy et al. 2019). Ling et al. (2016b) proposed a representative tensor-basis- +neural-network (TBNN) model with the multiplicative layer that predicts coefficients of the +basis tensors for the modeled Reynolds stress by taking velocity invariants as input to preserve +Galilean invariance. The TBNN architecture can accurately reconstruct the anisotropy of Reynolds +stress and predict the flow separation better than the baseline linear or nonlinear eddy-viscosity +model. Xie et al. (2020c) further developed the artificial-neural-network-based nonlinear algebraic +models yielding better predictions of LES statistics than classical dynamic SGS models. The gene- +expression-programming technique was proposed to acquire the explicit mathematical expression +of the unclosed SGS stress modeled by basis functions for LES using an evolutionary algorithm +(Schoepplein et al. 2018; Li et al. 2021; Wu et al. 2022). The multi-agent reinforcement-learning +framework was developed to discover Smagorinsky model coefficients using the control policy +rewarded by the statistical discrepancy of energy spectrum (Novati et al. 2021; Kurz et al. 2023), +and further applied to modeling the near-wall dynamics (Bae & Koumoutsakos 2022). +Although the machine-learning-based closure models can improve the a priori accuracy of +turbulence models fairly well, they have been reported to suffer from the ill-conditioned issues +in the a posteriori studies. The small a priori errors of the modeled Reynolds stress can be +significantly amplified and then propagated into the mean velocity field in the a posteriori +testings (Wu et al. 2019). Gamahara & Hattori (2017) established an artificial-neural-network +framework for the SGS closures of turbulent channel flows, which accurately predicts the unclosed +SGS stress in a priori studies, but shows no obvious advantages over the Smagorinsky model in +the reconstruction of the mean velocity profiles. The recurrent neural network was employed to + +4 +learn the coarse-grained discretization errors of LES and expected to construct the perfect LES +formulation (Beck et al. 2019). However, these perfect SGS closure terms also encounter serious +a posteriori instability issues, even though the a priori predictions show high correlations with +the exact unclosed terms (Beck et al. 2019). These results indicate that most current data-driven +closure approaches can acquire sufficiently high a priori accuracy after being trained by the +high-fidelity DNS or experimental data, but still lack indispensable extrapolation capabilities and +are difficult to be applied to the a posteriori testings of out-of-sampling flow scenarios. +The data-assimilation techniques can effectively remedy the deficiencies of insufficient a +posteriori accuracy of closure models by iteratively evaluating and minimizing the discrep- +ancies between coarse-grained a posteriori calculations and benchmark high-fidelity DNS or +experimental observations. The data-assimilation approaches can be generally classified into +three categories: ensemble-based statistical methods (Colburn et al. 2011; Zhang et al. 2022), +adjoint-based variational approaches (Bewley et al. 2001; Delport et al. 2009; Badreddine et al. +2014) and their mixed variants (Mons et al. 2021). The ensemble-based statistical techniques +use ensemble statistics to approximately measure the model uncertainty and continuously correct +the measurement errors of observations by the classical Kalman-filtering strategies or nudging +methods (Clark Di Leoni et al. 2020). These statistical assimilation methods allow the convenient +inference of flow states and statistics, without any detailed information of dynamical systems, +facilitating their wide application in complex practical scenarios. However, the state estimations +of these ensemble-based approaches frequently evaluate the matrix multiplication and inverse +operations, resulting in the massive computation expense and large memory usage for the high +degree-of-freedom turbulence problems at a high Reynolds number. In contrast, the adjoint- +based variational techniques employ the optimal control strategy to efficiently optimize the +model parameters or state variables by minimizing the discrepancies between the benchmark +observations and a posteriori predictions. Singh & Duraisamy (2016) proposed a field-inversion +procedure to infer model discrepancies in the source terms of Reynolds-averaged Navier–Stokes +(RANS) transport equations using Bayesian posterior estimation. He et al. (2018) simplified the +field-inversion strategy and employed the continuous adjoint formulation to optimize a spatially +varying turbulence production term in the Spalart–Allmaras model of RANS equations. +In comparison with the extensive studies of data-assimilation-based RANS models (Kato & +Obayashi 2013; Kato et al. 2015; Xiao et al. 2016; Li et al. 2017; Xiao & Cinnella 2019), +investigations on SGS models of LES assimilated with high-fidelity simulation data are still +preliminary. A spatially-varying parameter in a local uncertainty model and initial conditions +were optimized based on experimental observations of the cylindrical wake flow using the discrete +adjoint algorithm (Chandramouli et al. 2020). Mons et al. (2021) developed a non-intrusive +ensemble-variational approach (EnVar) to enhance the predictions of the mean flow and Reynolds +stresses by adjusting the wall-normal distribution of the Smagorinsky coefficient or injecting an +artificial steady force in the LES momentum equations. The SGS force modeled by the artificial +neural network was optimized by the point-to-point errors of the filtered velocity field using +the discrete adjoint method for the decaying isotropic turbulence and plane jet flows (Sirignano +et al. 2020; MacArt et al. 2021). However, these discrete adjoint or ensemble-based variational +methods require massive matrix operations with significant memory usage. +In this paper, a variational optimal mixed model (VOMM) is proposed to reconstruct the +unclosed SGS stress by assimilating the turbulence statistics of high-fidelity filtered DNS data +using the continuous adjoint approach. The main difference from the previous work is that we +derive adjoint LES equations with the general SGS model and conduct the energy budget analysis +of adjoint equations. The continuous adjoint algorithm can enhance the physical understanding of +the adjoint-based sensitivities and provide flexibility in selecting the discretization scheme for the +adjoint equations. The quadratic terms of shear strain rate in adjoint LES equations turn out to be +responsible for the exponential temporal growth of the adjoint-based gradients, giving rise to the + +5 +numerical divergence in a long time horizon for the chaotic turbulent flows. Hence, the stabilized +adjoint LES equations are correspondingly formulated to enhance the numerical stability of the +adjoint LES calculations. To the extent of the authors’ knowledge, few previous studies have given +detailed derivations of the adjoint LES equations with general SGS mixed models and formulated +the stabilized version for long-term gradient evaluations. In addition, the selected cost functional is +essential for the convergence and performance of adjoint-based gradient optimizations. Compared +to the previous studies, turbulence statistical discrepancies rather than the chaotic point-to-point +prediction errors are adopted to quantify the multiscale statistical behaviours of turbulence. +The a priori information about statistics of turbulence acquired from experimental data or DNS +results, including energy spectra, structure functions, and probability density functions of physical +quantities, can be used to determine or correct SGS model parameters to improve the a posteriori +accuracy of LES greatly. Turbulent statistical assimilation can effectively alleviate the impact of +chaotic field observations on the performance of data assimilation. Furthermore, the a posteriori +performance of VOMM model is comprehensively investigated and compared to classical SGS +models at multiple grid resolutions in different turbulence scenarios, including the forced and +decaying homogeneous isotropic turbulence, as well as the temporally evolving turbulent mixing +layer. +The remainder of this paper is structured as follows. Sec. 2 describes the governing equations +of the large-eddy simulation. The conventional subgrid-scale models, including DSM, DMM +and ADM models, are briefly introduced in Sec. 3. In Sec. 4, we first derive the adjoint LES +equations with a general form of mixed SGS models, then conduct the energy budget analysis of +adjoint equations, and correspondingly propose the stabilized adjoint LES equations. Afterwards, +the adjoint-based variational optimal mixed model is developed. Sec. 5 further investigates the +a posteriori performance of the VOMM model in comparison to the classical SGS models for +three turbulent flow scenarios, including the forced homogeneous isotropic turbulence, decaying +homogeneous isotropic turbulence, and temporally evolving turbulent mixing layer. Conclusions +are finally drawn in Sec. 6. +2. Governing equations of the large-eddy simulation +The three dimensional incompressible turbulence is governed by the Navier-Stokes equations +(Pope 2000), namely +𝜕𝑢𝑖 +𝜕𝑥𝑖 += 0, +(2.1) +𝜕𝑢𝑖 +𝜕𝑡 + 𝜕 �𝑢𝑖𝑢 𝑗 +� +𝜕𝑥 𝑗 += − 𝜕𝑝 +𝜕𝑥𝑖 ++ 𝜈 𝜕2𝑢𝑖 +𝜕𝑥 𝑗𝜕𝑥 𝑗 ++ F𝑖, +(2.2) +where 𝑢𝑖 is the 𝑖-th component of velocity, 𝑝 denotes the pressure divided by the constant density, +𝜈 is the kinematic viscosity, and F𝑖 represents the large-scale forcing on the fluid momentum +in the 𝑖-th coordinate direction. The summation convection for the repeated indices is adopted +by default for simplicity in this paper. Besides, the dimensionless governing parameter for the +incompressible turbulence, namely, the Taylor microscale Reynolds number 𝑅𝑒𝜆 is defined as +(Pope 2000) +𝑅𝑒𝜆 = 𝑢rms𝜆 +√ +3𝜈 +, +(2.3) +where 𝑢rms = +√︁ +⟨𝑢𝑖𝑢𝑖⟩ represents the root-mean-square (rms) value of the velocity magnitude, +and ⟨·⟩ represents a spatial average along the homogeneous direction (i.e., average over the entire +domain for the isotropic turbulence and the horizontal average for the temporally evolving mixing + +6 +layer). Here, 𝜆 = 𝑢rms√︁ +5𝜈/𝜀 is the Taylor microscale, where 𝜀 = 2𝜈 +� +𝑆𝑖 𝑗𝑆𝑖 𝑗 +� +represents the +average dissipation rate and 𝑆𝑖 𝑗 = 1 +2 +�𝜕𝑢𝑖/𝜕𝑥 𝑗 + 𝜕𝑢 𝑗/𝜕𝑥𝑖 +� denotes the strain-rate tensor. +To obtain the governing equations of the large-eddy simulation, a spatial filtering operation, +¯𝑓 (x) = +∫ +Ω +𝑓 (x′) 𝐺 �x − x′; ¯Δ� 𝑑x′ is applied to the Navier-Stokes equations. Here, an overbar +denotes the spatial filtering, Ω is the entire domain. 𝐺 and ¯Δ are the filter kernel and filter width, +respectively. The governing equations for the LES can be correspondingly derived as (Sagaut +2006) +𝜕 ¯𝑢𝑖 +𝜕𝑥𝑖 += 0, +(2.4) +𝜕 ¯𝑢𝑖 +𝜕𝑡 + 𝜕 � ¯𝑢𝑖 ¯𝑢 𝑗 +� +𝜕𝑥 𝑗 += − 𝜕 ¯𝑝 +𝜕𝑥𝑖 +− 𝜕𝜏𝑖 𝑗 +𝜕𝑥 𝑗 ++ 𝜈 𝜕2 ¯𝑢𝑖 +𝜕𝑥 𝑗𝜕𝑥 𝑗 ++ ¯F𝑖. +(2.5) +Here, the unclosed SGS stress tensor 𝜏𝑖 𝑗 = 𝑢𝑖𝑢 𝑗 − ¯𝑢𝑖 ¯𝑢 𝑗 cannot be directly calculated using the +resolved variables ¯𝑢𝑖, and additional SGS stress modeling is required to make the LES equations +solvable. +3. Conventional subgrid-scale models for LES +The SGS models aim to establish the approximate constitutive equation for SGS unclosed +terms using the known resolved variables, and reconstruct the nonlinear interactions between +the resolved large scales and unsolved small scales as accurately as possible (Moser et al. 2021; +Johnson 2022). The explicit SGS models consist of the functional and structural models. The +functional modeling adopts the eddy-viscosity forms to mimic the forward kinetic energy transfer +from the resolved large scales to the residual small scales, while the structural models can +accurately recover the unclosed SGS stress by the hypothesis of scale similarity or using the +truncated series expansions with high a priori accuracy (Sagaut 2006; Fowler et al. 2022). One +of the most widely-used functional models is the Smagorinsky model (Smagorinsky 1963; Lilly +1967), expressed as +𝜏𝐴 +𝑖 𝑗 = 𝜏𝑖 𝑗 − 𝛿𝑖 𝑗 +3 𝜏𝑘𝑘 = −2𝐶2 +𝑆 ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗, +(3.1) +where 𝛿𝑖 𝑗 denotes the Kronecker delta operator, ¯𝑆𝑖 𝑗 = +1 +2 +�𝜕 ¯𝑢𝑖/𝜕𝑥 𝑗 + 𝜕 ¯𝑢 𝑗/𝜕𝑥𝑖 +� is the filtered +strain-rate tensor and | ¯𝑆| = (2 ¯𝑆𝑖 𝑗 ¯𝑆𝑖 𝑗)1/2 represents the characteristic filtered strain rate. The +superscript “A” represents the trace-free anisotropic part of the arbitrary variables, namely, (•) 𝐴 +𝑖 𝑗 = +(•)𝑖 𝑗 − (•)𝑘𝑘𝛿𝑖 𝑗/3. The isotropic SGS stress 𝜏𝑘𝑘 is absorbed into the pressure term. 𝐶2 +𝑆 is the +Smagorinsky coefficient and can be determined empirically or by a theoretical analysis. The most +common approach is based on the least-squares dynamic procedure using the Germano identity, +giving rise to the dynamic Smagorinsky model (DSM), whose coefficient is given by (Germano +et al. 1991; Lilly 1992) +𝐶2 +𝑆 = +⟨𝐿 𝐴 +𝑖 𝑗M𝑖 𝑗⟩ +⟨M𝑘𝑙M𝑘𝑙⟩ , +(3.2) +where the Leonard stress 𝐿𝑖 𝑗 = � +¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗, 𝐿 𝐴 +𝑖 𝑗 = 𝐿𝑖 𝑗 − 1 +3𝛿𝑖 𝑗𝐿𝑘𝑘 and M𝑖 𝑗 = ˜𝛼𝑖 𝑗 − 𝛽𝑖 𝑗. Here, +a tilde stands for the test filtering operation at the double-filtering scale ˜Δ = 2 ¯Δ, the variables +𝛼𝑖 𝑗 = 2 ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗 and 𝛽𝑖 𝑗 = 2 ˜Δ2| ˜¯𝑆| ˜¯𝑆𝑖 𝑗. The scale-similarity model 𝜏𝑖 𝑗 = � +¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗 is a typical +structural model and can correctly reconstruct the SGS stress with high a priori accuracy. However, +these structural models often exhibit insufficient dissipation and numerical instability in the a +posteriori testings of LES due to the underestimation of the forward kinetic energy cascade. +The dynamic mixed model (DMM) combines the scale-similarity model with the dissipative + +7 +Smagorinsky term, and is given by (Liu et al. 1994; Shi et al. 2008) +𝜏𝑖 𝑗 = 𝐶1 ¯Δ2 �� ¯𝑆 +�� ¯𝑆𝑖 𝑗 + 𝐶2 +� +� +¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗 +� +. +(3.3) +Similar to the DSM model, model coefficients of the DMM model 𝐶1 and 𝐶2 are dynamically +determined by the least-squares algorithm using the Germano identity, expressed respectively as +(Xie et al. 2020c; Yuan et al. 2020) +𝐶1 = +� +𝑁2 +𝑖 𝑗 +� � +𝐿𝑖 𝑗 𝑀𝑖 𝑗 +� +− +� +𝑀𝑖 𝑗𝑁𝑖 𝑗 +� � +𝐿𝑖 𝑗𝑁𝑖 𝑗 +� +� +𝑁2 +𝑖 𝑗 +� � +𝑀2 +𝑖 𝑗 +� +− +� +𝑀𝑖 𝑗𝑁𝑖 𝑗 +�2 +, +(3.4) +𝐶2 = +� +𝑀2 +𝑖 𝑗 +� � +𝐿𝑖 𝑗𝑁𝑖 𝑗 +� +− +� +𝑀𝑖 𝑗𝑁𝑖 𝑗 +� � +𝐿𝑖 𝑗 𝑀𝑖 𝑗 +� +� +𝑁2 +𝑖 𝑗 +� � +𝑀2 +𝑖 𝑗 +� +− +� +𝑀𝑖 𝑗𝑁𝑖 𝑗 +�2 +, +(3.5) +where 𝑀𝑖 𝑗 = 𝐻1,𝑖 𝑗 − ˜ℎ1,𝑖 𝑗, and 𝑁𝑖 𝑗 = 𝐻2,𝑖 𝑗 − ˜ℎ2,𝑖 𝑗. Here, ℎ1,𝑖 𝑗 = −2 ¯Δ2 �� ¯𝑆 +�� ¯𝑆𝑖 𝑗, ℎ2,𝑖 𝑗 = � +¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗, +𝐻1,𝑖 𝑗 = −2 ˜Δ2 ��� ˜¯𝑆 +��� ˜¯𝑆𝑖 𝑗, and 𝐻2,𝑖 𝑗 = � +˜¯𝑢𝑖 ˜¯𝑢 𝑗 − ˆ˜¯𝑢𝑖 ˆ˜¯𝑢 𝑗. The hat stands for the test filtering at scale ˆΔ = 4 ¯Δ. +The unfiltered variables can be accurately recovered by the resolved filtered field using the +iterative approximate deconvolution procedure, namely (Stolz & Adams 1999; Stolz et al. 2001) +𝑢∗ +𝑖 = 𝐴𝐷 𝑁 ( ¯𝑢𝑖) = +𝑁 +∑︁ +𝑛=1 +(𝐼 − 𝐺)𝑛−1 ⊗ ¯𝑢𝑖, +(3.6) +where the asterisk represents the approximately unfiltered variables, 𝐴𝐷 𝑁 is the abbreviation of +the 𝑁-th order approximate deconvolution, 𝐼 is the identity, and the symbol “⊗” stands for the +spatial convolution operator. For any two functions 𝑓 and 𝑔, 𝑓 ⊗ 𝑔 = +∫ +∞ +−∞ 𝑓 (x′) 𝑔 (x − x′) 𝑑x′. +The unclosed SGS stress then can be recovered with the scale-similarity form by the approximate +deconvolution method (ADM), given by (Bardina et al. 1980) +𝜏𝑖 𝑗 = 𝑢∗ +𝑖 𝑢∗ +𝑗 − ¯𝑢∗ +𝑖 ¯𝑢∗ +𝑗. +(3.7) +The number of iterations for the ADM model is recommended to be 𝑁 =3 ∼ 5 (Stolz et al. 2001). +The accuracy of the ADM model becomes higher, while the numerical stability drops, as the +number of iterations increases. Hence, 𝑁 = 5 is selected in this paper. In order to maintain the +numerical stability of the a posteriori testings of LES [𝜕 ¯𝑢𝑖/𝜕𝑡 = ¯𝑅𝑖 ( ¯𝑢𝑖, 𝑡)], Stolz et al. (2001) and +Adams et al. (2004) introduced a secondary filtering relaxation term [𝜕 ¯𝑢𝑖/𝜕𝑡 = ¯𝑅𝑖 ( ¯𝑢𝑖, 𝑡)+ ¯𝑆𝑖 ( ¯𝑢𝑖)], +yielding +¯𝑆𝑖 ( ¯𝑢𝑖) = −𝜒 +� +𝐼 − 𝐺 ⊗ +𝑁 +∑︁ +𝑛=1 +(𝐼 − 𝐺)𝑛−1 +� +⊗ ¯𝑢𝑖, +(3.8) +where 𝜒 is the empirical regularization coefficient, which is approximately insensitive to the LES +results in previous studies, and we choose 𝜒 = 0 and 1 for comparisons in our paper. +4. Adjoint-based variational optimal mixed models (VOMM) +The mixed model is composed of the structural parts and the dissipative functional terms, and +its general form can be written as (Sagaut et al. 2000) +𝜏𝑖 𝑗 +�𝑢𝑖; ¯Δ� = +𝑁 +∑︁ +𝑛=1 +𝐶𝑛𝑇 (𝑛) +𝑖 𝑗 +� ¯𝑢𝑖; ¯Δ�, +(4.1) + +8 +where 𝑇 (𝑛) +𝑖 𝑗 +� ¯𝑢𝑖; ¯Δ� represents the 𝑛-th basis stress tensor. 𝐶𝑛 (𝑛 = 1, 2, ..., 𝑁) denotes the corre- +sponding model coefficient and 𝑁 is the number of basis stress tensors. The model coefficients +are generally respectively determined by the multivariate least-squares algorithm proposed by +Germano et al. (1991) and Lilly (1992). Many previous studies have shown that the dynamic +mixed models give rise to an excessive dissipation of energy in the transitional regions and +dissipation underestimation if the filter scales are sufficiently large, especially in situations of grid +anisotropy (Meneveau & Katz 2000; Moser et al. 2021). +In recent years, data-driven based high-accuracy SGS models are successively proposed (Kutz +2017; Duraisamy et al. 2019). Xie et al. (2019a) proposed an artificial-neural-network-based +mixed model which accurately recovers the unclosed SGS terms by estimating mixed model +coefficients with local flow characteristics as inputs of the machine-learning strategy, yielding +better predictions of LES statistics than the classical dynamic mixed model. The input features +of the data-driven closure models are crucial for the accuracy of SGS models (Gamahara & +Hattori 2017; Beck et al. 2019; Xie et al. 2019b; Park & Choi 2021). Incorporating the accurate +structural parts, i.e., filtered velocity gradients at the neighboring stencil turn out to improve +the performance of data-driven SGS models effectively (Xie et al. 2019b, 2020a). Moreover, the +spatial flow structures at scales between ¯Δ/2 and 2 ¯Δ are found to be essential for the SGS modeling +of LES at the filter scale ¯Δ (Xie et al. 2020b). The strategy of the blind deconvolution with the +artificial neural network was proposed to recover the unknown original unfiltered variables +from the known filtered quantities with high accuracy (Maulik & San 2017; Maulik et al. +2019). A deconvolutional-artificial-neural-network (DANN) framework was further proposed +to accurate reconstruct the SGS unclosed terms both in a priori and a posteriori analyses +of isotropic turbulence (Yuan et al. 2020, 2021a), and successfully applied to the chemically +reacting compressible turbulence (Teng et al. 2022). It was demonstrated that the DANN models +embed the properties of symmetry and realizability conditions, which preserve the physical +reliability of the DANN framework (Yuan et al. 2020). In order to enhance the interpretability of +black-box machine-learning SGS models, a semi-explicit ANN-based spatial gradient model and +constant-coefficient spatial gradient models are successively proposed by the elaborate Taylor +expansions of velocity gradients in the neighboring stencil locations (Wang et al. 2021, 2022b). +The machine-learning-based SGS models trained by high-fidelity simulation data can be regarded +as the structural models with high a priori accuracy, requiring additional indispensable dissipation +to account for the spatial discretization effect and ensure the numerical stability in the a posteriori +studies of LES. +In addition to the machine-learning-assisted SGS models, some a priori information about +statistics of turbulence acquired from experimental data or DNS results like energy spectra, +structure functions, and probability density functions of physical quantities can be used to +determine or correct the model coefficients of SGS models to improve the model accuracy greatly. +These a priori knowledge of turbulent statistical quantities can be dynamically assimilated into +the closure models via the data-assimilation based approaches. Among these data-assimilation +techniques, adjoint-based variational methods adopt the optimal control strategy to efficiently +calculate all the gradients of cost functionals for the model coefficients by solving the forward +governing equations and the backward adjoint equations (Bewley et al. 2001). Then, the model +coefficients of SGS models are iteratively updated using the gradient-based optimization algorithm +until the optimal values are obtained. The cost functionals measure the discrepancies of statistical +quantities in turbulence between the LES results and measurements from the experimental or DNS +data, which can greatly alleviate the impact of chaotic field observations on the performance of data +assimilation. In this work, we resort to the state-of-art adjoint-based data-assimilation approaches +to establish a general optimal SGS framework to determine model parameters adaptively for +various grid resolutions of LES in different turbulence scenarios. + +9 +4.1. Adjoint LES equations and gradient evaluations with the mixed model +We optimize the model coefficients of the SGS closure model to minimize the statistical +discrepancies between the LES calculations and the reference values acquired from the experi- +mental or DNS data, which can be defined as the minimal optimization problem constrained by +the governing equations (see Eqs. 2.4 and 2.5). The constrained optimization problem for the +turbulent closure modeling is expressed as +min +𝐶𝑛 +J +� +𝜙 ( ¯𝑢𝑖; 𝐶𝑛) , 𝜙 � ¯𝑢ref +𝑖 +�� +, +s.t. +𝑅0 ( ¯𝑢𝑖) = 𝜕 ¯𝑢𝑖 +𝜕𝑥𝑖 = 0, +𝑅𝑖 ( ¯𝑢𝑖, ¯𝑝) = 𝜕 ¯𝑢𝑖 +𝜕𝑡 + +𝜕( ¯𝑢𝑖 ¯𝑢𝑗) +𝜕𝑥𝑗 ++ 𝜕 ¯𝑝 +𝜕𝑥𝑖 − 𝜈 +𝜕2 ¯𝑢𝑖 +𝜕𝑥𝑗𝜕𝑥𝑗 − F 𝑖 + 𝜕𝜏𝑖 𝑗 +𝜕𝑥𝑗 = 0, +(4.2) +where J +� +𝜙 ( ¯𝑢𝑖; 𝐶𝑛) , 𝜙 � ¯𝑢ref +𝑖 +�� += +𝑇∫ +0 +∫ +Ω +𝐽 +� +𝜙 ( ¯𝑢𝑖; 𝐶𝑛, x, 𝑡) , 𝜙 � ¯𝑢ref +𝑖 ; x, 𝑡�� +𝑑x𝑑𝑡 denotes the total +cost functions, 𝐽 +� +𝜙 ( ¯𝑢𝑖; 𝐶𝑛, x, 𝑡) , 𝜙 � ¯𝑢ref +𝑖 ; x, 𝑡�� +is the discrepancy of statistical quantities 𝜙 (e.g. +kinetic energy spectra, structure functions, etc.) between the LES results ¯𝑢𝑖 and reference values +¯𝑢ref +𝑖 +(experimental or DNS data) at a certain state (𝐶𝑛, x, 𝑡). 𝐶𝑛 (𝑛 = 1, 2, ..., 𝑁) denotes model +coefficients of the SGS mixed model 𝜏𝑖 𝑗 = +𝑁� +𝑛=1 +𝐶𝑛𝑇 (𝑛) +𝑖 𝑗 , and 𝑡 ∈ [0,𝑇] is the time horizon. +Here, “s.t.” stands for the abbreviation of “subject to”. 𝑅0 and 𝑅𝑖 (𝑖 = 1, 2, 3) represent the LES +continuity equation and momentum equations, respectively. +The Lagrangian functional L is introduced to take the dynamics of LES variables ¯v = +[ ¯𝑝, ¯𝑢1, ¯𝑢2, ¯𝑢3]𝑇 into account and convert the constrained optimization (Eq. 4.2) into the un- +constrained optimization problem, namely (Lewis et al. 2006) +min +𝐶𝑛 +L (¯v; 𝐶𝑛) , where L = J +� +𝜙 (¯v; 𝐶𝑛) , 𝜙 +� +¯vref�� +− +3 +∑︁ +𝑘=0 +𝑇 +∫ +0 +∫ +Ω +𝑅𝑘 (¯v; 𝐶𝑛) · ¯𝑣† +𝑘𝑑x𝑑𝑡. +(4.3) +Here, ¯v† = +� +¯𝑝†, ¯𝑢† +1, ¯𝑢† +2, ¯𝑢† +3 +�𝑇 +are the adjoint LES variables of ¯v, where ¯𝑝† and ¯𝑢† +𝑖 are the adjoint +pressure and adjoint velocity, respectively. For the sake of brevity, the inner product of time +and space is defined by ⟨f, g⟩x,𝑡 = +𝑇∫ +0 +∫ +Ω +f (x, 𝑡) · g (x, 𝑡) 𝑑x𝑑𝑡, where f (x, 𝑡) and g (x, 𝑡) denote +the arbitrary physical variables. The Lagrangian functional L can be simplified as L (¯v; 𝐶𝑛) = +J (¯v; 𝐶𝑛)− +3� +𝑘=0 +� +𝑅𝑘 (¯v; 𝐶𝑛) , ¯v†� +x,𝑡. The sensitivity of the Lagrangian functional L can be derived +by +𝛿L (¯v; 𝐶𝑛) = 𝛿J (¯v; 𝐶𝑛) − +3 +∑︁ +𝑘=0 +� +𝑅𝑘 (𝛿¯v; 𝐶𝑛) , ¯v†� +x,𝑡 − +3 +∑︁ +𝑘=0 +� +𝑅𝑘 (¯v; 𝛿𝐶𝑛) , ¯v†� +x,𝑡, += 𝛿J (¯v; 𝐶𝑛) − +3 +∑︁ +𝑘=0 +� 𝜕𝑅𝑘 (¯v; 𝐶𝑛) +𝜕¯v +· 𝛿¯v, ¯v† +� +x,𝑡 +− +3 +∑︁ +𝑘=0 +� 𝜕𝑅𝑘 (¯v; 𝐶𝑛) +𝜕𝐶𝑛 +· 𝛿𝐶𝑛, ¯v† +� +x,𝑡 +, +(4.4) +where 𝜕𝑅𝑘/𝜕¯v and 𝜕𝑅𝑘/𝜕𝐶𝑛 +are the tangent operators of the governing equations +𝑅𝑘 +(𝑘 = 0, 1, 2, 3) for the variables ¯v and parameters 𝐶𝑛 with the perturbation field +𝛿¯v = ¯v (𝐶𝑛 + 𝛿𝐶𝑛) − ¯v (𝐶𝑛) , 𝑛 ∈ {1, 2, ..., 𝑁}. The first term in Eq. 4.4 is the sensitivity +of the cost functional J and calculated as the Gâteaux-Fréchet derivative (Bewley et al. 2001) + +10 +of J at 𝐶𝑛 in the direction 𝛿𝐶𝑛, namely +𝛿J (¯v; 𝛿𝐶𝑛) = lim +𝜀→0 +𝑑 +𝑑𝜀 J (¯v (𝐶𝑛 + 𝜀𝛿𝐶𝑛)) = +� 𝜕𝐽 +𝜕¯v, 𝛿¯v +� +x,𝑡 +. +(4.5) +The adjoint identity (Bewley et al. 2001) can be obtained via the integral by part, given by +� +R (¯v) , ¯v†� +x,𝑡 = +� +¯v, R† � +¯v†�� +x,𝑡 + +� ¯F, ¯v†� +𝑡 +�� +Γ + +�¯v, ¯v†� +x +��𝑇 +0 = +� +¯v, R† � +¯v†�� +x,𝑡 + 𝐵𝑇, +(4.6) +where the partial differential equations R (¯v) = 𝜕¯v/𝜕𝑡 + 𝜕 ¯F/𝜕x = 0 with the associated +adjoint operator R† �¯v†�, ¯F denotes the fluxes and Γ is the boundary of the domain Ω. Here, +𝐵𝑇 = +� ¯F, ¯v†� +𝑡 +�� +Γ + +�¯v, ¯v†� +x +��𝑇 +0 represents the boundary and temporal integral terms, which +determines the boundary and terminal conditions of the adjoint equations to give 𝐵𝑇 = 0. +⟨f, g⟩𝑡 = +𝑇∫ +0 +f (x, 𝑡) · g (x, 𝑡) 𝑑𝑡 and ⟨f, g⟩x = +∫ +Ω +f (x, 𝑡) · g (x, 𝑡) 𝑑x denote the temporal and spatial +inner products, respectively. The second term in Eq. 4.4 can be expressed with the adjoint identity, +namely (Bewley et al. 2001; Delport et al. 2009, 2011) +� 𝜕𝑅𝑘 +𝜕¯v · 𝛿¯v, ¯v† +� +x,𝑡 += +� +𝛿¯v, +� 𝜕𝑅𝑘 +𝜕¯v +�† +· ¯v† +� +x,𝑡 ++ 𝐵𝑇, +(4.7) +where (𝜕𝑅𝑘/𝜕¯v)† is the adjoint operator of the LES tangent Jacobian tensor 𝜕𝑅𝑘/𝜕¯v, +(𝑘 = 0, 1, 2, 3). Substitute the Fréchet derivative 𝛿J (Eq. 4.5) and the adjoint identity (Eq. 4.7) +into the sensitivity of the Lagrangian functional L (Eq. 4.4), and we get +𝛿L (¯v; 𝐶𝑛) = +� +𝜕𝐽 +𝜕¯v − +3 +∑︁ +𝑘=0 +� 𝜕𝑅𝑘 +𝜕¯v +�† +· ¯v†, 𝛿¯v +� +x,𝑡 +− +3 +∑︁ +𝑘=0 +� 𝜕𝑅𝑘 (¯v; 𝐶𝑛) +𝜕𝐶𝑛 +· 𝛿𝐶𝑛, ¯v† +� +x,𝑡 +− 𝐵𝑇, +(4.8) +To avoid calculating the perturbation field 𝛿¯v in the first term of Eq. 4.8, the inner product should +be equal to 0 and the corresponding adjoint LES equations can be derived by +3 +∑︁ +𝑘=0 +� 𝜕𝑅𝑘 +𝜕¯v +�† +· ¯v† − 𝜕𝐽 +𝜕¯v = 0. +(4.9) +Substitute the specific forms of the LES equations 𝑅𝑘 (𝑘 = 0, 1, 2, 3) (see Eq. 4.2), and the adjoint +LES equations can be written as +𝜕 ¯𝑢† +𝑖 +𝜕𝑥𝑖 += 0, +(4.10) +𝜕 ¯𝑢† +𝑖 +𝜕𝑡 + +� +𝜕 ¯𝑢† +𝑖 +𝜕𝑥 𝑗 ++ +𝜕 ¯𝑢† +𝑗 +𝜕𝑥𝑖 +� +¯𝑢 𝑗 + 𝜕 ¯𝑝† +𝜕𝑥𝑖 ++ 𝜈 𝜕2 ¯𝑢† +𝑖 +𝜕𝑥 𝑗𝜕𝑥 𝑗 ++ +𝜕𝜏† +𝑖 𝑗 +𝜕𝑥 𝑗 ++ 𝜕𝐽 +𝜕 ¯𝑢𝑖 += 0, +(4.11) +where 𝜏† +𝑖 𝑗 = +𝑁� +𝑛=1 +𝐶𝑛𝑇 (𝑛),† +𝑖 𝑗 +denotes the adjoint SGS mixed model and 𝑇 (𝑛),† +𝑖 𝑗 +is the 𝑛-th adjoint basis +stress tensor. The detailed derivation of the adjoint LES equations can refer to the Appendix A. +The terminal conditions of the adjoint LES equations is determined by the last term of adjoint +identity (Eq. 4.6), namely +�¯v†, 𝛿¯v +� +x +��𝑇 +0 = +�¯v† (𝑇) , 𝛿¯v (𝑇) +� +x − +�¯v† (0) , 𝛿¯v (0) +� +x = +�¯v† (𝑇) , 𝛿¯v (𝑇) +� +x, +(4.12) +where 𝛿¯v (0) = 0, since the unperturbed initial LES field is exactly given by the filtered DNS +(fDNS) data. The terminal conditions ¯v† (𝑇) = +� +¯𝑢† +𝑖 (𝑇) , ¯𝑝† (𝑇) +�𝑇 += 0 make the temporal integral + +11 +terms +�� +𝛿¯v, ¯v†� +x +�𝑇 +0 equal to zero and the calculation of the terminal perturbation 𝛿¯v (𝑇) is +obviated. The terminal conditions ( ¯𝑢† +𝑖 (𝑇) = 0, ¯𝑝† (𝑇) = 0) and boundary conditions of the +adjoint LES equations are identified by setting 𝐵𝑇 = 0 in Eq. 4.8. The sensitivity of the Lagrangian +functional L can be further expressed as +𝛿L (¯v; 𝐶𝑛) = − +3 +∑︁ +𝑘=0 +� 𝜕𝑅𝑘 (¯v; 𝐶𝑛) +𝜕𝐶𝑛 +· 𝛿𝐶𝑛, ¯v† +� +x,𝑡 +, +(4.13) +where 𝜕𝑅0/𝜕𝐶𝑛 = 0, and 𝜕𝑅𝑖/𝜕𝐶𝑛 = +𝜕 +𝜕𝐶𝑛 +� 𝜕𝜏𝑖 𝑗 +𝜕𝑥𝑗 +� += 𝜕𝑇 (𝑛) +𝑖 𝑗 /𝜕𝑥 𝑗 (𝑛 = 1, 2, ..., 𝑁) denotes the 𝑛-th +SGS basis force. Once the LES equations (Eqs. 2.4 and 2.5) temporally advances forward in +the time horizon 𝑡 ∈ [0,𝑇] and the adjoint LES equations (Eqs. 4.10 and 4.11) are integrated +backward with zero terminal conditions, the gradients of Lagrangian functional for the SGS +model coefficients can be calculated efficiently by +𝜕L +𝜕𝐶𝑛 += 𝛿L (¯v; 𝐶𝑛) +𝛿𝐶𝑛 += − +� 𝜕𝑇 (𝑛) +𝑖 𝑗 +𝜕𝑥 𝑗 +, ¯𝑢† +𝑖 +� +x,𝑡 +, (𝑛 = 1, 2, ..., 𝑁) . +(4.14) +The adjoint-based gradient evaluations are independent of the parameter perturbations +𝛿𝐶𝑛 (𝑛 = 1, 2, ..., 𝑁), which are very efficient compared to the finite difference algorithm and +forward sensitivity analysis with at least 𝑁 parameter perturbations and 𝑁 + 1 LES equation +calculations for each optimization iteration (Chandramouli et al. 2020; Sirignano et al. 2020; +MacArt et al. 2021). +4.2. Energy budget analysis of the adjoint LES equations +Before proceeding to the introduction of the variational optimal mixed models, it is essential +to analyze the energy budget of the adjoint LES equations. The adjoint LES kinetic energy +( ¯E† = ¯𝑢† +𝑖 ¯𝑢† +𝑖 /2) equation is derived through multiplying the adjoint velocity ¯𝑢† +𝑖 on both sides of +the adjoint LES momentum equations (Eq. 4.11), namely +𝜕 ¯E† +𝜕𝑡 + 𝜕 ¯P𝑗 +𝜕𝑥 𝑗 += ¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 + ¯𝐷† − ¯Π† − ¯𝐽†, +(4.15) +where ¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 denotes the adjoint energy production term due to the shear strain rate ¯𝑆𝑖 𝑗. Here, +¯P𝑗 is the adjoint spatial transport flux, ¯𝐷 is the adjoint viscous dissipation term, ¯Π† is the adjoint +variable of the SGS energy flux ¯Π = −𝜏𝑖 𝑗 ¯𝑆𝑖 𝑗 and ¯𝐽† is the energy injected from the discrepancy +between LES results and reference data. These terms are respectively defined by +¯P𝑗 = ¯E† ¯𝑢 𝑗 + +� +¯𝑝† + ¯𝑢𝑖 ¯𝑢† +𝑖 +� +¯𝑢† +𝑗 + +� +𝜈 𝜕 ¯𝑢† +𝑖 +𝜕𝑥 𝑗 ++ 𝜏† +𝑖 𝑗 +� +¯𝑢† +𝑖 , +(4.16) +¯𝐷 = 𝜈 𝜕 ¯𝑢† +𝑖 +𝜕𝑥 𝑗 +𝜕 ¯𝑢† +𝑖 +𝜕𝑥 𝑗 +, +(4.17) +¯Π† = −𝜏† +𝑖 𝑗 ¯𝑆† +𝑖 𝑗, +(4.18) +¯𝐽† = ¯𝑢† +𝑖 +𝜕𝐽 +𝜕 ¯𝑢𝑖 +, +(4.19) + +12 +where ¯𝑆† +𝑖 𝑗 = +� +𝜕 ¯𝑢† +𝑖 /𝜕𝑥 𝑗 + 𝜕 ¯𝑢† +𝑗/𝜕𝑥𝑖 +� +/2 represents the adjoint strain-rate tensor. The backward +evolution of the adjoint volume-averaged kinetic energy can be written as +− 𝜕 +� ¯E†� +𝜕𝑡 += − +� +¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 +� +− +� ¯𝐷†� ++ +� ¯Π†� ++ +� ¯𝐽†� +, +(4.20) +where +� ¯𝐷†� +is pure dissipation term that drains out the adjoint energy. +� ¯Π†� +denotes the adjoint +SGS energy transport term which represents the forward adjoint energy transfer from large +scales to unsolved residual scales if +� ¯Π†� +> 0, otherwise stands for the adjoint SGS energy +backscatter. The accurate reconstruction of +� ¯Π†� +is crucial for the SGS modeling of LES and +gradient evaluations with respect to the SGS model coefficients. +� ¯𝐽†� +is the loss-induced adjoint +energy injection term. +� ¯𝐷†� +is the viscous dissipation which enhances the numerical stability +of the adjoint LES field. +� ¯𝐽†� +is the adjoint energy production due to the discrepancy between +LES evaluation and reference data, which dominates the accuracy of the sensitivity calculations. +The large-scale strain-rate tensor ¯𝑆𝑖 𝑗 can be decomposed into its principal components using the +eigendecomposition approach, such that (Wang & Gao 2013) +¯𝑆𝑖 𝑗=𝜆1𝑞(1) +𝑖 +𝑞(1) +𝑗 ++ 𝜆2𝑞(2) +𝑖 +𝑞(2) +𝑗 ++ 𝜆3𝑞(3) +𝑖 +𝑞(3) +𝑗 += +3 +∑︁ +𝑘=1 +𝜆𝑘𝑞(𝑘) +𝑖 +𝑞(𝑘) +𝑗 +, +(4.21) +where 𝜆1, 𝜆2 and 𝜆3 are the eigenvalues of the shear strain rate, with 𝑞(1) +𝑖 +, 𝑞(2) +𝑖 +and 𝑞(3) +𝑖 +being the +associated eigenvectors. Here,𝜆1+𝜆2+𝜆3 = 0 for the trace-free strain rate ¯𝑆𝑖 𝑗 in the incompressible +turbulent flows. Hence, the quadratic term − +� +¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 +� +in Eq. 4.20 is further expressed as +− +� +¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 +� += − +3 +∑︁ +𝑘=1 +� +𝜆𝑘 +� +𝑞(𝑘) +𝑖 +¯𝑢† +𝑖 +� � +𝑞(𝑘) +𝑗 +¯𝑢† +𝑗 +�� += − +3 +∑︁ +𝑘=1 +� +𝜆𝑘 +� +𝑞(𝑘) +𝑖 +¯𝑢† +𝑖 +�2� +. +(4.22) +The sign of the eigenvalues 𝜆𝑘, (𝑘 = 1, 2, 3) determines the contribution of the adjoint energy +from the quadratic term − +� +¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 +� +is productive or dissipative. The quadratic terms with +negative eigenvalues of the shear strain rate produce the positive adjoint energy production, +while those with positive eigenvalues drain out the adjoint energy. In previous studies of chaotic +adjoint methods, the adjoint-based gradients are found to grow exponentially with time and finally +numerically diverge in a long time horizon for the chaotic flows (Wang & Gao 2013; Ashley et al. +2019; Garai & Murman 2021). The terms +� ¯𝐷†� +, +� ¯Π†� +and +� ¯𝐽†� +in the volume-averaged adjoint +energy equation (Eq. 4.20) are less likely to cause the exponential growth of the adjoint energy, +since the adjoint energy term +� ¯E†� +does not appears explicitly in these terms. It can be further +shown that the quadratic term − +� +¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 +� +plays the dominant role in the exponential growth +of the adjoint variables. We apply the Cauchy-Schwarz inequality to the inner product terms in +Eq. 4.22, such that (Talnikar et al. 2017) +� +𝑞(𝑘) +𝑖 +¯𝑢† +𝑖 +�2 +⩽ +� +𝑞(𝑘) +𝑖 +𝑞(𝑘) +𝑖 +� � +¯𝑢† +𝑖 ¯𝑢† +𝑖 +� += 2 +���q(𝑘)��� E +† +(𝑘 = 1, 2, 3) , +(4.23) +where “∥·∥” denotes the L2 norm of the vectors. For the quadratic terms with negative eigenvalues +(adjoint energy production), the evolution of the adjoint energy can be approximated using the +leading principal vectors as +− +𝜕 +� +E +†� +𝜕𝑡 +≈ 2|𝜆|∞∥q∥∞ +� +E +†� +, +(4.24) + +13 +where |𝜆|∞ = max +Ω {−𝜆1, −𝜆2, −𝜆3} denotes the magnitude of the leading negative eigenvalue in +the entire domain Ω and ∥q∥∞ represents the corresponding eigenvector magnitude. The adjoint +energy is then calculated by the backward time interval, namely +� +E +†� +(𝑡) ≈ +� +E +†� +(𝑇) exp [2|𝜆|∞∥q∥∞ (𝑇 − 𝑡)] , 𝑡 ∈ [0,𝑇] . +(4.25) +The quadratic term − +� +¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 +� +with negative eigenvalues makes the adjoint energy grow +exponentially over time and numerically unstable if it cannot be suppressed by the adjoint +dissipation in a long time horizon 𝑡 ∈ [0,𝑇]. In order to stabilize the adjoint equations during +every iteration, an additional symmetric tensor ¯𝑆𝑎 +𝑖 𝑗 (Ashley et al. 2019; Garai & Murman 2021) +is introduced to maintain the numerical stability of the adjoint momentum (Eq. 4.11), and the +stabilized adjoint momentum equations are then expressed as +𝜕 ¯𝑢† +𝑖 +𝜕𝑡 + +� +𝜕 ¯𝑢† +𝑖 +𝜕𝑥 𝑗 ++ +𝜕 ¯𝑢† +𝑗 +𝜕𝑥𝑖 +� +¯𝑢 𝑗 + ¯𝑆𝑎 +𝑖 𝑗 ¯𝑢† +𝑗 + 𝜕 ¯𝑝† +𝜕𝑥𝑖 ++ 𝜈 𝜕2 ¯𝑢† +𝑖 +𝜕𝑥 𝑗𝜕𝑥 𝑗 ++ +𝜕𝜏† +𝑖 𝑗 +𝜕𝑥 𝑗 ++ 𝜕𝐽 +𝜕 ¯𝑢𝑖 += 0. +(4.26) +Consequently, the stabilized adjoint kinetic energy equation is written by +𝜕E +† +𝜕𝑡 + 𝜕P 𝑗 +𝜕𝑥 𝑗 += ¯𝑢† +𝑖 +� +¯𝑆𝑖 𝑗 − ¯𝑆𝑎 +𝑖 𝑗 +� +¯𝑢† +𝑗 + ¯𝐷† − ¯Π† − ¯𝐽†. +(4.27) +Here, the quadratic term ¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 < 0 +� +− ¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 > 0 +� +is responsible for the exponential growth +of the adjoint energy, and the minimal artificial symmetric tensor is added to keep the adjoint +variables numerically stable in advancing backward of the adjoint LES equations. The artificial +symmetric tensor ¯𝑆𝑎 +𝑖 𝑗 can be optimized by the suboptimal minimization problem (Ashley et al. +2019; Garai & Murman 2021), such that +min +¯𝑆𝑎 +𝑖 𝑗 +1 +2 ¯𝑆𝑎 +𝑖 𝑗 ¯𝑆𝑎 +𝑖 𝑗, +𝑠.𝑡. +¯𝑢† +𝑖 +� +¯𝑆𝑖 𝑗 − ¯𝑆𝑎 +𝑖 𝑗 +� +¯𝑢† +𝑗 ⩾ 0. +(4.28) +We use the sequential quadratic programming (SQP) approach (Boggs & Tolle 1995; Chung +& Freund 2022) to efficiently solve the suboptimal problem, and the augmented Lagrangian +functional L is applied to the constrained minimization problem, namely +L = 1 +2 +¯𝑆𝑎 +𝑖 𝑗 ¯𝑆𝑎 +𝑖 𝑗 + 𝜆 +� +¯𝑢† +𝑖 +� +¯𝑆𝑖 𝑗 − ¯𝑆𝑎 +𝑖 𝑗 +� +¯𝑢† +𝑗 +� +, +(4.29) +where 𝜆 is the Lagrangian multiplier. The Karush–Kuhn–Tucker (KKT) optimal conditions (Kuhn +& Tucker 1951; Blonigan & Wang 2018) are obtained by taking the derivatives of the cost +functional with respect to the augmented optimal variables ( ¯𝑆𝑎 +𝑖 𝑗 and 𝜆), derived by +𝜕L +𝜕 ¯𝑆𝑎 +𝑖 𝑗 += ¯𝑆𝑎 +𝑖 𝑗 − 𝜆 +� +¯𝑢† +𝑖 ¯𝑢† +𝑗 +� += 0 ⇒ ¯𝑆𝑎 +𝑖 𝑗 = 𝜆 +� +¯𝑢† +𝑖 ¯𝑢† +𝑗 +� +, +(4.30) +𝜕L +𝜕𝜆 = ¯𝑢† +𝑖 +� +¯𝑆𝑖 𝑗 − ¯𝑆𝑎 +𝑖 𝑗 +� +¯𝑢† +𝑗 = 0 ⇒ ¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 = ¯𝑢† +𝑖 ¯𝑆𝑎 +𝑖 𝑗 ¯𝑢† +𝑗. +(4.31) +By multiplying Eq. 4.30 by ¯𝑢† +𝑖 from the left and right by ¯𝑢† +𝑗 , and then substituting it into Eq. 4.31, +the Lagrangian multiplier 𝜆 is calculated by +𝜆 = +¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 +� +¯𝑢† +𝑘 ¯𝑢† +𝑘 +�2 = +¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 +4E +†2 . +(4.32) + +14 + +Initial SGS parameters + + + +SGS parameters + + + +SGS model + + + +Forward LES +equations + +LES statistics + + +Loss function +Stop criterion + + +Adjoint SGS model + + + +Stop +L-BFGS gradient + Optimization +Gradient Calculations +Stop + +Reference statistics + + +No +No +Yes +Start +Yes + +Initial velocity field + + +L-BFGS gradient + Optimization + +Terminal condition + + +Loss sensitivity + +Backward adjoint + LES equations + + + + +Backward adjoint + LES equations + + + +Stop criterion + + + +Figure 1: Schematic diagram of the adjoint-based variational optimal mixed models. +The minimal artificial symmetric tensor ¯𝑆𝑎 +𝑖 𝑗 can be obtained by substituting Eq. 4.32 into Eq. 4.30, +yielding +¯𝑆𝑎 +𝑖 𝑗 = +��� +��� +¯𝑢† +𝑚 ¯𝑆𝑚𝑛 ¯𝑢† +𝑛 +4E +†2 +¯𝑢† +𝑖 ¯𝑢† +𝑗, +if ¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 < 0, +0 +, +if ¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 ⩾ 0. +(4.33) +The artificial momentum term ¯𝑆𝑎 +𝑖 𝑗 ¯𝑢† +𝑗 is thus additionally calculated in the stabilized adjoint +momentum equations (Eq. 4.26), namely +¯𝑆𝑎 +𝑖 𝑗 ¯𝑢† +𝑗 = +�� +�� +¯𝑢† +𝑚 ¯𝑆𝑚𝑛 ¯𝑢† +𝑛 +2E +† +¯𝑢† +𝑖 , +if ¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 < 0, +0 +, +if ¯𝑢† +𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† +𝑗 ⩾ 0. +(4.34) +The minimal stabilization term ¯𝑆𝑎 +𝑖 𝑗 ¯𝑢† +𝑗 can efficiently maintain the numerical stability of LES +adjoint variables in the long-term chaotic turbulent calculations as much as possible, without +deteriorating the correct evaluations of the adjoint-based gradient. +4.3. Adjoint-based variational optimal mixed models (VOMM) +In this research, we select the mixed model comprised of the Smagorinsky dissipative term +(Eq. 3.1) and approximate deconvolution model (ADM, Eq. 3.6) in the scale-similarity form, +expressed as (Sagaut 2006) +𝜏𝑖 𝑗 = 𝐶1 +� +¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗 +� ++ 𝐶2 +� +𝑢∗ +𝑖 𝑢∗ +𝑗 − 𝑢∗ +𝑖 𝑢∗ +𝑗 +� +, +(4.35) +where 𝑢∗ +𝑖 denotes the approximate unfiltered velocity recovered by the iterative van Cittert +procedure (Eq. 3.6). In previous studies (Yuan et al. 2020), we have conducted error analyses to +validate that deconvolutional-type SGS models with scale-similarity form (Eq. 3.7) perform better +than those with theconventionaldirect-modelingform(𝜏𝑖 𝑗 = 𝑢∗ +𝑖 𝑢∗ +𝑗− ¯𝑢𝑖 ¯𝑢 𝑗),satisfyingtheproperties +of symmetry and realizability conditions. The model coefficients 𝐶1 and 𝐶2 are optimally +identified by minimizing the discrepancy between statistical quantities calculated by the LES +results and those measured by the filtered DNS (fDNS) data. The selected statistics should be able +to sufficiently quantify the multiscale transport behaviours of turbulence, meanwhile facilitating +the practical measurements. The SGS stress 𝜏𝑖 𝑗 and SGS force 𝜕𝜏𝑖 𝑗/𝜕𝑥 𝑗 are intermediate variables, + +15 +𝑅𝑒𝜆 +𝐸𝑘 +𝑘max𝜂 +𝜂/ℎDNS +𝐿𝐼 /𝜂 +𝜆/𝜂 +𝑢rms +𝜔rms +𝜀 +252 +2.63 +2.11 +1.01 +235.2 +31.2 +2.30 +26.90 +0.73 +Table 1: One-point statistics for the DNS of forced homogeneous isotropic turbulence with +grid resolution of 10243. +and their statistics are relatively difficult to be obtained through the actual observations. In +contrast, the statistics of velocity are more convenient to measure and the velocity spectrum +clearly quantifies the turbulent kinetic energy distributions at different scales. The SGS modeling +is especially concerned with the accurate reconstruction of small scales near the filter width, +therefore we select the dissipation spectrum as the optimization statistical quantities 𝜙 ( ¯𝑢𝑖) to +increase the weights of small scales, namely (Pope 2000) +𝜙 ( ¯𝑢𝑖, 𝑘, 𝑡) = 𝐷 (𝑘, 𝑡) = +∫ +k +𝜈𝑘2 ¯𝑣∗ +𝑖 (k, 𝑡) ¯𝑣𝑖 (k, 𝑡) 𝛿 (|k| − 𝑘) 𝑑k, +(4.36) +where 𝛿 (·) denotes the Dirac delta function and the star symbol represents complex conjugate. +𝑘 and k stand for the wavenumber magnitude and wavenumber vectors, respectively. Here, +¯𝑣 𝑗 (𝜅, 𝑡) = F +� +¯𝑢 𝑗 (x, 𝑡) +� += � +k +¯𝑢 𝑗 (x, 𝑡) 𝑒−𝑖k·x is the 𝑗-th velocity component in Fourier space, +where F {·} represents the 3D Fourier transform, and 𝑖 is the imaginary unit with 𝑖2 = −1. +The optimization problem constrained by the governing equations for the SGS parameters 𝐶1 +and 𝐶2 is defined in Eq. 4.2, where the cost functional for the dissipation spectrum 𝐷 (𝑘, 𝑡) is +given by +J +� +𝜙, 𝜙fDNS� += +𝑇 +∫ +0 +𝑘max +∑︁ +𝑘=1 +𝐽 +� +𝐷 (𝑘, 𝑡) , 𝐷fDNS (𝑘, 𝑡) +� +𝑑𝑡, +(4.37) +where 𝑘max = 𝑁LES/3 is the effective maximum wavenumber, 𝑁LES is the number of LES grids, +and the discrepancy function 𝐽 +� +𝐷 (𝑘, 𝑡) , 𝐷fDNS (𝑘, 𝑡) +� += +� +𝐷 (𝑘, 𝑡) − 𝐷fDNS (𝑘, 𝑡) +�2 takes the L2 +norm of the prediction error. +The gradients of the loss function with respect to the model coefficients 𝐶1 and 𝐶2 are evaluated +by Eq. 4.14, where the adjoint variables ¯𝑢† +𝑖 are calculated by backward advancing the stabilized +adjoint LES equations (Eqs. 4.10 and 4.26) with zero terminal conditions. The sensitivity term +𝜕𝐽/𝜕 ¯𝑢𝑖 is calculated by the chain rule, namely +𝜕𝐽 +𝜕 ¯𝑢𝑖 += 𝜕𝐽 +𝜕𝐷 +𝜕𝐷 +𝜕 ¯𝑢𝑖 += 2 +� +𝐷 − 𝐷fDNS� +F−1 � +2𝜈𝑘2 ¯𝑣𝑖 (k, 𝑡) 𝛿 (|k| − 𝑘) +� +, +(4.38) +where F−1 {·} denotes the 3D inverse Fourier transform. In the stabilized adjoint momentum +equations (Eq. 4.26), the adjoint SGS stress is given by 𝜏† +𝑖 𝑗 = 𝐶1𝑇 (1),† +𝑖 𝑗 ++ 𝐶2𝑇 (2),† +𝑖 𝑗 +, where the +associated adjoint basis stress tensors 𝑇 (1),† +𝑖 𝑗 +and 𝑇 (2),† +𝑖 𝑗 +are expressed in detail as +𝑇 (1),† +𝑖 𝑗 += − ¯Δ2 +� +�� ¯𝑆 +�� ¯𝑆† +𝑖 𝑗 + 2 +¯𝑆𝑘𝑙 ¯𝑆† +𝑘𝑙 +�� ¯𝑆 +�� +¯𝑆𝑖 𝑗 +� +, +(4.39) +𝑇 (2),† +𝑖 𝑗 += +𝑁 +∑︁ +𝑛=1 +(𝐼 − 𝐺)𝑛−1 ⊗ +� +¯𝑢† +𝑖 𝑢∗ +𝑗 − ¯𝑢† +𝑖 𝑢∗ +𝑗 +� +, +(4.40) + +16 +(𝑎) +�� +� +�� +� +�� +� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +���� +� +���� +� +� +� +��� +DNS +��� +���� +� +� +� �� +(𝑏) +�� +� +�� +� +�� +� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +���� +� +���� +� +� +� +��� +DNS +��� +���� +� +� +� �� +Figure 2: Velocity and dissipation spectra of DNS and filtered DNS in forced +homogeneous isotropic turbulence with grid resolution of 10243: (𝑎) velocity spectra, and +(𝑏) dissipation spectra. Diamond represent the cutoff wavenumber 𝑘𝑐=16 ( ¯Δ = 32ℎDNS). +where 𝑁 = 5 denotes the number of iterations for the AD procedure. The detailed derivation of the +adjoint SGS stress tensors for the VOMM model can refer to the Appendix B. To our knowledge, +few previous works have studied the mixed SGS models and given the detailed derivations of the +adjoint SGS models. +Once the gradients of the cost functional for the model coefficients are obtained by successively +solving the forward LES equations and backward stabilized adjoint LES equations, a gradient- +based iterative optimization procedure can be established, namely (Liu & Nocedal 1989; +Badreddine et al. 2014) +𝐶 (𝑘+1) +𝑛 += 𝐶 (𝑘) +𝑛 ++ 𝛾(𝑘)𝑑 (𝑘) +𝑛 , (𝑛 = 1, 2, · · · , 𝑁) , +(4.41) +where 𝐶 (𝑘) +𝑛 +is the 𝑛-th model coefficient during the 𝑘-th gradient-based optimal iteration, 𝑑 (𝑘) +𝑛 +denotes the updated direction of the 𝑛-th model coefficient and 𝛾(𝑘) represents the step size. We +use a popular quasi-Newton method named limited-memory Broyden–Fletcher–Goldfarb–Shanno +(L-BFGS) algorithm to update the directions 𝑑 (𝑘) +𝑛 +(Liu & Nocedal 1989). The step size 𝛾(𝑘) is +calculated by the backtracking-Armijo line search method in the L-BFGS algorithm (Armijo +1966). +In summary, the diagram of the VOMM model is illustrated in Fig. 1, and the calculation steps +are listed as follows. +(1) We first select the pure structural ADM model without the dissipative Smagorinsky term +as the initial SGS model (Eq. 4.35) with model coefficients 𝐶 (0) +1 += 0 and 𝐶 (0) +2 += 1. +(2) The LES transient statistics (e.g. the dissipation spectrum shown in Eq. 4.36) is then +evaluated by forward calculating the LES equations (Eqs. 2.4 and 2.5) initialized by the filtered +DNS velocity field. The statistical discrepancy (Eq. 4.37) between the LES statistics and the a +priori measurable benchmark data (fDNS data) is measured to evaluate the performance of the +SGS model with current parameters. +(3) Afterwards, the stabilized adjoint LES equations (Eqs. 4.10 and 4.26) are integrated back- +ward with zero terminal conditions, driven by the loss sensitivity (Eq. 4.38) and corresponding +adjoint SGS model (Eqs. 4.39 and 4.40). The adjoint-based gradients of augmented functional +with respect to the model coefficients (Eq. 4.14) are sequentially evaluated using the adjoint +variables and the SGS basis forces. +(4) The L-BFGS gradient-based optimization algorithm (Eq. 4.41) is adopted to iteratively +update the SGS model parameters by repeating the above calculations until the stopping criteria +are satisfied. + +17 +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +��� +���������� +� +��� +��� +��� +��� +��� +��� +��� +��� +��� +� +J�J0 +� +� +� +��� +DNS +��� +� +�� +� +� +�� +� +��� +� +�� +� +� +�� +� +��� +� +�� +� +� +��� +� +Figure 3: The evolution of the normalized cost function in forced homogeneous isotropic +turbulence. +FGR +LES Resolution +𝐶 (0) +1 +𝐶 (0) +2 +𝐶opt +1 +𝐶opt +2 +1 +323 +0 +1 +-0.0529 +1.229 +2 +643 +0 +1 +-0.0101 +1.027 +4 +1283 +0 +1 +-0.0030 +1.000 +Table 2: The initial and optimal parameters of the VOMM model for LES computations +with the filter width ¯Δ = 32ℎDNS in forced homogeneous isotropic turbulence. +The stop criteria for the VOMM model for the optimization iterations are summarized as +follows: +(a) the number of iterations reaches the maximum number of iterations; +(b) the ratio of the current loss to the initial loss is smaller than a given error threshold 𝜖0 (e.g., +𝜖0 = 1%) , namely, J (𝑘)/J (0) ⩽ 𝜖0; +(c) the difference of model coefficients between two successive iterations is negligible, namely, +���𝐶 (𝑘+1) +𝑛 +− 𝐶 (𝑘) +𝑛 +��� / +���𝐶 (0) +𝑛 +��� ⩽ 𝜖0. +Eventually, the optimal parameters of the VOMM model are automatically obtained after +reaching the given stopping optimization criteria. +5. A posteriori studies of the VOMM models +In order to examine the performance of the proposed VOMM model, the a posteriori evaluations +are respectively carried out for the forced, decaying homogeneous isotropic turbulence and +temporally evolving turbulent mixing layer in this paper. The results of the filtered direct numerical +simulation (DNS) are the benchmark for the performance evaluations of the large-eddy simulation +(LES). We first introduce the detailed settings of DNS for these three turbulent problems. The +DNS data are then explicitly filtered by the commonly-used Gaussian filter, which is expressed + +18 +Model(FGR=1,𝑁 = 323) +DSM +DMM +ADM(𝜒=0) +ADM(𝜒=1) +VOMM +t(CPU·s) +0.142 +0.243 +0.056 +0.056 +0.066 +t/tDMM +0.584 +1 +0.231 +0.230 +0.273 +Model(FGR=2,𝑁 = 643) +DSM +DMM +ADM(𝜒=0) +ADM(𝜒=1) +VOMM +t(CPU·s) +0.870 +1.465 +0.368 +0.361 +0.418 +t/tDMM +0.594 +1 +0.251 +0.246 +0.285 +Model(FGR=4,𝑁 = 1283) +DSM +DMM +ADM(𝜒=0) +ADM(𝜒=1) +VOMM +t(CPU·s) +6.512 +10.103 +2.517 +2.588 +3.240 +t/tDMM +0.645 +1 +0.249 +0.256 +0.321 +Table 3: The average computational cost of SGS stress modeling 𝜏𝑖 𝑗 for LES computations +with the filter width ¯Δ = 32ℎDNS in forced homogeneous isotropic turbulence. +as (Pope 2000; Sagaut 2006) +𝐺 �r; ¯Δ� = +� 6 +𝜋 ¯Δ2 +�1/2 +exp +� +−6r2 +¯Δ2 +� +. +(5.1) +The filter scale ¯Δ = 32ℎDNS is selected for both the forced and decaying homogeneous isotropic +turbulence, while ¯Δ = 8ℎDNS for the temporally evolving turbulent mixing layer, where ℎDNS +denotes the grid spacing of DNS. Three conventional SGS models, i.e., the dynamic Smagorinsky +model (DSM, Eq. 3.1), the dynamic mixed model (DMM, Eq. 3.3) and the approximate +deconvolution model with standard secondary filtering regularization (ADM, Eqs. 3.6 ∼ 3.8) +are adopted to compare against the VOMM model. The consistent instantaneous snapshots of the +filtered DNS data are used to initialize the LES calculations for different SGS models. Both the +turbulent statistics and transient contours are evaluated and compared with different SGS models +for the a posteriori testings of the three canonical turbulent flows. +5.1. Forced homogeneous isotropic turbulence +We perform the direct numerical simulation of forced incompressible isotropic turbulence +using the uniform grid resolution 𝑁 = 10243 in a cubic box of (2𝜋)3 with periodic boundary +conditions (ℎDNS = 2𝜋/1024) (Xie et al. 2020a,c; Yuan et al. 2020). The pseudo-spectral method +is used for the spatial discretization of the governing equations (Canuto et al. 1988; Peyret 2002). +The nonlinear advection terms are fully dealiased by the two-thirds dealiasing rule (Canuto et al. +1988). A second-order two-step Adams-Bashforth explicit scheme is used for time integration +(Chen et al. 1993). +The kinematic viscosity is chosen as 𝜈 = 0.001, and large-scale forcing is applied to the two +lowest wavenumber shells to maintain the turbulence in statistical equilibrium, giving rise to the +Taylor Reynolds number Re𝜆 ≈ 250 (Wang et al. 2010; Yuan et al. 2020). The detailed one-point +statistics of DNS data for the forced isotropic turbulence are summarized in Table 1 (Yuan et al. +2022). Here, 𝑘max = +2𝜋 +3ℎDNS denotes the largest effective wavenumber after the fully dealiasing, +and 𝜔rms = +√︁ +⟨𝜔𝑖𝜔𝑖⟩ represents the root-mean-square value of the vorticity magnitude, where +𝜔 = ∇ × u stands for the vorticity which is the curl of the velocity field. The Kolmogorov length +scale 𝜂 and the integral length scale 𝐿𝐼 stand for the smallest resolved scale and the largest + +19 +(𝑎) +�� +� +�� +� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +���� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +� +�� +�� +�� +� +�� +�� +�� +�� +�� +� +�� +� +���� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +�� +� +�� +� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +���� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑑) +� +� +�� +�� +�� +�� +� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +���� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑒) +�� +� +�� +� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +���� +� +� +��� +� +� +� +� +� +��� +DNS +��� +� +� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +( 𝑓 ) +� +�� +�� +�� +�� +�� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +���� +� +� +��� +� +� +� +� +� +��� +DNS +��� +� +� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 4: Velocity spectra for different SGS models in the a posteriori analysis of forced +homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS: (a) log-log for +FGR=1, 𝑁 = 323; (b) semi-log for FGR=1, 𝑁 = 323; (c) log-log for FGR=2, 𝑁 = 643; (d) +semi-log for FGR=2, 𝑁 = 643; (e) log-log for FGR=4; 𝑁 = 1283; and (f) semi-log for +FGR=4, 𝑁 = 1283. +characteristic scale of turbulence, and are defined respectively by +𝜂 = +� 𝜈3 +𝜀 +�1/4 +, +(5.2) +𝐿𝐼 = +3𝜋 +2(𝑢rms)2 +∫ +∞ +0 +𝐸 (𝑘) +𝑘 +𝑑𝑘, +(5.3) +where 𝜀 is the spatial average dissipation rate of kinetic energy. The total turbulent kinetic energy +𝐸𝑘 = ⟨𝑢𝑖𝑢𝑖⟩ /2 = +∫ +∞ +0 +𝐸 (𝑘) 𝑑𝑘, and 𝐸 (𝑘) represents the velocity spectrum. The resolution +parameters 𝑘max𝜂 ⩾ 2.1 and 𝜂/ℎDNS ⩾ 1 indicate that the grid resolution is sufficient to capture + +20 +(𝑎) +� +� +� +� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +�� +�� +�� +�� +� +�� +� +� +� +� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +� +� +� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +�� +�� +�� +�� +� +�� +� +� +� +� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +� +� +� +� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +�� +�� +�� +�� +� +�� +� +� +� +� +� +� +��� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 5: Second-order structure functions of the filtered velocity for LES in the a +posteriori analysis of forced homogeneous isotropic turbulence with the same filter scale +¯Δ = 32ℎDNS: (a) FGR=1, 𝑁 = 323; (b) FGR=2, 𝑁 = 643; and (c) FGR=4, 𝑁 = 1283. +the smallest turbulent eddy scales and ensure the convergence of turbulent kinetic energy at all +scales (Ishihara et al. 2007, 2009). In order to alleviate the impact of initial conditions, the forced +homogeneous isotropic turbulence is run for a long period after the flow gradually reaches a +statistically steady state (more than 50 large-eddy turnover times 𝜏 = 𝐿𝐼 /𝑢rms). We select data of +the last ten large-eddy turnover times as a benchmark for LES comparisons (total forty flow-field +snapshots of DNS data). +In this paper, the Gaussian filter (Eq. 5.1) is used as the explicit filter to calculate the filtered +physical variables. Theselected filter width ¯Δ = 32ℎDNS and the correspondingcutoff wavenumber +is 𝑘𝑐 = 𝜋/ ¯Δ = 16. The velocity and dissipation spectra of the DNS and filtered DNS at ¯Δ = 32ℎDNS +are illustrated in Fig. 2. The filtered velocity spectrum nearly overlaps with the DNS data in a +Kolmogorov scaling law of 𝑘−5/3 at the low wavenumber region, while it drops significantly at +the region larger than the truncated wavenumber 𝑘𝑐. Overall 12% of the turbulent kinetic energy +is filtered out in the residual velocity field at the filter scale ¯Δ = 32ℎDNS. In contrast, the filtered +dissipation spectrum gradually grows with the power of law scaling 𝑘1/3 at the low-wavenumber +inertial region, and drops sharply where the cutoff wavenumber exceeds. The small scales near +the truncated wavenumbers are essential for the reconstruction of the filtered dissipation spectrum +and also very important for the residual SGS modeling. However, these small scales account for a +very small proportion of the turbulent kinetic energy, almost several orders of magnitude smaller +than the large scales. Thus, the dissipation spectrum instead of the kinetic energy spectrum is +chosen as the optimization objective function of the proposed VOMM model in the paper. +The a posteriori testings of LES are essential to validate the practical performance of the SGS +models. LES calculations use the same kinematic viscosity (𝜈 = 0.001) with the DNS. The filter +width is fixed to ¯Δ = 32ℎDNS and the impact of the spatial discretization errors on the SGS +models is investigated by changing the grid resolution of LES. Three different filter-to-grid ratios +FGR= ¯Δ/ℎLES=1, 2 and 4 are chosen to study the influence of spatial discretization on the SGS +modeling, and the corresponding grid points of LES are 𝑁 = 323, 643 and 1283, respectively. The +proposed VOMM model (Eq. 4.35) is compared against the classical SGS models, including the +dynamic Smagorinsky model (DSM, Eq. 3.1), the dynamic mixed model (DMM, Eq. 3.3) and +the standard approximate deconvolution model with secondary filtering regularization (ADM, +Eqs. 3.7 and 3.8). The relaxation factors of ADM model 𝜒=0 and 1 are chosen for comparisons. +The ratios of the time steps for LES and DNS are Δ𝑡LES/Δ𝑡DNS = {10, 10, 5} for different grids +(FGR=1, 2 and 4 with 𝑁 = 323, 643 and 1283). Among the filtered DNS data of the ten large- +eddy turnover periods, the data of the first two large-eddy turnover times are used for the adjoint +optimization of the VOMM model (only the dissipation spectrum is used, stored once every 0.1𝜏, +twenty sets in total), and the remaining data of the last eight large-eddy turnover times are used +for the a posteriori accuracy validation of the LES models. + +21 +(𝑎) +� +� +� +� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +�� +�� +�� +�� +� +�� +� +� +� +� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +� +� +� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +�� +�� +�� +�� +� +�� +� +� +� +� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +� +� +� +� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +�� +�� +�� +�� +� +�� +� +� +� +� +� +� +��� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 6: Fourth-order structure functions of the filtered velocity for LES in the a +posteriori analysis of forced homogeneous isotropic turbulence with the same filter scale +¯Δ = 32ℎDNS: (a) FGR=1, 𝑁 = 323; (b) FGR=2, 𝑁 = 643; and (c) FGR=4, 𝑁 = 1283. +At the adjoint-based optimization stage of the VOMM model, the calculations of the adjoint +equations are consistent with the primary LES equations. We adopt the same pseudo-spectral +numerical scheme to spatially discrete the stabilized adjoint momentum equations (Eq. 4.26). +A second-order two-step Adams-Bashforth explicit scheme is applied for the time backward +integration with zero terminal conditions. Since the large-scale forcing is assumed to be nearly +independent of the filtered velocity, the large-scale forcing term does not appear in the adjoint +momentum equations. During the adjoint optimization stage (see Fig. 1) of the VOMM model, +the pure structural ADM model without the dissipative Smagorinsky term is selected as the +initial SGS model with model coefficients 𝐶 (0) +1 += 0 and 𝐶 (0) +2 += 1. The LES forward evolution +is initialized by the filtered DNS velocity field and the dissipation spectrum is calculated when +the filtered DNS data are available (every 0.1𝜏). The statistical discrepancy of the dissipation +spectrum between the LES and fDNS data is evaluated and recorded as the cost functional. The +adjoint-based gradients of the cost functional with respect to the model coefficients are calculated +through backward integrating the stabilized adjoint LES equations (Eqs. 4.10 and 4.26) with zero +terminal conditions. The SGS model coefficients are then iteratively updated by the gradient-based +L-BFGS optimization algorithm (Eq. 4.41) until reaching the stopping criteria. +Figure 3 shows the evolution of the cost function normalized by the initial discrepancy during +the adjoint-based optimization in forced homogeneous isotropic turbulence. The loss functions +(prediction errors of dissipation spectra between LES and fDNS data) for all three different +filter-to-grid ratio cases (FGR=1,2 and 4) gradually converge and become stationary within less +than twenty iterations. The error is significantly reduced by nearly an order of magnitude for the +cases of FGR=1 and 2 within about ten iterations, and is drastically reduced to 20% of the initial +state at FGR=4. These results indicate that the adjoint-based L-BFGS gradient optimization is +very efficient and effectively obtains the optimal model coefficients within several iterations. The +optimal parameters of the VOMM model are summarized in Table 2. The magnitude of the eddy- +viscosity coefficient (( +���𝐶opt +1 +���) ) dramatically reduces from 0.0529 to 0.003 with the increasing of +FGR and LES resolutions, while the coefficient of the ADM part (𝐶opt +2 ) gradually approaches +unity, which is identical to the theoretical value derived from the Taylor series expansions. Once +the optimal model coefficients are obtained, we further examine the a posteriori performance of +the VOMM model using the filtered DNS data of the last eight large-eddy turnover periods. +Table 3 gives the average computational cost for the SGS stress modeling at the same filter width +¯Δ = 32ℎDNS. For all three different grid resolutions, the computation time of the VOMM model +is only about 30% of that of the DMM model, without significantly increasing the computational +cost in comparison to the ADM models (𝜒 = 0 and 1). +The velocity spectra of different SGS models with the filter scale ¯Δ = 32ℎDNS in comparison to + +22 +(𝑎) +� +� +� +� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +�� +� +� +� +� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +� +� +� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +�� +�� +�� +�� +� +�� +� +�� +� +� +� +� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +� +� +� +� +� +�� +�� +�� +�� +�� +�� +�� +� +� +�� +�� +�� +�� +�� +� +�� +� +�� +� +� +� +� +� +� +��� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 7: Sixth-order structure functions of the filtered velocity for LES in the a posteriori +analysis of forced homogeneous isotropic turbulence with the same filter scale +¯Δ = 32ℎDNS: (a) FGR=1, 𝑁 = 323; (b) FGR=2, 𝑁 = 643; and (c) FGR=4, 𝑁 = 1283. +those of the DNS and filtered DNS (fDNS) data are shown in Fig. 4. The velocity spectrum of DNS +data exhibits a sufficiently long inertial range with a typical 𝑘−5/3 scaling. The spectrum of fDNS +almost overlaps with that of DNS at the low-wavenumber region, but is obviously lower than that +of DNS near the truncated wavenumber since the small-scale kinetic energy at high wavenumbers +is filtered out. LES only solves the large-scale variables with the filtered Navier-Stokes equations +(Eqs. 2.4 and 2.5), leaving the effect of residual small scales to be approximately reconstructed by +the SGS model. Therefore the statistics of an ideal LES would overlap with that of the fDNS data +as closely as possible. When the grid resolution of LES is sufficiently coarse and the grid spacing +of LES is equal to the filter scale (FGR=1, c.f. Figs 4a and 4b), the spatial discretization error +is significant and deteriorates the accuracy of the SGS stress modeling. LES calculations with +traditional SGS models are very difficult to obtain accurate predictions of the turbulent kinetic +energy cascade at FGR=1. The velocity spectra predicted by the ADM models with 𝜒 = 0 and 1 +exhibit numerical unstable, and kinetic energy at high wavenumbers is obviously overestimated +due to the insufficient dissipation. DSM and DMM models also have dramatic overestimations +at high-wavenumber regions, with predictions even larger than that of the DNS data. In contrast, +VOMM model predicts the velocity spectra most accurately among these SGS models whose +results nearly coincide with that of fDNS. +For the cases of fine grid resolutions (FGR=2 and 4), the pure ADM model (𝜒 = 0) is +still numerically unstable since the pure structural model itself cannot produce sufficient SGS +dissipation. The ADM model with the standard secondary-filtering regularization (𝜒 = 1) exhibits +excessively dissipative, and the small-scale kinetic energy at high wavenumbers is extremely +exhausted and much lower than that of fDNS. The predictions of DSM and DMM models illustrate +the obviously tilted distribution, where kinetic energy at low wavenumbers is accumulated, while +that near the truncated wavenumber is diminished. The dynamic least-square procedure for both +DSM and DMM models would overestimate the eddy-viscosity coefficient for the cases of fine +grid resolutions (FGR=2 and 4), and small-scale flow structures near the truncated wavenumbers +are exhausted by the excessive dissipation. The turbulent kinetic energy is transferred from large +scales to small scales through the forward energy cascade process of the nonlinear advection. The +lack of the sufficient flow structures near the cutoff wavenumber leads to the energy accumulation +in the intermediate wavenumber region. In contrast, the VOMM model is superior to the other +SGS models and can accurately predict the velocity spectra at all different grid resolutions of +LES, with the predictions very close to the fDNS data. +In order to further examine the reconstruction of multiscale properties of turbulence by the +SGS models, we calculate the longitudinal structure functions of the filtered velocity, namely + +23 +(𝑎) +�� +�� +�� +� +� +� +� +� +� +� +� +�� +� +� +��� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +��� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +�� +�� +�� +�� +� +� +� +� +� +� +� +� +� + � +� + +��� +�� +� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +��� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +�� +�� +�� +�� +�� +� +� +� +� +� +� +� +� +� +� +!� +� +! +�� +�� +� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +��� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑑) +�� +�� +�� +�� +�� +� +� +� +� +� +� +� +� +� +� +!� +� +! +�� +�� +� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� +� +��� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 8: PDFs of the normalized velocity increments 𝛿r ¯𝑢/ ¯𝑢rms for LES at grid resolution +of 323 in the a posteriori analysis of forced homogeneous isotropic turbulence with the +same filter scale ¯Δ = 32ℎDNS: (a) r = ¯Δ; (b) r = 2 ¯Δ; (c) r = 3 ¯Δ; (d) r = 4 ¯Δ. +(Xie et al. 2018, 2019a) +¯𝑆𝑛(𝑟) = +����� +𝛿𝑟 ¯𝑢 +¯𝑢rms +���� +𝑛� +, +(5.4) +where 𝑛 represents the order of structure function and 𝛿𝑟 ¯𝑢 = [¯u (x + r) − ¯u (x)] · ˆr denotes +the longitudinal velocity increment at the separation r with the unit distance vector ˆr = r/|r|. +Figures 5, 6 and 7 respectively compare the second-order, fourth-order and sixth-order structure +functions of the filtered velocity for different SGS models with the filtered DNS data. For +all three grid resolutions of LES (FGR=1, 2 and 4), all SGS models predict the lower-order +structure functions (Fig. 5) much better than the higher-order structure functions (Figs. 6 and 7). +Besides, the predictions of structure functions are improved greatly with the increasing of the +grid resolution, and those of all SGS models almost coincide with each other at large separations. +The ADM models (both 𝜒 = 0 and 1) give the worst predictions and obviously overestimate the +structure function at small distances r. DSM and DMM models also predict the structure functions +greater than the fDNS data at small separations but underestimate the structure functions at large +distances. In contrast, the VOMM model can accurately reconstruct the structure functions with +different orders at both small and large separations, almost overlapping with those of the filtered +DNS. +We then evaluate the probability density functions (PDFs) of the filtered velocity increments to +measure the spatial correlations of turbulence, as shown in Fig. 8, where the velocity increments +𝛿𝑟 ¯𝑢/ ¯𝑢rms are normalized by the root-mean-square value of velocity. The cases of fine grid +resolutions (FGR=2 and 4) are very similar to that of FGR=1 and not shown in the paper. +The PDFs of the velocity increments exhibit approximately symmetrical distribution, relatively + +24 +(a) fDNS +(b) DMM +(c) ADM (𝜒=0) +(d) ADM (𝜒=1) +(e) DSM +(f) VOMM +Figure 9: Contours of the normalized vorticity ¯𝜔/ ¯𝜔rms +fDNS at an arbitrary 𝑥1-𝑥2 plane at +𝑡/𝜏 ≈ 4 for LES at a grid resolution of 643 (FGR=2) in forced homogeneous isotropic +turbulence with the filter width ¯Δ = 32ℎDNS: (a) fDNS, (b) DMM, (c) ADM(𝜒=0), (d) +ADM(𝜒=1), (e) DMM, and (f) VOMM. + +rms +0 +0.5 +1.5 +2 +2.5 +3 +0.8 +0.6 +0.4 +0.2 +0 +0.2 +0.4 +0.6 +0.8 +C1/2πrms +0 +0.5 +1.5 +2 +2.5 +3 +0.8 +0.6 +0.4 +0.2 +0 +0.2 +0.4 +0.6 +0.8 +C1/2πrms +0 +0.5 +1.5 +2 +2.5 +3 +0.8 +0.6 +0.4 +0.2 +0.2 +0.4 +0.6 +0.8 +C1/2πrms +0 +0.5 +1.5 +2 +2.5 +3 +0.8 +0.6 +0.4 +0.2 +0.2 +0.4 +0.6 +0.8 +C1/2πrms +0 +0.5 +1.5 +2 +2.5 +3 +0.8 +0.6 +0.4 +0.2 +0.2 +0.4 +0.6 +0.8 +C1/2πrms +0 +0.5 +1.5 +2 +2.5 +3 +0.8 +0.6 +0.4 +0.2 +0.2 +0.4 +0.6 +0.8 +C1/2π25 +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +��� +���������� +� +��� +��� +��� +��� +��� +��� +��� +��� +��� +� +J�J0 +� +� +� +��� +DNS +��� +� +�� +� +� +�� +� +��� +� +�� +� +� +�� +� +��� +� +�� +� +� +��� +� +Figure 10: The evolution of the normalized cost function in decaying homogeneous +isotropic turbulence. +FGR +LES Resolution +𝐶 (0) +1 +𝐶 (0) +2 +𝐶opt +1 +𝐶opt +2 +1 +323 +0 +1 +-0.0398 +3.150 +2 +643 +0 +1 +-0.0094 +1.326 +4 +1283 +0 +1 +-0.0020 +1.101 +Table 4: The initial and optimal parameters of the VOMM model for LES computations +with the filter width ¯Δ = 32ℎDNS in decaying homogeneous isotropic turbulence. +concentrated at small distances while gradually becoming wider as the distance increases. The +PDFs predicted by the ADM, DSM and DMM models are significantly wider than those of +the fDNS. In comparison with these traditional SGS models, the VOMM model gives the most +accurate prediction of the velocity increments for different distances, which are in reasonable +agreement with the fDNS data. +We finally examine the reconstruction of instantaneous spatial flow structures by plotting the +contours of the normalized vorticity magnitude as shown in Fig. 9. The vorticity contours are +consistently extracted on an arbitrary 𝑥1-𝑥2 plane for the isotropic turbulence at the same time +with approximately four large-eddy turnover periods ( 𝑡/𝜏 ≈ 4) at a grid resolution of 643. It +is noteworthy that the exact point-to-point correlations are difficult to achieve under the long- +term forecasting of LES due to the chaotic nature of the turbulence and extreme sensitivity to +perturbations (Pope 2000; Wang et al. 2022c, 2023). The pure ADM model overpredicts some +unrealistic small-scale structures, which are obviously different from the band-like or strip-like +spatial structures of the fDNS data. The DSM, DMM and ADM (𝜒 = 1) models only predict +the large-scale vorticity structures and some small scales are excessively dissipated. Compared +to these traditional SGS models, the VOMM model predicts the vortex structures very similar to +the fDNS data. + +26 +Model(FGR=1,𝑁 = 323) +DSM +DMM +ADM(𝜒=0) +ADM(𝜒=1) +VOMM +t(CPU·s) +0.153 +0.259 +0.065 +0.062 +0.070 +t/tDMM +0.590 +1 +0.249 +0.239 +0.269 +Model(FGR=2,𝑁 = 643) +DSM +DMM +ADM(𝜒=0) +ADM(𝜒=1) +VOMM +t(CPU·s) +1.026 +1.857 +0.567 +0.563 +0.589 +t/tDMM +0.553 +1 +0.306 +0.303 +0.317 +Model(FGR=4,𝑁 = 1283) +DSM +DMM +ADM(𝜒=0) +ADM(𝜒=1) +VOMM +t(CPU·s) +6.026 +10.287 +2.521 +2.531 +3.393 +t/tDMM +0.586 +1 +0.245 +0.246 +0.330 +Table 5: The average computational cost of SGS stress modeling 𝜏𝑖 𝑗 for LES computations +with the filter width ¯Δ = 32ℎDNS in decaying homogeneous isotropic turbulence. +(𝑎) +� +� +� +� +� +� +� +��� +� +���� +���� +���� +���� +��� +���� +���� +� +� +��� +� +�� +� +� +�� +� +� +� +� +� +��� +DNS +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +� +� +� +� +� +� +��� +� +���� +���� +���� +���� +��� +���� +���� +� +� +��� +� +�� +� +� +�� +� +� +� +� +� +��� +DNS +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +� +� +� +� +� +� +� +��� +� +���� +���� +���� +���� +��� +���� +���� +� +� +��� +� +�� +� +� +��� +� +� +� +� +� +��� +DNS +���� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 11: Temporal evolutions of the turbulent kinetic energy 𝐸𝑘 for LES in the a +posteriori analysis of decaying homogeneous isotropic turbulence with the same filter +scale ¯Δ = 32ℎDNS: (a) FGR=1, 𝑁 = 323; (b) FGR=2, 𝑁 = 643; and (c) FGR=4, 𝑁 = 1283. +5.2. Decaying homogeneous isotropic turbulence +In order to investigate the impact of turbulent unsteady evolution on SGS stress modeling, +the numerical simulation of decaying homogeneous isotropic turbulence in a cubic box of (2𝜋)3 +with periodic boundary conditions is investigated in this subsection. The numerical simulation +method is consistent with the forced homogeneous isotropic turbulence. We spatially discretize +the governing equations using the pseudo-spectral method with the two-thirds dealiasing rule at +a uniform grid resolution of 𝑁 = 10243. The temporal discretization scheme adopts the second- +order two-step Adams-Bashforth explicit method. The statistically steady isotropic turbulence +data of the forced isotropic turbulence (detailed statistics see Fig. 1) is used as the initial field +for DNS decaying turbulence without the large-scale forcing. The kinematic viscosity is set to +𝜈 = 0.001 and the initial Taylor Reynolds number is Re𝜆 ≈ 250. We calculate the DNS data of +decaying turbulence for about six large-eddy turnover times (𝜏 = 𝐿𝐼 /𝑢rms), the first two of which +are used for the adjoint-based optimization to determine the model coefficients of VOMM model +(only the dissipation spectrum is used, stored once every 0.1𝜏, twenty sets in total). +The a posteriori studies of LES adopt the consistent kinematic viscosity (𝜈 = 0.001) with +the DNS. The Gaussian filter (Eq. 5.1) is selected as the explicit filter with the given filter +width ¯Δ = 32ℎDNS. Similar to the forced isotropic turbulence, three different filter-to-grid ratios +FGR= ¯Δ/ℎLES=1,2 and 4 are chosen to investigate the impact of the spatial discretization on the +SGS stress modeling with the corresponding grid resolutions of LES 𝑁 = 323, 643 and 1283. The + +27 +(𝑎) +� +� +� +� +� +� +� +��� +� +��� +� +��� +� +��� +� +� +��� +� +�� +� +� +�� +� +� +� +� +� +��� +DNS +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +� +� +� +� +� +� +��� +� +��� +� +��� +� +��� +� +� +��� +� +�� +� +� +�� +� +� +� +� +� +��� +DNS +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +� +� +� +� +� +� +� + �� +� +��� +� +��� +� +��� +� +� +��� +� +�� +� +� +��� +� +� +� +� +� +��� +DNS +���� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 12: Temporal evolutions of the average dissipation rate ¯𝜀 for LES in the a +posteriori analysis of decaying homogeneous isotropic turbulence with the same filter +scale ¯Δ = 32ℎDNS: (a) FGR=1, 𝑁 = 323; (b) FGR=2, 𝑁 = 643; and (c) FGR=4, 𝑁 = 1283. +adjoint-based optimization of the VOMM model (c.f. Fig. 1) is first performed to determine the +optimal model coefficients using the dissipation spectra as the cost functional. The pure ADM +model without the Smagorinsky part is used as the initial SGS model with parameters 𝐶 (0) +1 += 0 +and 𝐶 (0) +2 += 1. The adjoint-based gradients of the cost functional for the model coefficients are +evaluated by successively forward solving the LES equations (Eqs. 2.4 and 2.5) and backward +integrating the stabilized adjoint LES equations (Eqs. 4.10 and 4.26). The gradient-based L-BFGS +optimization algorithm (Eq. 4.41) is used for iteratively updating the SGS model parameters until +reaching the stopping criteria. +The evolution of the cost function normalized by the initial loss during the adjoint-based +optimization for the decaying isotropic turbulence is displayed in Fig. 10. The loss functions for +all three cases of different grid resolutions (FGR=1,2 and 4) drop rapidly at the beginning and +gradually reach a plateau within approximately twenty iterations. The prediction errors of the +optimization objective are considerably reduced to 10% of the initial state for both FGR=1 and +2, and substantially decreased to about 20% of the original value at FGR=4. The adjoint-based +gradient optimization can quickly obtain the optimal model parameters within a limited number +of iterations (less than 100 optimization iterations, namely, 200 LES evaluations). Table 4 gives +the optimal parameters of the VOMM model. The magnitude of the dissipative Smagorinsky +coefficient ( +���𝐶opt +1 +���) significantly drops from 0.0398 to 0.002 as the LES resolution increases, +which is slightly lower than that in forced homogeneous isotropic turbulence. In contrast, the +coefficient of the structural part (𝐶opt +2 ) is asymptotically close to unity as the grid spacing of LES +becomes smaller, similar to the results of forced isotropic turbulence. +The a posteriori performance of the VOMM model is further validated after determining the +optimal SGS model coefficients by the adjoint-based gradient optimization. We compare the +proposed VOMM model (Eq. 4.35) with the classical SGS models including the DSM model +(Eq. 3.1), DMM model (Eq. 3.3) and the ADM model regularized by the standard secondary- +filtering technique (Eqs. 3.7 and 3.8). The time steps of LES are given as Δ𝑡LES/Δ𝑡DNS = +{10, 10, 5} for different grid resolutions (FGR=1, 2 and 4 with 𝑁 = 323, 643 and 1283). The +average computational costs for the SGS stress modeling with different grid resolutions using +different SGS models at the same filter scale ¯Δ = 32ℎDNS are summarized in Table 5. The +computation time of the VOMM model only accounts for approximately 30% of the time of +DMM model and slightly increases in computational cost compared to the ADM models with +𝜒 = 0 and 1. +Figures 11 and 12 respectively compare the temporal evolutions of the turbulent kinetic energy +and the resolved dissipation rate ( ¯𝜀 = 2𝜈 +� ¯𝑆𝑖 𝑗 ¯𝑆𝑖 𝑗 +� +) of different SGS models with the filtered DNS +(fDNS) data. The turbulent kinetic energy gradually decays from the initial statistically steady +state over time, since there are no additional forcing driving the dissipative turbulent system. All +the classical SGS models (DSM, DMM and ADM models) clearly overestimate the kinetic energy + +28 +(𝑎) +�� +� +�� +� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +� +���� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +�� +��� +� +��� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +�� +� +�� +� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +� +���� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +�� +��� +� +��� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +�� +� +�� +� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +���� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +�� + �� +� +��� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑑) +�� +� +�� +� +�� +� +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +���� +� +� +�� +� +� +� +� +� +��� +DNS +��� +� +�� + �� +� +��� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑒) +�� +� +�� +� +�� +� + +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� � +� +� +��� +� +� +� +� +� +��� +DNS +��� +� +�� +!�� +� +��� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +( 𝑓 ) +�� +� +�� +� +�� +� + +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +� +�� � +� +� +��� +� +� +� +� +� +��� +DNS +��� +� +�� +!�� +� +��� +��� +���� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 13: Velocity spectra for different SGS models in the a posteriori analysis of +decaying homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS at +𝑡/𝜏 ≈ 2 and 4: (a) FGR=1, 𝑁 = 323 at 𝑡/𝜏 ≈ 2; (b) FGR=1, 𝑁 = 323 at 𝑡/𝜏 ≈ 4; (c) +FGR=2, 𝑁 = 643 at 𝑡/𝜏 ≈ 2; (d) FGR=2, 𝑁 = 643 at 𝑡/𝜏 ≈ 4; (e) FGR=4, 𝑁 = 1283 at +𝑡/𝜏 ≈ 2; and (f) FGR=4, 𝑁 = 1283 at 𝑡/𝜏 ≈ 4. +throughout the time, which differs significantly from the benchmark fDNS data. In contrast, the +VOMM model gives reasonable predictions of the turbulent kinetic energy, which is the closest +to the fDNS data. The average dissipation rate displays a decline trend with time, similar to that +of the turbulent kinetic energy. However, all conventional SGS models wrongly predict the non- +monotonic tendency of the average dissipation rate over time. For the case of sufficiently coarse +grid resolution of LES (FGR=1 with 𝑁 = 323), DSM, DMM and ADM models overpredict the +dissipation rate with an erroneous temporal evolution that first increases and then decreases. When +the grid resolution of LES becomes fine (FGR=2 and 4 with 𝑁 = 643 and 1283), DSM and DMM +models obviously underestimate the dissipative rate at the early stage of decaying turbulence +(𝑡/𝜏 ⩽ 3), then DMM model gradually becomes closer to the fDNS data while DSM model + +29 +(𝑎) +� +��� +� +��� +� +��� +� +� +�� +� +� +#"$ + ��� +� +��� +��� +��� +��� +��� +��� +��� +� +� +�� +� +� +� +� +� +��! +DNS +��� +� +�� +%�� +� +��� + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +��� +� +��� +� +��� +� +� +�� +� +� +#"$ + ��� +� +��� +��� +��� +��� +��� +��� +��� +� +� +�� +� +� +� +� +� +��! +DNS +��� +� +�� +%�� +� +��� + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +� +��� +� +��� +� +��� +� +� +�� +� +� +#"$ + ��� +� +��� +��� +��� +��� +��� +��� +��� +� +� +��� +� +� +� +� +� +��! +DNS +��� +� +�� +%�� +� +��� + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 14: PDFs of the normalized vorticity ¯𝜔/ ¯𝜔rms +fDNS for LES in the a posteriori analysis +of decaying homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS at +𝑡/𝜏 ≈ 4: (a) FGR=1, 𝑁 = 323; (b) FGR=2, 𝑁 = 643; and (c) FGR=4, 𝑁 = 1283. +overestimates the dissipation rate with the decaying of turbulence. The pure ADM model ( 𝜒 = 0 +) always gives the overestimations of the dissipation rate for all three different grid resolutions +of LES, even though the pure ADM model can accurately predict the turbulent kinetic energy +at a sufficiently high grid resolution (FGR=4). These results demonstrate that the pure structural +ADM model without any dissipative terms might not accurately predict all physical quantities +of LES (i.e., the average dissipation rate), even if the grid resolution is high enough compared +to the filter scale (FGR=4). The ADM model with standard secondary-filtering regularization +(𝜒 = 1) provides excessive dissipation similar to the DSM model with mispredictions of first +underestimating and then overestimating the average dissipation rate over time at FGR=2 and 4. +In comparison to these classical SGS models, the VOMM model accurately predicts the temporal +evolutions of average dissipation rate for all three different grid resolutions, which agrees fairly +well with the benchmark filtered DNS data. +The transient velocity spectra of different SGS models at the filter width ¯Δ = 32ℎDNS with two +different time instants 𝑡/𝜏 ≈ 2 and 4 are further illustrated in Fig. 13. The velocity spectra exhibit +an overall decrease, and the kinetic energy at all wavenumbers declines with the decaying of +turbulence. All the classical SGS models (DSM, DMM and ADM models) overpredict the kinetic +energy at high wavenumbers for the coarse grid-resolution case (FGR=1 with 𝑁 = 323 ) and the +excessive kinetic energy stacked at small scales leads to the numerical instability of LES, which +gradually intensifies with the evolution of time. The conventional SGS models provide insufficient +model dissipation to balance the discretization errors and the small-scale kinetic energy cannot +be effectively dissipated in time at FGR=1. For the fine grid-resolution cases (FGR=2 and 4 with +𝑁 = 643 and 1283), the dissipation of the traditional SGS models (DSM, DMM models, and ADM +model with 𝜒 = 1) is too strong to diminish most small-scale flow structures near the truncated +wavenumber, which hinders the normal transmission of turbulent kinetic energy cascades from +large scales to small scales. Therefore, the kinetic energy of classical SGS models accumulates in +the region of intermediate wavenumbers, leading to the overestimations of the turbulent kinetic +energy with time (Fig. 11) at FGR=2 and 4 with 𝑁 = 643 and 1283. LES using the pure ADM +model with 𝜒 = 0 is always numerically unstable and lacks necessary SGS dissipation to drain +out the small-scale kinetic energy for all different grid resolutions. Compared to these classical +SGS models, the VOMM model can accurately reconstruct the kinetic energy cascade with the +predictions that nearly coincide with those of fDNS at all three different grid resolutions. +Furthermore, we compare the PDFs of the normalized vorticity magnitude at the dimensionless +time 𝑡/𝜏 ≈ 4 as shown in Fig. 14. The vorticity is normalized by the root-mean-square values +of the vorticity calculated by the fDNS data for comparisons of different grid resolutions. The +pure ADM models with 𝜒 = 0 gives the worst prediction of the vorticity with erroneous peaks + +30 +(a) fDNS +(b) DMM +(c) ADM (𝜒=0) +(d) ADM (𝜒=1) +(e) DSM +(f) VOMM +Figure 15: Contours of the normalized vorticity ¯𝜔/ ¯𝜔rms +fDNS at an arbitrary 𝑥1-𝑥2 plane at +𝑡/𝜏 ≈ 4 for LES at a grid resolution of 643 (FGR=2) in decaying homogeneous isotropic +turbulence with the filter width ¯Δ = 32ℎDNS: (a) fDNS, (b) DMM, (c) ADM(𝜒=0), (d) +ADM(𝜒=1), (e) DMM, and (f) VOMM. + +0 +0.5 +1.5 +2 +0.8 +0.6 +0.4 +0.2 +0 +0.2 +0.4 +0.6 +0.8 +C1/2πrms +0 +0.5 +1.5 +2 +0.8 +0.6 +0.4 +0.2 +0 +0.2 +0.4 +0.6 +0.8 +C1/2πrms +0 +0.5 +1.5 +2 +0.8 +0.6 +0.4 +0.2 +0 +0.2 +0.4 +0.6 +0.8 +C1/2πrms +0 +0.5 +1.5 +2 +0.8 +0.6 +2 +0.4 +0.2 +0 +0.2 +0.4 +0.6 +0.8 +C1 /2πrms +0 +0.5 +1.5 +2 +0.8 +0.6 +0.4 +0.2 +0 +0 +0.2 +0.4 +0.6 +0.8 +C1/2πrms +0 +0.5 +1.5 +2 +0.8 +0.6 +0.4 +0.2 +0.2 +0.4 +0.6 +0.8 +C1/2π31 +(𝑎) +(𝑏) +π +Figure 16: Diagram of the temporally evolving mixing layer with the mean velocity +profile: (a) schematic of the mixing layer, (b) mean streamwise velocity profile ⟨𝑢1⟩ along +the normal (𝑥2) direction. +of PDFs significantly different from the fDNS data for all three grid resolutions. The secondary +filtering technique (𝜒 = 1) of the ADM model cannot improve the prediction of vorticity very +well, whose estimations are still obviously different from the benchmark fDNS data. DSM and +DMM models underestimate the PDF of vorticity and have wrong predictions of the PDF peak at +the coarse grid-resolution case (FGR=1 with 𝑁 = 323 ), while greatly improving the predictions +of PDFs with the increasing of the grid resolution (FGR=2 and 4 with 𝑁 = 643 and 1283). In +contrast, the VOMM model outperforms these classical SGS models at all three different grid +resolutions, which gives a reasonably good prediction for both the locations and the peaks of the +PDFs of the vorticity. +The reconstruction of transient spatial vorticity structures are finally demonstrated by the +contours of the normalized vorticity magnitude shown in Fig. 15. The instantaneous snapshots +are selected on an arbitrary 𝑥1-𝑥2 slice at the consistent time instant 𝑡/𝜏 ≈ 4. The pure ADM +model predicts the excessive stochastic small-scale structures, which significantly differ from the +fDNS data. The other SGS models can predict the large-scale vorticity structures quite well, but +the VOMM model reconstruct the spatial vortex structures very similar to the benchmark fDNS +data. The VOMM model can accurately recover more flow structures and the temporal evolution +of the vortex with suitable SGS dissipation and accurate structural modeling. +5.3. Temporally evolving turbulent mixing layer +The turbulent mixing layer is one of the cardinal flows in the fluid-mechanics community, +which is widely applied to the investigation of turbulent combustion, chemical reaction mixing +process, and fundamental studies of flow instabilities. The turbulent mixing layer involves +the unsteady shear process of vortex shedding and transition from laminar to turbulent flows, +which are remarkably suitable for investigating the impact of non-uniform turbulent shear and +mixing on the SGS models. The temporally evolving turbulent mixing layer characterized by +the Kelvin–Helmholtz instability induced by the initial velocity difference is considered in this +paper (Vreman et al. 1997; Sharan et al. 2019; Wang et al. 2022a). The free-shear mixing layer +is governed by the same Navier-Stokes equations (Eqs. 2.1 and 2.2) without the forcing term. +Figure 16 illustrates the diagram of the flow configuration for the temporally evolving turbulent +mixing layer with the initial hyperbolic tangent streamwise velocity profile. The numerical +simulation of mixing layer is performed in a cuboid domain with lengths 𝐿1×𝐿2×𝐿3 = 8𝜋×8𝜋×4𝜋 +at the uniform grid resolution of 𝑁1 × 𝑁2 × 𝑁3 = 512 × 512 × 256 where 𝑥1 ∈ [−𝐿1/2, 𝐿1/2], +𝑥2 ∈ [−𝐿2/2, 𝐿2/2] and 𝑥3 ∈ [−𝐿3/2, 𝐿3/2] denote the streamwise, transverse and spanwise +directions, respectively. To enable a periodic configuration in the normal direction, the initial + +32 +𝑁1 × 𝑁2 × 𝑁3 +𝐿1 × 𝐿2 × 𝐿3 +𝜈∞ +𝑅𝑒𝜃 +𝛿0 +𝜃 +Δ𝑈 +Δ𝑑/ℎDNS +ℎDNS +Δ𝑡DNS +512 × 512 × 256 +8𝜋 × 8𝜋 × 4𝜋 +5 × 10−4 +4000 +0.08 +2 +8 +𝜋/64 +0.002 +Table 6: Numerical parameters for the DNS of the temporally evolving mixing layer. +mean streamwise velocity (c.f. Fig. 16b) is given by (Sharan et al. 2019; Wang et al. 2022a) +⟨𝑢1⟩ = Δ𝑈 +2 +� +tanh +� +𝑥2 +2𝛿0 +𝜃 +� +− tanh +� +𝑥2 + 𝐿2/2 +2𝛿0 +𝜃 +� +− tanh +� +𝑥2 − 𝐿2/2 +2𝛿0 +𝜃 +�� +, for − 𝐿2 +2 ⩽ 𝑥2 ⩽ 𝐿2 +2 , +(5.5) +where Δ𝑈 = 2 is the velocity difference between two equal and opposite free streams across +the shear layer, 𝛿0 +𝜃 = 0.08 denotes the initial momentum thickness, and ⟨·⟩ stands for a spatial +average over all the homogeneous directions (i.e., 𝑥1 and 𝑥3 directions for the mixing layer). The +initial mean transverse and spanwise velocities are both set to zero, namely, ⟨𝑢2⟩ = ⟨𝑢3⟩ = 0. +Since the initial mean velocity field is periodic in all three directions, the triply periodic boundary +conditions are adopted and the pseudo-spectral method with the two-thirds dealiasing rule is +used for the spatial discretization. An explicit two-step Adam-Bashforth scheme is selected as +the time-advancing scheme. In order to effectively suppress the influence of the top and bottom +boundaries on the central mixing layer, two numerical diffusion buffer zones are applied near the +vertical edges of domain (Wang et al. 2022a). The thickness of the buffer layer is set to 15𝛿0 +𝜃 +in the paper, which is sufficiently large and has a negligible effect on the calculations of mixing +layer (Wang et al. 2022a). +The digital filter method is used to generate the spatially-correlated initial perturbation imposed +on the mean velocities with the digital filter width Δ𝑑 = ¯Δ = 8ℎDNS consistent to the filter scale of +LES (Klein et al. 2003; Wang et al. 2022b). The initial Reynolds stress distribution (𝑅𝑖 𝑗 = +� +𝑢′ +𝑖𝑢′ +𝑗 +� +where 𝑢′ +𝑖 = 𝑢𝑖 − ⟨𝑢𝑖⟩ represents the fluctuated velocity) of the digital filter method is assumed +as a vertical distribution of 𝑅𝑖 𝑗 = 𝐴 +� +1 − ⟨𝑢1⟩2� +𝐼𝑖 𝑗 with the identity 𝐼𝑖 𝑗 and peak amplitude +𝐴 = 0.025Δ𝑈. The kinematic viscosity of shear layer is set to 𝜈∞ = 5 × 10−4. The momentum +thickness quantifies the range of turbulence region in the mixing layer, which is defined by (Rogers +& Moser 1994; Sharan et al. 2019) +𝛿𝜃 = +𝐿2/4 +∫ +−𝐿2/4 +� +1 +4 − +� ⟨ ¯𝑢1⟩ +Δ𝑈 +�2� +𝑑𝑥2. +(5.6) +Correspondingly, the Reynolds number based on the momentum thickness 𝑅𝑒𝜃 is expressed as +𝑅𝑒𝜃 = Δ𝑈𝛿𝜃 +𝜈∞ +. +(5.7) +Here, the initial momentum thickness Reynolds number is 𝑅𝑒0 +𝜃 = 320. The detailed numerical +parameters of DNS for the temporally evolving mixing layer is summarized in Table 6. +We calculate the DNS of the mixing layer for total of eight hundred time units (𝑡/𝜏𝜃 = 800) +normalized by 𝜏𝜃 = 𝛿0 +𝜃/Δ𝑈. In order to reduce the impact of initial random disturbances on +the temporal development of the shear layer, six numerical experiments with different random +initializations are performed, one of which is adopted for the parameter optimization of the VOMM +model, while the remaining five are used to evaluate the ensemble-averaged physical quantities. +The a posteriori studies of LES are conducted using the explicit Gaussian filter (Eq. 5.1) with + +33 +� +�� +�� +�� +�� +�� +�� +�� +�� +�� +��� +���������� +� +��� +��� +��� +��� +��� +��� +��� +��� +��� +� +J�J0 +� +� +� +�� +DNS +��� +� +�� +� +� +�� +� +� +�� +��� +� +�� +� +� +��� +� +� +�� +Figure 17: The evolution of the normalized cost function in temporally evolving turbulent +mixing layer. +FGR +LES Resolution +𝐶 (0) +1 +𝐶 (0) +2 +𝐶opt +1 +𝐶opt +2 +1 +642 × 32 +0 +1 +-0.0637 +1.188 +2 +1282 × 64 +0 +1 +-0.0126 +1.000 +Table 7: The initial and optimal parameters of the VOMM model for LES computations +with the filter width ¯Δ = 8ℎDNS in temporally evolving mixing layer. +the given filter scale ¯Δ = 8ℎDNS and initialized by the same instantaneous velocity field of the +filtered DNS at 𝑡/𝜏𝜃 = 50. Two different filter-to-grid ratios FGR= ¯Δ/ℎLES=1 and 2 are selected +to study the influence of the spatial resolution or discretization error on the SGS stress modeling +with the corresponding grid resolutions of LES: 𝑁 = 642 × 32 and 1282 × 64. The results from +the previous two turbulence problems (forcing and decaying homogenous isotropic turbulence) +indicate that the statistics of turbulence are very close and similar when the grid resolution is +sufficiently fine (FGR=2 and 4) and the discretization error is considered negligible. However, +the statistics of LES with a relatively coarse grid resolution (FGR=1) are distinctly different from +those of LES with satisfactory grid resolutions (FGR=2 and 4), since the spatial discretization +error of FGR=1 is considerably significant and dominates the SGS modelling error. Therefore, +the a posteriori testings of LES at both FGR=1 and 2 are essential for performance evaluations +of the SGS model. +The dissipation spectrum of the filtered DNS is consistently used as the objective function to +optimize the model parameters of the VOMM model during the period (assess every 𝑡/𝜏𝜃 = 10 +with total thirty-six groups at 50 ⩽ 𝑡/𝜏𝜃 ⩽ 400) of the adjoint-based optimization (c.f. Fig. 1). +The pure ADM model without the dissipative term is adopted as the initial SGS model with +coefficients 𝐶 (0) +1 += 0 and 𝐶 (0) +2 += 1. We calculate the adjoint-based gradients of the cost functional +for the model parameters by backward integrating the stabilized adjoint LES equations (Eqs. 4.10 +and 4.26). The SGS model coefficients are iteratively updated by the L-BFGS optimization method +(Eq. 4.41) until the stopping criterion is ultimately satisfied. Figure 17 gives the optimization + +34 +Model(FGR=1,𝑁 = 642 × 32) +DSM +DMM +ADM(𝜒=0) +ADM(𝜒=1) +VOMM +t(CPU·s) +0.646 +1.096 +0.254 +0.247 +0.362 +t/tDMM +0.590 +1 +0.232 +0.225 +0.330 +Model(FGR=2,𝑁 = 1282 × 64) +DSM +DMM +ADM(𝜒=0) +ADM(𝜒=1) +VOMM +t(CPU·s) +3.756 +6.370 +1.465 +1.460 +1.908 +t/tDMM +0.590 +1 +0.230 +0.229 +0.300 +Table 8: The average computational cost of SGS stress modeling 𝜏𝑖 𝑗 for LES computations +with the filter width ¯Δ = 8ℎDNS in temporally evolving turbulent mixing layer. +process of the cost function during the adjoint-based optimization for the temporally evolving +mixing layer. The loss functions for both FGR=1 and 2 drop dramatically and reach a steady plateau +within less than ten iterations. The cost function of FGR=1 shows a more distinct reduction with +approximately 8% of the initial level than that of FGR=2 decreasing to the 10% of original +value. The optimal parameters of VOMM model are quickly obtained by the effective gradient- +based optimization within a limited number of iterations (around 10 optimization evaluations, +namely, 20 LES calculations). Table 7 summarizes the optimal parameters of the VOMM model. +The parameter magnitude of the dissipative Smagorinsky term ( +���𝐶opt +1 +���) obviously decreases from +0.0637 to 0.0126 when the FGR increases from 1 to 2, while the ADM coefficient (𝐶opt +2 ) generally +tends towards unity, similar to the cases of isotropic turbulence. +We then examine the a posteriori performance of the proposed VOMM model once the SGS +model coefficients are determined by the adjoint-based gradient optimization strategy. In order +to demonstrate the generality of the optimal model parameters that are insensitive to the initial +perturbations, ensemble-averaged quantities are evaluated by five numerical experiments with +different initial random disturbances from the optimization process. The time steps of LES are set +as Δ𝑡LES/Δ𝑡DNS = {10, 5} to guarantee the consistent CFL number for different grid resolutions +(FGR=1 and 2 with 𝑁 = 642 × 32 and 1282 × 64). The VOMM model is compared with the +conventional SGS models (DSM, DMM and ADM models), and the average modeling costs for +different SGS models are listed in Table 8. The VOMM model evaluates efficiently with about +30% computational cost of the DMM model which is similar to those of the ADM models. +Figure 18 illustrates the temporal evolutions of the momentum thickness 𝛿𝜃 in LES calculations +of different SGS models compared to the benchmark fDNS data. At the case of coarse +grid resolution (FGR=1 with 𝑁 = 642 × 32), all conventional SGS models underpredict the +momentum thickness at the early stage of shear layer development (𝑡/𝜏𝜃 ⩽ 300) but give +obvious overestimations in the linear growth region. For the fine-grid-resolution case (FGR=2 +with 𝑁 = 1282 × 64), DMM and ADM (𝜒=1) models can capture the growth rate of momentum +thickness well at the beginning of temporal development, but still overpredict the thickness with +the developing of shear layer. The prediction of the pure ADM model with 𝜒 = 0 is irregular and +nonlinear all the time without an apparent linear self-similar region. The DSM model at different +grid resolutions gives the clearly tilted temporal evolutions, where the momentum thickness is +underestimated at the beginning of transition region and overpredicted in the region of linear +growth. In contrast, the predictions of the VOMM model always coincide well with those of +fDNS, and they accurately capture the temporal growth rate in the linear region at both grid +resolutions. +Furthermore, the evolutions of the turbulent kinetic energy in the streamwise and spanwise + +35 +(𝑎) +� +��� +��� +��� +��� +��� +��� +��� +��� +#�� +� +� +��� +��� +��� +��� +��� +� +� +��� +� +�� +� +� +�� +� + +��� +� +� +� +�" +DNS +!��� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +��� +��� +��� +��� +��� +��� +��� +��� +#�� +� +� +��� +��� +��� +��� +��� +� +� +��� +� +�� +� +� +��� +� + +��� +� +� +� +�" +DNS +!��� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 18: Temporal evolutions of the momentum thickness 𝛿𝜃 for LES in the a posteriori +analysis of temporally evolving turbulent mixing layer with the same filter scale +¯Δ = 8ℎDNS: (a) FGR=1, 𝑁 = 642 × 32; (b) FGR=2, 𝑁 = 1282 × 64. +(𝑎) +� +��� +��� +��� +��� +��� +��� +��� +��� +#�� +� +� +����� +���� +����� +���� +� +"� +��� +� +�� +� +� +�� +� +� +��� +� +� +� +�! +DNS + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +��� +��� +��� +��� +��� +��� +��� +��� +#�� +� +� +����� +���� +����� +���� +� +"� +��� +� +�� +� +� +��� +� +� +��� +� +� +� +�! +DNS + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 19: Temporal evolutions of the streamwise turbulent kinetic energy 𝐸𝑘1 for LES in +the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter +scale ¯Δ = 8ℎDNS: (a) FGR=1, 𝑁 = 642 × 32; (b) FGR=2, 𝑁 = 1282 × 64. +directions are displayed in Figs. 19 and 20, respectively. The comparisons of transverse turbulent +kinetic energy for different SGS models are very similar to those in the spanwise direction, +not shown in the paper. The turbulent kinetic energy of DNS in different directions gradually +increases with the developing of the shear layer, since the initial perturbated velocity field is +approximately laminar and steadily transitions to turbulence. The temporal development of the +streamwise kinetic energy can be approximately regarded as a linear growth with time, which is +distinctly different from that of spanwise kinetic energy. All classical SGS models predict both +streamwise and spanwise kinetic energy much larger than the benchmark fDNS results at both grid +resolutions of LES, except that the pure ADM model gives underestimations of kinetic energy +in the fine-grid-resolution case (FGR=2). Compared to these traditional models, the VOMM +model accurately predicts the kinetic energy at different grid resolutions in both streamwise and +spanwise directions, and is the closest to the fDNS data. +The profiles of the resolved Reynolds shear stress component ¯𝑅12 = +� +¯𝑢′ +1 ¯𝑢′ +2 +� +at time instants +𝑡/𝜏𝜃 ≈ 500 and 800 are illustrated in Fig. 21, which is the dominant Reynolds stress term due to +the intense mixing along the streamwise and normal directions (Vreman et al. 1997; Sharan et al. +2019). The normal distribution of the Reynolds stress is a second-order statistic of turbulence +which has high requirements for the accuracy of SGS modeling of LES. The ADM models +underpredict the Reynolds stress, while DSM and DMM models give obvious overestimations at + +36 +(𝑎) +� +��� +��� +��� +��� +��� +��� +��� +��� +#�� +� +� +����� +���� +����� +� +"� +��� +� +�� +� +� +�� +� +� +��� +� +� +� +�! +DNS + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +� +��� +��� +��� +��� +��� +��� +��� +��� +#�� +� +� +����� +���� +����� +� +"� +��� +� +�� +� +� +��� +� +� +��� +� +� +� +�! +DNS + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 20: Temporal evolutions of the spanwise turbulent kinetic energy 𝐸𝑘3 for LES in +the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter +scale ¯Δ = 8ℎDNS: (a) FGR=1, 𝑁 = 642 × 32; (b) FGR=2, 𝑁 = 1282 × 64. +(𝑎) +���� +���� +���� +� +��� +��� +��� +x +2 +/4π +����� +� +���� +���� +���� +���� +���� +���� +� +� +� +�� +��� +� +�� +� +� +��� +� +� +��� +� +� +� +�! +DNS +� +"�� +� +� +��� + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +���� +���� +���� +� +��� +��� +��� +x +2 +/4π +����� +� +���� +���� +���� +���� +���� +���� +� +� +� +�� +��� +� +�� +� +� +��� +� +� +��� +� +� +� +�! +DNS +� +"�� +� +� +��� + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 21: The transient profile of the resolved Reynolds shear stress ¯𝑅12 = +� +¯𝑢′ +1 ¯𝑢′ +2 +� +along +the cross-stream direction for LES in the a posteriori analysis of temporally evolving +turbulent mixing layer with filter scale ¯Δ = 8ℎDNS at grid resolution of 𝑁 = 1282 × 64: (a) +𝑡/𝜏𝜃 ≈ 500; (b) 𝑡/𝜏𝜃 ≈ 800. +different times. Compared to these classical SGS models, the VOMM model gives the prediction +closest to the fDNS results, and accurately recovers the transient profiles of Reynolds stress. +We further compare the velocity spectra of different SGS models with the DNS and filtered +DNS data at time instants 𝑡/𝜏𝜃 ≈ 500 and 800, as shown in Fig. 22. The spectra of DNS at +𝑡/𝜏𝜃 ≈ 500 and 800 are very similar since the instantaneous velocity fields at different moments +are both at the self-similar stage of mixing layer. For the coarse grid-resolution case at FGR=1 +with 𝑁 = 642 × 32, the conventional SGS models (DSM, DMM and ADM models) always give +the overestimations of the small-scale kinetic energy at high wavenumbers, and the excess kinetic +energy accumulates at small scales and exacerbates the numerical instability of LES over time. +The SGS dissipation provided by these conventional SGS models is insufficient to stabilize the +numerical perturbations induced by the spatial discretization errors, which cannot effectively drain +out the small-scale kinetic energy in time at FGR=1. For the case of fine grid resolution at FGR=2 +with 𝑁 = 1282×64, the pure ADM model is still numerically unstable, whose prediction distinctly +deviates from the fDNS data. And the velocity spectra predicted by the other conventional SGS +models (DSM, DMM and ADM with 𝜒=0) diminish at high-wavenumber regions and accumulate +in the region of intermediate wavenumbers, since these traditional SGS models are too dissipative +at the fine grid-resolution case to recover the effect of small-scale flow structures near the cutoff +wavenumber, giving rise to the blockage of the kinetic energy cascade from large scales to small +scales. In contrast, the kinetic energy cascade can be correctly constructed with high accuracy by + +37 +(𝑎) +�� +� +� � �� +� +�� +� +�� +�� +" +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +��"� +��� +� +�� +� +� +�� +� +� +��� +� +� +� +�! +DNS +� +#�� +� +� +��� +��� + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑏) +�� +� +� � �� +� +�� +� +�� +�� +" +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +��"� +��� +� +�� +� +� +�� +� +� +��� +� +� +� +�! +DNS +� +#�� +� +� +��� +��� + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑐) +�� +� +� � �� +� +�� +� +�� +�� +" +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +��"� +��� +� +�� +� +� +��� +� +� +��� +� +� +� +�! +DNS +� +#�� +� +� +��� +��� + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +(𝑑) +�� +� +� � �� +� +�� +� +�� +�� +" +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +��"� +��� +� +�� +� +� +��� +� +� +��� +� +� +� +�! +DNS +� +#�� +� +� +��� +��� + ��� +��� +��� +����� +� +�� +����� +� +�� +���� +Figure 22: Velocity spectra for different SGS models in the a posteriori analysis of +temporally evolving turbulent mixing layer with the same filter scale ¯Δ = 8ℎDNS at 𝑡/𝜏𝜃 ≈ +500 and 800: (a) FGR=1, 𝑁 = 642 × 32 at 𝑡/𝜏𝜃 ≈ 500; (b) FGR=1, 𝑁 = 642 × 32 at 𝑡/𝜏𝜃 ≈ +800; (c) FGR=2, 𝑁 = 1282 × 64 at 𝑡/𝜏𝜃 ≈ 500; (d) FGR=2, 𝑁 = 1282 × 64 at 𝑡/𝜏𝜃 ≈ 800. +the VOMM model, and the predictions are always in reasonable agreement with those of fDNS +at different grid resolutions and time instants. +The reconstruction of vortex structures is finally compared with different SGS models by +displaying the iso-surface of the Q-criterion. The Q-criterion is a useful visualization tool for +observing vortex structures in turbulent flows, and is the second invariant of velocity gradient +tensor, namely (Hunt et al. 1988; Dubief & Delcayre 2000; Zhan et al. 2019) +𝑄 = 1 +2 +� ¯Ω𝑖 𝑗 ¯Ω𝑖 𝑗 − ¯𝑆𝑖 𝑗 ¯𝑆𝑖 𝑗 +� , +(5.8) +where ¯Ω𝑖 𝑗 = 1 +2 +�𝜕 ¯𝑢𝑖/𝜕𝑥 𝑗 − 𝜕 ¯𝑢 𝑗/𝜕𝑥𝑖 +� represents the rotation-rate tensor. The instantaneous iso- +surface of Q at 𝑡/𝜏𝜃 ≈ 500 is illustrated in Fig. 23 during the self-similar stage of the mixing layer +for Q=0.2 colored by the streamwise velocity. The Q iso-surface of fDNS contains a large number +of elaborate vortex structures near the middle 𝑥1-𝑥3 plane of the shear layer, including the rib-like +vortices, hairpin vortices and complex helical vortices, etc. DSM, DMM and ADM (𝜒=1) models +exhibit an excessive dissipation that only large-scale rib-like vortex structures remain, while the +pure ADM model with 𝜒=0 suffers from numerical instability of LES and overpredicts many +nonphysical small-scale structures caused by numerical noise. In contrast, the VOMM model can +accurately reconstruct much more vortex structures, highlighting its advantage in improving the +accuracy of LES. + +38 +(a) fDNS +(b) DMM +(c) ADM (𝜒=0) +(d) ADM (𝜒=1) +(e) DSM +(f) VOMM +Figure 23: The iso-surface of the Q-criterion at 𝑄=0.2 colored by the streamwise velocity +at 𝑡/𝜏𝜃 ≈ 500 in the a posteriori analysis of temporally evolving turbulent mixing layer +with filter scale ¯Δ = 8ℎDNS at grid resolution of 𝑁 = 1282 × 64: (a) fDNS, (b) DMM, (c) +ADM(𝜒=0), (d) ADM(𝜒=1), (e) DMM, and (f) VOMM. + +u1 +I +0.8 +0.5 +0.6 +0.4 +/4元 +0.2 +0 +-0.2 +0 +-0.4 +-0.6 +-0.8 +-1 +-0.5 +0 +-0.5 +14元 +0.5u1 +1 +0.8 +0.5 +0.6 +0.4 +/4元 +0.2 +0 +-0.2 +0 +-0.4 +-0.6 +-0.8 +-1 +-0.5 +-0.5 +14元 +0.5u1 +1 +0.8 +0.5 +0.6 +0.4 +/4元 +0.2 +0 +-0.2 +0 +-0.4 +-0.6 +-0.8 +-1 +-0.5 +0 +-0.5 +14元 +0.5u1 +1 +0.8 +0.5 +0.6 +0.4 +/4元 +0.2 +0 +-0.2 +0 +-0.4 +-0.6 +-0.8 +1 +-0.5 +-0.5 +14元 +0.5u1 +1 +0.8 +0.5 +0.6 +0.4 +14元 +0.2 +0 +-0.2 +0 +-0.4 +-0.6 +-0.8 +-1 +-0.5 +-0.5 +14元 +0.5u1 +1 +0.8 +0.5 +0.6 +0.4 +/4元 +0.2 +0 +-0.2 +0 +-0.4 +-0.6 +-0.8 +-1 +-0.5 +-0.5 +14元 +0.539 +6. Conclusion +In this work, an adjoint-based variational optimal mixed model (VOMM) is developed for the +large-eddy simulation of turbulence. We first derive the original adjoint LES equations with the +general SGS model, and then carry out the energy budget analysis of adjoint equations. These +detailed derivations demonstrate that the quadratic term with negative eigenvalues of the shear +strain rate is responsible for the exponential temporal growth of the adjoint-based gradients, +giving rise to the numerical divergence in a long time horizon for the chaotic turbulent flows. +This issue might greatly limits the application of the adjoint-based variational methods and +optimal control strategy in turbulence problems. An additional stabilization term is introduced to +maintain the numerical stability of the adjoint LES equations and is efficiently calculated by the +sequential quadratic programming (SQP) approach, without degrading the accuracy of gradient +evaluations for the SGS model parameters. Subsequently, the stabilized adjoint LES equations +are correspondingly formulated. +The approximate deconvolution model (ADM) in the scale-similarity form and the dissipative +Smagorinsky term are selected as the basis tensors of the proposed VOMM model. The parameters +of the VOMM model are optimally identified by minimizing the statistical discrepancies between +dissipation spectra of the LES and those of the benchmark filtered DNS data. The adjoint-based +gradients of cost functional for model coefficients are efficiently evaluated by successively forward +solving the LES equations and backward integrating the stabilized adjoint LES equations. The +gradient-based L-BFGS optimization algorithm is adopted for iteratively updating the VOMM +model parameters until the optimal values are obtained. +Three turbulent flow scenarios including the forced homogeneous isotropic turbulence, de- +caying homogeneous isotropic turbulence and temporally evolving turbulent mixing layer are +investigated to examine the a posteriori performance of the VOMM model. The pure structural +ADM model without the dissipative Smagorinsky term is selected as the initial SGS model +for the parameter optimization. The loss functions of the dissipation spectra can dramatically +converge and reach the optimal state of only about 10% of the initial value within less than twenty +iterations (about forty LES evaluations) during the adjoint-based gradient optimization at different +grid resolutions for these three types of turbulence. These results indicate that the adjoint-based +gradient optimization is an effective tool to obtain the optimal parameters of VOMM model +within only a few iterations. Meanwhile, the computational efficiency of the proposed method is +independent of the number of parameters. +Once the optimal SGS model coefficients are determined by the adjoint-based gradient +optimization, the a posteriori accuracy of the VOMM model is further tested in comparison +with the classical SGS models, including the dynamic Smagorinsky model (DSM), dynamic +mixed model (DMM), the pure ADM model and ADM model with the standard secondary- +filtering regularization, respectively. The various statistics of turbulence and the instantaneous +flow structures are comprehensively compared for LES calculations of different SGS models with +the benchmark filtered DNS data at different grid resolutions of three turbulent flow scenarios. +In the cases of forced and decaying homogeneous isotropic turbulence, the filter scale is fixed +to ¯Δ = 32ℎDNS and the impact of the spatial discretization errors on the SGS modeling is studied +by changing the grid resolution of LES with three different filter-to-grid ratios FGR=1, 2 and +4. The a posteriori performance of the proposed VOMM model is systematically evaluated by +comparison to the conventional SGS models (DSM, DMM and ADM models) in terms of the +velocity spectra, structure functions with different orders, PDFs of the velocity increments and +vorticity, temporal evolutions of the turbulent kinetic energy and average dissipation rate, as +well as the instantaneous vorticity contours at different grid resolutions. The pure ADM model +always exhibits numerical instability due to the insufficient sufficient SGS dissipation for all grid- +resolution cases. The dynamic models and standard regularized ADM model underpredict the + +40 +model dissipation in the case of coarse grid resolution (FGR=1), with the excess kinetic energy +accumulated at small scales leading to the numerical instability of LES. The SGS dissipation +imposed by these classical SGS models is insufficient to suppress the numerical perturbations +dominated by the spatial discretization, and it cannot effectively drain out the small-scale kinetic +energy in time at FGR=1. However, the traditional SGS models are too dissipative that most small- +scale flow structures near the truncated wavenumber are diminished, giving rise to the blockage +of the kinetic energy cascade from large scales to small scales at situations of satisfactory grid +resolutions. In contrast, the VOMM model can correctly reconstruct the kinetic energy cascade +and the evolution of dissipation rate with high accuracy, which is essential for the isotropic +turbulence. In addition, the VOMM model accurately predicts various flow statistics and transient +spatial flow structures, which are always in reasonable agreement with the benchmark filtered +DNS results at different grid resolutions and times. +In the context of the temporally evolving turbulent mixing layer, the unsteady evolution of +the shear layer from the initial perturbed velocity field gradually transitions to fully developed +turbulence is challenging for the SGS modeling of LES. The VOMM model can accurately +reconstruct the temporal evolutions of characteristic physical quantities of the mixing layer, +including the momentum thickness, turbulent kinetic energy in different directions and transient +velocity spectra at different times. The corresponding predictions of VOMM are closest to the +filtered DNS results and superior to these conventional SGS models (DSM, DMM and ADM +models). The profiles of Reynolds shear stress at the self-similar stage of the shear layer are critical +for the development of mixing layer, and all conventional SGS models are not able to accurately +predict the vertical distributions with significant deviations from the benchmark fDNS result. +In contrast, the VOMM model predicts the Reynolds stress fairly well at different time instants. +Besides, it can be clearly observed from the iso-surface of Q-criterion that the VOMM model +accurately recovers the diverse spatial vortex structures very similar to the benchmark fDNS data +in comparison to the classical SGS models. +Furthermore, for the cases of three turbulent flow scenarios with different grid resolutions, +the computational cost of the proposed VOMM model is only about 30% the time of the DMM +model, which is very efficient and competitive compared to the classical SGS models. These +results suggest that the proposed VOMM model has high a posteriori accuracy and computational +efficiency by assimilating the a priori knowledge of turbulence statistics, and can be a promising +tool to develop advanced SGS models in the LES of turbulence. +Eventually, it is noteworthy that fine-tuning a small number of model parameters of some +traditional SGS models can significantly improve the a posteriori accuracy of LES using the +proposed adjoint-based optimization framework. In addition, the predictions of LES in complex +turbulent flows using the VOMM model might be dramatically accurate as the number of model +coefficients increases, while the computational cost of the adjoint-based approach hardly varies +with to the number of parameters. Although the high-fidelity turbulence statistics is provided +by DNS data in the current study, the experimental measurements can also be assimilated using +the same optimization procedure to increase the accuracy of LES modeling for a particular type +of complex turbulent flow. In future work, we would further apply the VOMM model with the +existing optimal parameters to more complex turbulent flows and generalize to turbulence with +different filter scales. +Funding. This work was supported by the National Natural Science Foundation of China +(NSFC Grants No. 91952104, No. 92052301, No. 12172161, and No. 12161141017), by the +National Numerical Windtunnel Project (No. NNW2019ZT1-A04), by the NSFC Basic Science +Center Program (Grant No. 11988102), by the Shenzhen Science and Technology Program +(Grants No. KQTD20180411143441009), by Key Special Project for Introduced Talents Team +of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (Grant No. + +41 +GML2019ZD0103), and by Department of Science and Technology of Guangdong Province +(Grants No. 2019B21203001). This work was also supported by Center for Computational Science +and Engineering of Southern University of Science and Technology. +Declaration of interests. The authors report no conflict of interest. +Appendix A. Derivation of the adjoint large-eddy simulation equations +The large-eddy simulation (LES) equations are expressed as (Pope 2000; Sagaut 2006) +𝑅0 ( ¯𝑢𝑖) = 𝜕 ¯𝑢𝑖 +𝜕𝑥𝑖 += 0, +(A 1) +𝑅𝑖 ( ¯𝑢𝑖, ¯𝑝) = 𝜕 ¯𝑢𝑖 +𝜕𝑡 + 𝜕 � ¯𝑢𝑖 ¯𝑢 𝑗 +� +𝜕𝑥 𝑗 ++ 𝜕 ¯𝑝 +𝜕𝑥𝑖 +− 𝜈 𝜕2 ¯𝑢𝑖 +𝜕𝑥 𝑗𝜕𝑥 𝑗 +− F 𝑖 + 𝜕𝜏𝑖 𝑗 +𝜕𝑥 𝑗 += 0, +(A 2) +where an overbar denotes the filtered variables with filter scale ¯Δ, ¯𝑢𝑖 and ¯𝑝 denote the filtered +velocity and pressure, respectively. Here, 𝜈 is the kinematic viscosity, and ¯F𝑖 represents the +large-scale forcing. The unclosed SGS stress 𝜏𝑖 𝑗 = 𝑢𝑖𝑢 𝑗 − ¯𝑢𝑖 ¯𝑢 𝑗 is modeled by the 𝑁-parameter +mixed model 𝜏𝑖 𝑗 = +𝑁� +𝑛=1 +𝐶𝑛𝑇 (𝑛) +𝑖 𝑗 +� ¯𝑢𝑖; ¯Δ� with the basis stress tensors 𝑇 (𝑛) +𝑖 𝑗 +and model coefficients +𝐶𝑛 (𝑛 = 1, 2, ..., 𝑁). The sensitivities of the governing equations for the LES variables ¯v = +[ ¯𝑝, ¯𝑢1, ¯𝑢2, ¯𝑢3]𝑇 are given by +𝛿𝑅𝑘 = 𝜕𝑅𝑘 +𝜕¯v · 𝛿¯v = +� +𝜕𝛿 ¯𝑢𝑖 +𝜕𝑥𝑖 +𝜕𝛿 ¯𝑢𝑖 +𝜕𝑡 ++ +𝜕( ¯𝑢𝑗 𝛿 ¯𝑢𝑖) +𝜕𝑥𝑗 ++ +𝜕( ¯𝑢𝑖 𝛿 ¯𝑢𝑗) +𝜕𝑥𝑗 ++ 𝜕𝛿 ¯𝑝 +𝜕𝑥𝑖 − 𝜈 𝜕2 𝛿 ¯𝑢𝑖 +𝜕𝑥𝑗𝜕𝑥𝑗 + 𝜕𝛿𝜏𝑖 𝑗 +𝜕𝑥𝑗 +� += 0. +(A 3) +The adjoint LES equations are derived by the adjoint identity acting on the adjoint variables +¯v† = +� +¯𝑝†, ¯𝑢† +1, ¯𝑢† +2, ¯𝑢† +3 +�𝑇 +, namely +� 𝜕𝑅𝑘 +𝜕¯v · 𝛿¯v, ¯v† +� +x,𝑡 += +� +𝛿¯v, +� 𝜕𝑅𝑘 +𝜕¯v +�† +· ¯v† +� +x,𝑡 ++ 𝐵𝑇, +(A 4) +where 𝐵𝑇 denotes the boundary and temporal integral terms, and 𝐵𝑇 = 0 can identify the boundary +and terminal conditions of the adjoint equations. The corresponding adjoint LES equations can +be expressed as +3 +∑︁ +𝑘=0 +� 𝜕𝑅𝑘 +𝜕¯v +�† +· ¯v† − 𝜕𝐽 +𝜕¯v = 0, +(A 5) +where 𝜕𝐽/𝜕¯v = +� +0, 𝜕𝐽 +𝜕 ¯𝑢1 , 𝜕𝐽 +𝜕 ¯𝑢2 , 𝜕𝐽 +𝜕 ¯𝑢3 +�𝑇 +denotesthesensitivityofthecostfunctional 𝐽 � ¯𝑢𝑖, ¯𝑢ref +𝑖 ; 𝐶𝑛, x, 𝑡� +which quantifies the discrepancy between ¯𝑢𝑖 and the reference data ¯𝑢ref +𝑖 +in the LES calculations +under the given parameters 𝐶𝑛 (𝑛 = 1, 2, ..., 𝑁) at a certain space-time state (x, 𝑡). Here, the +terms (𝜕𝑅𝑘/𝜕¯v)† · ¯v† (𝑘 = 0, 1, 2, 3) are derived by multiplying the perturbation LES equations +(Eq. A 3) with the adjoint LES variables ¯v†, and then integrating by parts to rearrange all of the + +42 +differential operators without 𝛿¯v onto the adjoint variables ¯v† , yielding +𝜕𝛿 ¯𝑢𝑖 +𝜕𝑥𝑖 ¯𝑝† + +� +𝜕𝛿 ¯𝑢𝑖 +𝜕𝑡 ++ +𝜕( ¯𝑢𝑗 𝛿 ¯𝑢𝑖) +𝜕𝑥𝑗 ++ +𝜕( ¯𝑢𝑖 𝛿 ¯𝑢𝑗) +𝜕𝑥𝑗 ++ 𝜕𝛿 ¯𝑝 +𝜕𝑥𝑖 − 𝜈 𝜕2 𝛿 ¯𝑢𝑖 +𝜕𝑥𝑗𝜕𝑥𝑗 + 𝜕𝛿𝜏𝑖 𝑗 +𝜕𝑥𝑗 +� +¯𝑢† +𝑖 = +− +� +𝜕 ¯𝑢† +𝑖 +𝜕𝑥𝑖 +� +𝛿 ¯𝑝 − +� +𝜕 ¯𝑢† +𝑖 +𝜕𝑡 + +� +𝜕 ¯𝑢† +𝑖 +𝜕𝑥𝑗 + +𝜕 ¯𝑢† +𝑗 +𝜕𝑥𝑖 +� +¯𝑢 𝑗 + 𝜈 +𝜕2 ¯𝑢† +𝑖 +𝜕𝑥𝑗𝜕𝑥𝑗 + +𝜕 +𝜕𝑥 𝑗 +� +¯𝑢† +𝑘 +𝜕𝜏𝑗𝑘 +𝜕 ¯𝑢𝑖 +� +− ¯𝑢† +𝑘 +𝜕2𝜏𝑗𝑘 +𝜕 ¯𝑢𝑖𝜕𝑥𝑗 +� +𝛿 ¯𝑢𝑖+ ++ +𝜕 +� +¯𝑢† +𝑖 𝛿 ¯𝑢𝑖 +� +𝜕𝑡 +���������������� +terminal condition ++ 𝜕 +𝜕𝑥 𝑗 +�� +¯𝑢† +𝑖 ¯𝑢 𝑗 + 𝜈 𝜕 ¯𝑢† +𝑖 +𝜕𝑥 𝑗 ++ ¯𝑢† +𝑘 +𝜕𝜏𝑗𝑘 +𝜕 ¯𝑢𝑖 +� +𝛿 ¯𝑢𝑖 − 𝜈 ¯𝑢† +𝑖 +𝜕𝛿 ¯𝑢𝑖 +𝜕𝑥 𝑗 +� ++ 𝜕 +𝜕𝑥𝑖 +� +¯𝑢† +𝑖 𝛿 ¯𝑝 + +� +¯𝑝† + ¯𝑢 𝑗 ¯𝑢† +𝑗 +� +𝛿 ¯𝑢𝑖 +� +�������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� +boundary condition +. +(A 6) +The adjoint LES equations are written in detail as +𝜕 ¯𝑢† +𝑖 +𝜕𝑥𝑖 += 0, +(A 7) +𝜕 ¯𝑢† +𝑖 +𝜕𝑡 + +� +𝜕 ¯𝑢† +𝑖 +𝜕𝑥 𝑗 ++ +𝜕 ¯𝑢† +𝑗 +𝜕𝑥𝑖 +� +¯𝑢 𝑗 + 𝜈 𝜕2 ¯𝑢† +𝑖 +𝜕𝑥 𝑗𝜕𝑥 𝑗 ++ +𝜕 +𝜕𝑥 𝑗 +� +¯𝑢† +𝑘 +𝜕𝜏𝑗𝑘 +𝜕 ¯𝑢𝑖 +� +− ¯𝑢† +𝑘 +𝜕2𝜏𝑗𝑘 +𝜕 ¯𝑢𝑖𝜕𝑥 𝑗 ++ 𝜕𝐽 +𝜕 ¯𝑢𝑖 += 0. +(A 8) +It is worth noting that the adjoint SGS term ¯𝑢† +𝑘 +𝜕2𝜏𝑗𝑘 +𝜕 ¯𝑢𝑖𝜕𝑥𝑗 can lead to the non-conservation of the adjoint +momentum and deteriorate the evaluation of the adjoint-based gradients. To our knowledge, few +previous studies have addressed this critical issues that make the LES adjoint field prone to +numerical instability and eventual divergence. In order to maintain the momentum conservation +in the adjoint equations, we remove ¯𝑢† +𝑘 +𝜕2𝜏𝑗𝑘 +𝜕 ¯𝑢𝑖𝜕𝑥𝑗 from Eq. A 8, and the conservative adjoint LES +equations are obtained as +𝜕 ¯𝑢† +𝑖 +𝜕𝑥𝑖 += 0, +(A 9) +𝜕 ¯𝑢† +𝑖 +𝜕𝑡 + +� +𝜕 ¯𝑢† +𝑖 +𝜕𝑥 𝑗 ++ +𝜕 ¯𝑢† +𝑗 +𝜕𝑥𝑖 +� +¯𝑢 𝑗 + 𝜈 𝜕2 ¯𝑢† +𝑖 +𝜕𝑥 𝑗𝜕𝑥 𝑗 ++ +𝜕𝜏† +𝑖 𝑗 +𝜕𝑥 𝑗 ++ 𝜕𝐽 +𝜕 ¯𝑢𝑖 += 0, +(A 10) +where 𝜏† +𝑖 𝑗 = ¯𝑢† +𝑘 +𝜕𝜏𝑗𝑘 +𝜕 ¯𝑢𝑖 is the adjoint SGS stress. If the unclosed SGS terms is modeled by the +𝑁-parameter mixed model 𝜏𝑖 𝑗 = +𝑁� +𝑛=1 +𝐶𝑛𝑇 (𝑛) +𝑖 𝑗 +� ¯𝑢𝑖; ¯Δ� with the basis stress tensors 𝑇 (𝑛) +𝑖 𝑗 +and model +coefficients 𝐶𝑛, the adjoint SGS stresses are correspondingly represented as 𝜏† +𝑖 𝑗 = +𝑁� +𝑛=1 +𝐶𝑛𝑇 (𝑛),† +𝑖 𝑗 +with the associated adjoint basis stress tensors 𝑇 (𝑛),† +𝑖 𝑗 +(𝑛 = 1, 2, ..., 𝑁). +Appendix B. Derivation of the adjoint SGS stress for the VOMM model +The present variational optimal mixed model (VOMM) combines the approximate deconvolu- +tion model (ADM) in the scale-similarity form with the dissipative Smagorinsky part, expressed +as +𝜏𝑖 𝑗 = 𝐶1𝑇 (1) +𝑖 𝑗 ++ 𝐶2𝑇 (2) +𝑖 𝑗 , with 𝑇 (1) +𝑖 𝑗 += ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗, 𝑇 (2) +𝑖 𝑗 += 𝑢∗ +𝑖 𝑢∗ +𝑗 − 𝑢∗ +𝑖 𝑢∗ +𝑗, +(B 1) +where 𝑢∗ +𝑖 = +𝑁� +𝑛=1 +(𝐼 − 𝐺)𝑛−1 ⊗ ¯𝑢𝑖 stands for the 𝑖-th approximate unfiltered velocity component +recovered by the iterative van Cittert procedure, 𝑁 is the number of iterations for the AD procedure, +𝐼 is the identity, and the symbol “⊗” is the spatial convolution operator. Here, 𝐶1 and 𝐶2 are SGS + +43 +model coefficients. The variation of the first basis SGS tensor 𝑇 (1) +𝑖 𝑗 +with respect to the velocity, is +derived by +𝛿𝑇 (1) +𝑖 𝑗 += ¯Δ2 � +| ¯𝑆|𝛿 ¯𝑆𝑖 𝑗 + �𝛿| ¯𝑆|� ¯𝑆𝑖 𝑗 +� += ¯Δ2 +� +| ¯𝑆| 𝜕 ¯𝑆𝑖 𝑗 +𝜕 ¯𝑢𝑘 ++ 𝜕| ¯𝑆| +𝜕 ¯𝑢𝑘 +¯𝑆𝑖 𝑗 +� +𝛿 ¯𝑢𝑘, +(B 2) +where the derivatives of the shear strain-rate tensor and characteristic strain rate for the velocity +are further written as +𝜕 ¯𝑆𝑖 𝑗 +𝜕 ¯𝑢𝑘 += 1 +2 +𝜕 +𝜕 ¯𝑢𝑘 +� 𝜕 ¯𝑢𝑖 +𝜕𝑥 𝑗 ++ 𝜕 ¯𝑢 𝑗 +𝜕𝑥𝑖 +� += 1 +2 +� 𝜕𝛿𝑖𝑘 +𝜕𝑥 𝑗 ++ 𝜕𝛿 𝑗𝑘 +𝜕𝑥𝑖 +� +, +(B 3) +and +𝜕| ¯𝑆| +𝜕 ¯𝑢𝑘 += 𝜕| ¯𝑆| +𝜕 ¯𝑆𝑖 𝑗 +𝜕 ¯𝑆𝑖 𝑗 +𝜕 ¯𝑢𝑘 += +¯𝑆𝑖 𝑗 +| ¯𝑆| +� 𝜕𝛿𝑖𝑘 +𝜕𝑥 𝑗 ++ 𝜕𝛿 𝑗𝑘 +𝜕𝑥𝑖 +� +. +(B 4) +The inner product between the variation of the first basis SGS force and the adjoint velocity is +derived by +𝜕𝛿𝑇 (1) +𝑖 𝑗 +𝜕𝑥𝑗 +¯𝑢† +𝑖 = − +𝜕 ¯𝑢† +𝑖 +𝜕𝑥𝑗 𝛿𝑇 (1) +𝑖 𝑗 ++ +𝜕 +𝜕𝑥𝑗 +� +¯𝑢† +𝑖 𝛿𝑇 (1) +𝑖 𝑗 +� += − ¯Δ2 +2 +�� +| ¯𝑆| +𝜕 ¯𝑢† +𝑖 +𝜕𝑥𝑗 +� � +𝜕𝛿𝑖𝑘 +𝜕𝑥𝑗 + 𝜕𝛿 𝑗𝑘 +𝜕𝑥𝑖 +� ++ +� +𝜕𝛿𝑚𝑘 +𝜕𝑥𝑛 + 𝜕𝛿𝑛𝑘 +𝜕𝑥𝑚 +� � +2 ¯𝑆𝑚𝑛 +| ¯𝑆| ¯𝑆𝑖 𝑗 +𝜕 ¯𝑢† +𝑖 +𝜕𝑥𝑗 +�� +𝛿 ¯𝑢𝑘 + +𝜕 +𝜕𝑥𝑗 +� +¯𝑢† +𝑖 𝛿𝑇 (1) +𝑖 𝑗 +� += − ¯Δ2 +2 +� +𝜕 +𝜕𝑥 𝑗 +� +| ¯𝑆| +� +𝜕 ¯𝑢† +𝑘 +𝜕𝑥𝑗 + +𝜕 ¯𝑢† +𝑗 +𝜕𝑥𝑘 +�� ++ +𝜕 +𝜕𝑥𝑗 +� 2 ¯𝑆𝑗𝑘 +| ¯𝑆| ¯𝑆𝑚𝑛 +� +𝜕 ¯𝑢† +𝑚 +𝜕𝑥𝑛 + 𝜕 ¯𝑢† +𝑛 +𝜕𝑥𝑚 +��� +𝛿 ¯𝑢𝑘 + +𝜕 +𝜕𝑥𝑗 +� +¯𝑢† +𝑖 𝛿𝑇 (1) +𝑖 𝑗 +� +, +(B 5) +Here, the adjoint strain-rate tensor ¯𝑆† +𝑖 𝑗 = +� +𝜕 ¯𝑢† +𝑖 /𝜕𝑥 𝑗 + 𝜕 ¯𝑢† +𝑗/𝜕𝑥𝑖 +� +/2, and the inner product term +can be further expressed as +¯𝑢† +𝑖 +𝜕𝛿𝑇 (1) +𝑖 𝑗 +𝜕𝑥 𝑗 += +� +𝜕 +𝜕𝑥 𝑗 +� +− ¯Δ2 +� +| ¯𝑆| ¯𝑆† +𝑖 𝑗 + +2 ¯𝑆𝑘𝑙 ¯𝑆† +𝑘𝑙 +| ¯𝑆| +¯𝑆𝑖 𝑗 +��� +𝛿 ¯𝑢𝑖 + +𝜕 +𝜕𝑥 𝑗 +� +¯𝑢† +𝑖 𝛿𝑇 (1) +𝑖 𝑗 +� +. +(B 6) +Thus, the adjoint basis stress tensor 𝑇 (1),† +𝑖 𝑗 +is given by +𝑇 (1),† +𝑖 𝑗 += − ¯Δ2 +� +| ¯𝑆| ¯𝑆† +𝑖 𝑗 + +2 ¯𝑆𝑘𝑙 ¯𝑆† +𝑘𝑙 +| ¯𝑆| +¯𝑆𝑖 𝑗 +� +. +(B 7) +The common filter function 𝐺 (e.g. top-hat, Gaussian and spectral filters) is symmetric spatial +filter, and is self-adjoint, namely (Vreman 2004) +⟨𝐺 ⊗ 𝑓 , 𝑔⟩x = ⟨ 𝑓 , 𝐺 ⊗ 𝑔⟩x, +(B 8) +where 𝑓 (x) and 𝑔 (x) are arbitrary variables. The 𝐺𝑛 filter with spatially filtering 𝑛 times +(𝐺𝑛 = 𝐺 ⊗ 𝐺 ⊗ · · · ⊗ 𝐺) also satisfies the self-adjoint property proved by the mathematical +induction method, expressed as +⟨𝐺𝑛 ⊗ 𝑓 , 𝑔⟩x = +� +𝐺 ⊗ 𝐺𝑛−1 ⊗ 𝑓 , 𝑔 +� +x = +� +𝐺𝑛−1 ⊗ 𝑓 , 𝐺 ⊗ 𝑔 +� +x = · · · = ⟨ 𝑓 , 𝐺𝑛 ⊗ 𝑔⟩x. +(B 9) +The (𝐼 − 𝐺) filter is also a symmetric filter, and the approximate deconvolution procedure 𝐻 = +𝑁� +𝑛=1 +(𝐼 − 𝐺)𝑛−1 is thus the self-adjoint filter. The second basis SGS tensor 𝑇 (2) +𝑖 𝑗 can be described +using the AD abbreviated notation, namely +𝑇 (2) +𝑖 𝑗 += 𝑢∗ +𝑖 𝑢∗ +𝑗 − 𝑢∗ +𝑖 𝑢∗ +𝑗 = 𝐺 ⊗ +� +(𝐻 ⊗ ¯𝑢𝑖) �𝐻 ⊗ ¯𝑢 𝑗 +�� +− [𝐺 ⊗ (𝐻 ⊗ ¯𝑢𝑖)] +� +𝐺 ⊗ �𝐻 ⊗ ¯𝑢 𝑗 +�� +. +(B 10) + +44 +The variation of the second basis SGS tensor 𝑇 (2) +𝑖 𝑗 +with respect to the velocity, expressed as +𝛿𝑇 (2) +𝑖 𝑗 += 𝐺 ⊗ +� +(𝐻 ⊗ 𝛿 ¯𝑢𝑖) 𝑢∗ +𝑗 +� ++𝐺 ⊗ +� +𝑢∗ +𝑖 +�𝐻 ⊗ 𝛿 ¯𝑢 𝑗 +�� +−[𝐺 ⊗ (𝐻 ⊗ 𝛿 ¯𝑢𝑖)] 𝑢∗ +𝑗 −𝑢∗ +𝑖 +� +𝐺 ⊗ �𝐻 ⊗ 𝛿 ¯𝑢 𝑗 +�� +. +(B 11) +The inner product between the variation of the second basis SGS force and the adjoint velocity is +given by +𝜕𝛿𝑇 (2) +𝑖 𝑗 +𝜕𝑥 𝑗 +¯𝑢† +𝑖 = − +𝜕 ¯𝑢† +𝑖 +𝜕𝑥𝑗 𝛿𝑇 (2) +𝑖 𝑗 ++ +𝜕 +𝜕𝑥𝑗 +� +¯𝑢† +𝑖 𝛿𝑇 (2) +𝑖 𝑗 +� += −2 ¯𝑆† +𝑖 𝑗 +� +𝐺 ⊗ +� +(𝐻 ⊗ 𝛿 ¯𝑢𝑖) 𝑢∗ +𝑗 +�� ++ 2 ¯𝑆† +𝑖 𝑗 [𝐺 ⊗ (𝐻 ⊗ 𝛿 ¯𝑢𝑖)] 𝑢∗ +𝑗 + +𝜕 +𝜕𝑥𝑗 +� +¯𝑢† +𝑖 𝛿𝑇 (2) +𝑖 𝑗 +� +. +(B 12) +The inner product term can be further simplified by the self-adjoint property, such that +𝜕𝛿𝑇 (2) +𝑖 𝑗 +𝜕𝑥𝑗 +¯𝑢† +𝑖 = −2 +� +𝐺 ⊗ ¯𝑆† +𝑖 𝑗 +� � +(𝐻 ⊗ 𝛿 ¯𝑢𝑖) 𝑢∗ +𝑗 +� ++ 2 +� +𝐺 ⊗ +� +¯𝑆† +𝑖 𝑗𝑢∗ +𝑗 +�� +(𝐻 ⊗ 𝛿 ¯𝑢𝑖) + +𝜕 +𝜕𝑥𝑗 +� +¯𝑢† +𝑖 𝛿𝑇 (2) +𝑖 𝑗 +� += 𝐻 ⊗ +� +−2 ¯𝑆† +𝑖 𝑗𝑢∗ +𝑗 + 2 ¯𝑆† +𝑖 𝑗𝑢∗ +𝑗 +� +𝛿 ¯𝑢𝑖 + +𝜕 +𝜕𝑥𝑗 +� +¯𝑢† +𝑖 𝛿𝑇 (2) +𝑖 𝑗 +� += +� +𝜕 +𝜕𝑥𝑗 +� +𝐻 ⊗ +� +¯𝑢† +𝑖 𝑢∗ +𝑗 − ¯𝑢† +𝑖 𝑢∗ +𝑗 +�� ++ 𝐻 ⊗ +� +𝜕 ¯𝑢† +𝑗 +𝜕𝑥𝑖 𝑢∗ +𝑗 − +𝜕 ¯𝑢† +𝑗 +𝜕𝑥𝑖 𝑢∗ +𝑗 +�� +𝛿 ¯𝑢𝑖 + +𝜕 +𝜕𝑥𝑗 +� +¯𝑢† +𝑖 𝛿𝑇 (2) +𝑖 𝑗 +� +. +(B 13) +It is quite notable that the second adjoint SGS term makes the non-conservation of the adjoint +momentum, therefore we discard the second adjoint SGS term. Thus, the second adjoint basis +stress tensor 𝑇 (2),† +𝑖 𝑗 +can be written as +𝑇 (2),† +𝑖 𝑗 += 𝐻 ⊗ +� +¯𝑢† +𝑖 𝑢∗ +𝑗 − ¯𝑢† +𝑖 𝑢∗ +𝑗 +� += +𝑁 +∑︁ +𝑛=1 +(𝐼 − 𝐺)𝑛−1 ⊗ +� +¯𝑢† +𝑖 𝑢∗ +𝑗 − ¯𝑢† +𝑖 𝑢∗ +𝑗 +� +. +(B 14) +In summary, the adjoint SGS stress of the proposed VOMM model is represented by +𝜏† +𝑖 𝑗 = 𝐶1𝑇 (1),† +𝑖 𝑗 ++ 𝐶2𝑇 (2),† +𝑖 𝑗 +, +(B 15) +where the adjoint basis stress tensors are 𝑇 (1),† +𝑖 𝑗 += − ¯Δ2 +� +| ¯𝑆| ¯𝑆† +𝑖 𝑗 + +2 ¯𝑆𝑘𝑙 ¯𝑆† +𝑘𝑙 +| ¯𝑆| +¯𝑆𝑖 𝑗 +� +and 𝑇 (2),† +𝑖 𝑗 += +𝑁� +𝑛=1 +(𝐼 − 𝐺)𝑛−1 ⊗ +� +¯𝑢† +𝑖 𝑢∗ +𝑗 − ¯𝑢† +𝑖 𝑢∗ +𝑗 +� +. +REFERENCES +Abkar, Mahdi, Bae, Hyun J & Moin, Parviz 2016 Minimum-dissipation scalar transport model for +large-eddy simulation of turbulent flows. Phys. Rev. Fluids 1 (4), 041701. +Adams, N. A., Hickel, S. & Franz, S. 2004 Implicit subgrid-scale modeling by adaptive deconvolution. J. +Comput. Phys. 200 (2), 412–431. +Armijo, Larry 1966 Minimization of functions having Lipschitz continuous first partial derivatives. Pacific +J. 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Fluids 195, 104319. + diff --git a/4NFAT4oBgHgl3EQfExwU/content/tmp_files/load_file.txt b/4NFAT4oBgHgl3EQfExwU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a295371f9dcd31e21446bb3c9389a22af10cc52 --- /dev/null +++ b/4NFAT4oBgHgl3EQfExwU/content/tmp_files/load_file.txt @@ -0,0 +1,3600 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf,len=3599 +page_content='Under consideration for publication in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1 Adjoint-based variational optimal mixed models for large-eddy simulation of turbulence Zelong Yuan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yunpeng Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xiaoning Wang and Jianchun Wang† 1Department of Mechanics and Aerospace Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Southern University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Shenzhen 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' People’s Republic of China 2Guangdong–Hong Kong–Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Southern University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Shenzhen 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' People’s Republic of China (Received xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' accepted xx) An adjoint-based variational optimal mixed model (VOMM) is proposed for subgrid-scale (SGS) closure in large-eddy simulation (LES) of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The stabilized adjoint LES equations are formulated by introducing a minimal regularization to address the numerical instabilities of the long-term gradient evaluations in chaotic turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model parameters are optimized by minimizing the discrepancy of energy dissipation spectra between LES calculations and a priori knowledge of direct numerical simulation (DNS) using the gradient- based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori performance of the VOMM model is comprehensively examined in LES of three turbulent flows, including the forced homogeneous isotropic turbulence, decaying homogenous isotropic turbulence, and temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model outperforms the dynamic Smagorinsky model (DSM), dynamic mixed model (DMM) and approximate deconvolution model (ADM) in predictions of various turbulence statistics, including the velocity spectrum, structure functions, statistics of velocity increments and vorticity, temporal evolutions of the turbulent kinetic energy, dissipation rate, momentum thickness and Reynolds stress, as well as the instantaneous vortex structures at different grid resolutions and times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the VOMM model only takes up 30% time of the DMM model for all flow scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results demonstrate that the proposed VOMM model improves the numerical stability of LES and has high a posteriori accuracy and computational efficiency by incorporating the a priori information of turbulence statistics, highlighting that the VOMM model has a great potential to develop advanced SGS models in the LES of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Key words: subgrid-scale model, variational optimal models, adjoint-based optimization, large- eddy simulation, incompressible turbulence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Introduction Large-eddy simulation (LES) has become an effective tool for the investigation of turbulent flows, and has been widely applied to many industrial problems including the aeroacoustics, combustions, meteorological physics, interfacial mixing, etc (Sagaut 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dominant large-scale motions of turbulence are directly resolved by the LES, leaving the effects of residual subgrid scales (SGS) on the resolved large scales modeled by the SGS models (Lesieur & Metais 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Meneveau & Katz 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, direct numerical simulation (DNS) of turbulence requires a sufficiently high mesh resolution to fully resolve all flow scales down † Email address for correspondence: wangjc@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='08423v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='flu-dyn] 20 Jan 2023 2 to the size of the Kolmogorov eddies, whose computational cost is prohibitively expensive at a high Reynolds number (Pope 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Therefore, LES is much more computationally efficient than the DNS by significantly reducing the degrees of freedom of turbulence, meanwhile accurately reconstructing large-scale flow structures (Pope 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sagaut 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Durbin 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The modeling of the unclosed SGS stress is crucial for the accuracy of predictions in LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' SGS models can be generally categorized into functional models, structural models and mixed models (Sagaut 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The functional SGS models utilize the explicit dissipative terms to correctly reconstruct the forward kinetic energy cascade from large scales to small scales (Rozema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Abkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Smagorinsky model is one of the most popular functional SGS models and is favored for its substantial numerical stability and excellent robustness of LES calculations (Smagorinsky 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Lilly 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the functional SGS models generally exhibit excessive dissipation and fail to predict the sophisticated small-scale flow structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the structural SGS models recover the unclosed SGS stress with high a priori accuracy by exactly truncating the Taylor series expansions or the assumption of scale similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These structural models include the approximate deconvolution method (Stolz & Adams 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001), scale-similarity model (Bardina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1994), velocity gradient model (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1979), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The structural SGS models can accurately capture the spatial distribution of SGS energy flux and backscatter of the kinetic energy, but suffer from the numerical instability without sufficient SGS dissipation in the a posteriori studies of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The mixed models consist of the structural models and functional eddy-viscosity models to balance the numerical stability and accuracy of LES and compensate their inherent model deficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Clark model combines the velocity gradient model with the Smagorinsky eddy viscosity (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Erlebacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (1992) proposed a mixed model which consists of the scale-similarity model and the dissipative Smagorinsky term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In the early stage, the SGS model parameters were either theoretically derived from the isotropic turbulent flows (Lilly 1967) or estimated by the a priori analysis of DNS and experimental observations (Deardorff 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1979), yielding poor predictions in the a posteriori LES (Lesieur & Metais 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Meneveau & Katz 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A pioneering dynamical procedure with the Germano identity was developed to determine the Smagorinsky coefficient adaptively by the least-squares algorithm (Germano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Lilly 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Subsequently, the dynamic versions of mixed models were successively proposed, including the one-parameter (Zang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1992) and two-parameter dynamical mixed models (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2008), the dynamic Clark model (Vreman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1994) and dynamic ADM model (Habisreutinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2007), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The coefficients of a general multi- parameter dynamic mixed model (DMM) can be conveniently determined by the Germano- identity-based dynamic approach (Sagaut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, extensive previous studies have shown that these DMM models are excessively dissipative in the transitional regions, but underestimate the SGS dissipation in situations of coarse mesh resolutions and grid anisotropy (Meneveau & Katz 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Moser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the dissipative Smagorinsky part in the DMM models is usually dominant over the structural part, leading to little advantage in the high a priori accuracy of structural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The basis tensors of the DMM model, comprising the functional eddy-viscosity and the accurate structural part, give a complete representation of the SGS stress and SGS energy flux (SGS dissipation), which is essential for the SGS modeling of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2022) preliminarily explored a scale-similarity dynamic procedure (SSD) with a dynamic nonlinear algebraic model, yielding more accurate predictions of various turbulence statistics and instantaneous vortex structures for both a priori and a posteriori analyses of LES than the Germano-identity-based dynamic (GID) approach in the homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the SSD procedure still suffers from the numerical instability at coarse-grid-resolution cases, where the spatial discretization error dominates the SGS modeling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It might be challenging to develop a general dynamic framework for the model coefficient determination at various grid resolutions applicable to different types of turbulence problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results 3 demonstrate that the adjustment of SGS model parameters can effectively improve the accuracy of SGS modeling and enhance the predictions of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Besides, additional artificial viscous or penalized regularization terms have been also in- troduced to enhance the a posteriori stability of structural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A secondary filtering regularization technique was proposed by Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2001) and Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2004) to maintain the numerical stability of ADM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Vollant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2016) efficiently regularized the velocity gradient model by dynamically clipping the SGS backscatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A spectral-vanishing- viscosity method (Tadmor 1989) was proposed to effectively suppress the Gibbs oscillations at high wavenumbers (Cerutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2000) and has been successfully applied to the prediction of turbulent channel flows (Karamanos & Karniadakis 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2020a) used a hyperviscosity term to address the stability issue of the spatial-artificial-neural-network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The effective hyperviscosity term was further applied to other data-driven SGS models (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2021b) developed a small-scale eddy-viscosity model to enhance the a posteriori stability of dynamic iterative approximate deconvolution models, without affecting the accurate predictions of large-scale flow structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A kinetic-energy-flux constrained SGS model proposed by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2022) regularizes the DSM model by the correct kinetic energy flux approximated by the tensor-diffusivity model and accurately predicts the transition to turbulence of a compressible flat-plate boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It is noteworthy that additional numerical parameters would be introduced for most regularization techniques, which are sensitive to the grid resolution of LES, requiring multiple tedious testings for different turbulence scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To our knowledge, there might not be a unified adaptive regularization framework proposed for the stability of structural SGS models that can be universally applied to various types of turbulence with different grid resolutions of LES calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dependence of SGS model parameters on grid resolutions of LES might be effectively addressed by incorporating the a priori knowledge of DNS or experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In recent years, many data-driven closure approaches (Tracey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Maulik & San 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Park & Choi 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Guan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022) have been extensively developed to improve the modeling of unclosed terms in turbulence, as more high-fidelity DNS or experimental data become available (Kutz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Duraisamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2016b) proposed a representative tensor-basis- neural-network (TBNN) model with the multiplicative layer that predicts coefficients of the basis tensors for the modeled Reynolds stress by taking velocity invariants as input to preserve Galilean invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The TBNN architecture can accurately reconstruct the anisotropy of Reynolds stress and predict the flow separation better than the baseline linear or nonlinear eddy-viscosity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2020c) further developed the artificial-neural-network-based nonlinear algebraic models yielding better predictions of LES statistics than classical dynamic SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The gene- expression-programming technique was proposed to acquire the explicit mathematical expression of the unclosed SGS stress modeled by basis functions for LES using an evolutionary algorithm (Schoepplein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The multi-agent reinforcement-learning framework was developed to discover Smagorinsky model coefficients using the control policy rewarded by the statistical discrepancy of energy spectrum (Novati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Kurz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2023), and further applied to modeling the near-wall dynamics (Bae & Koumoutsakos 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Although the machine-learning-based closure models can improve the a priori accuracy of turbulence models fairly well, they have been reported to suffer from the ill-conditioned issues in the a posteriori studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The small a priori errors of the modeled Reynolds stress can be significantly amplified and then propagated into the mean velocity field in the a posteriori testings (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Gamahara & Hattori (2017) established an artificial-neural-network framework for the SGS closures of turbulent channel flows, which accurately predicts the unclosed SGS stress in a priori studies, but shows no obvious advantages over the Smagorinsky model in the reconstruction of the mean velocity profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The recurrent neural network was employed to 4 learn the coarse-grained discretization errors of LES and expected to construct the perfect LES formulation (Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, these perfect SGS closure terms also encounter serious a posteriori instability issues, even though the a priori predictions show high correlations with the exact unclosed terms (Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results indicate that most current data-driven closure approaches can acquire sufficiently high a priori accuracy after being trained by the high-fidelity DNS or experimental data, but still lack indispensable extrapolation capabilities and are difficult to be applied to the a posteriori testings of out-of-sampling flow scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The data-assimilation techniques can effectively remedy the deficiencies of insufficient a posteriori accuracy of closure models by iteratively evaluating and minimizing the discrep- ancies between coarse-grained a posteriori calculations and benchmark high-fidelity DNS or experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The data-assimilation approaches can be generally classified into three categories: ensemble-based statistical methods (Colburn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022), adjoint-based variational approaches (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Delport et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Badreddine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2014) and their mixed variants (Mons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ensemble-based statistical techniques use ensemble statistics to approximately measure the model uncertainty and continuously correct the measurement errors of observations by the classical Kalman-filtering strategies or nudging methods (Clark Di Leoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These statistical assimilation methods allow the convenient inference of flow states and statistics, without any detailed information of dynamical systems, facilitating their wide application in complex practical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the state estimations of these ensemble-based approaches frequently evaluate the matrix multiplication and inverse operations, resulting in the massive computation expense and large memory usage for the high degree-of-freedom turbulence problems at a high Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the adjoint- based variational techniques employ the optimal control strategy to efficiently optimize the model parameters or state variables by minimizing the discrepancies between the benchmark observations and a posteriori predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Singh & Duraisamy (2016) proposed a field-inversion procedure to infer model discrepancies in the source terms of Reynolds-averaged Navier–Stokes (RANS) transport equations using Bayesian posterior estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2018) simplified the field-inversion strategy and employed the continuous adjoint formulation to optimize a spatially varying turbulence production term in the Spalart–Allmaras model of RANS equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In comparison with the extensive studies of data-assimilation-based RANS models (Kato & Obayashi 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xiao & Cinnella 2019), investigations on SGS models of LES assimilated with high-fidelity simulation data are still preliminary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A spatially-varying parameter in a local uncertainty model and initial conditions were optimized based on experimental observations of the cylindrical wake flow using the discrete adjoint algorithm (Chandramouli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Mons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2021) developed a non-intrusive ensemble-variational approach (EnVar) to enhance the predictions of the mean flow and Reynolds stresses by adjusting the wall-normal distribution of the Smagorinsky coefficient or injecting an artificial steady force in the LES momentum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS force modeled by the artificial neural network was optimized by the point-to-point errors of the filtered velocity field using the discrete adjoint method for the decaying isotropic turbulence and plane jet flows (Sirignano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' MacArt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, these discrete adjoint or ensemble-based variational methods require massive matrix operations with significant memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In this paper, a variational optimal mixed model (VOMM) is proposed to reconstruct the unclosed SGS stress by assimilating the turbulence statistics of high-fidelity filtered DNS data using the continuous adjoint approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The main difference from the previous work is that we derive adjoint LES equations with the general SGS model and conduct the energy budget analysis of adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The continuous adjoint algorithm can enhance the physical understanding of the adjoint-based sensitivities and provide flexibility in selecting the discretization scheme for the adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The quadratic terms of shear strain rate in adjoint LES equations turn out to be responsible for the exponential temporal growth of the adjoint-based gradients, giving rise to the 5 numerical divergence in a long time horizon for the chaotic turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Hence, the stabilized adjoint LES equations are correspondingly formulated to enhance the numerical stability of the adjoint LES calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To the extent of the authors’ knowledge, few previous studies have given detailed derivations of the adjoint LES equations with general SGS mixed models and formulated the stabilized version for long-term gradient evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the selected cost functional is essential for the convergence and performance of adjoint-based gradient optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to the previous studies, turbulence statistical discrepancies rather than the chaotic point-to-point prediction errors are adopted to quantify the multiscale statistical behaviours of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a priori information about statistics of turbulence acquired from experimental data or DNS results, including energy spectra, structure functions, and probability density functions of physical quantities, can be used to determine or correct SGS model parameters to improve the a posteriori accuracy of LES greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Turbulent statistical assimilation can effectively alleviate the impact of chaotic field observations on the performance of data assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Furthermore, the a posteriori performance of VOMM model is comprehensively investigated and compared to classical SGS models at multiple grid resolutions in different turbulence scenarios, including the forced and decaying homogeneous isotropic turbulence, as well as the temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2 describes the governing equations of the large-eddy simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The conventional subgrid-scale models, including DSM, DMM and ADM models, are briefly introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4, we first derive the adjoint LES equations with a general form of mixed SGS models, then conduct the energy budget analysis of adjoint equations, and correspondingly propose the stabilized adjoint LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Afterwards, the adjoint-based variational optimal mixed model is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5 further investigates the a posteriori performance of the VOMM model in comparison to the classical SGS models for three turbulent flow scenarios, including the forced homogeneous isotropic turbulence, decaying homogeneous isotropic turbulence, and temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Conclusions are finally drawn in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Governing equations of the large-eddy simulation The three dimensional incompressible turbulence is governed by the Navier-Stokes equations (Pope 2000), namely 𝜕𝑢𝑖 𝜕𝑥𝑖 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) 𝜕𝑢𝑖 𝜕𝑡 + 𝜕 �𝑢𝑖𝑢 𝑗 � 𝜕𝑥 𝑗 = − 𝜕𝑝 𝜕𝑥𝑖 + 𝜈 𝜕2𝑢𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + F𝑖, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) where 𝑢𝑖 is the 𝑖-th component of velocity, 𝑝 denotes the pressure divided by the constant density, 𝜈 is the kinematic viscosity, and F𝑖 represents the large-scale forcing on the fluid momentum in the 𝑖-th coordinate direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The summation convection for the repeated indices is adopted by default for simplicity in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Besides, the dimensionless governing parameter for the incompressible turbulence, namely, the Taylor microscale Reynolds number 𝑅𝑒𝜆 is defined as (Pope 2000) 𝑅𝑒𝜆 = 𝑢rms𝜆 √ 3𝜈 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) where 𝑢rms = √︁ ⟨𝑢𝑖𝑢𝑖⟩ represents the root-mean-square (rms) value of the velocity magnitude, and ⟨·⟩ represents a spatial average along the homogeneous direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', average over the entire domain for the isotropic turbulence and the horizontal average for the temporally evolving mixing 6 layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝜆 = 𝑢rms√︁ 5𝜈/𝜀 is the Taylor microscale, where 𝜀 = 2𝜈 � 𝑆𝑖 𝑗𝑆𝑖 𝑗 � represents the average dissipation rate and 𝑆𝑖 𝑗 = 1 2 �𝜕𝑢𝑖/𝜕𝑥 𝑗 + 𝜕𝑢 𝑗/𝜕𝑥𝑖 � denotes the strain-rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To obtain the governing equations of the large-eddy simulation, a spatial filtering operation, ¯𝑓 (x) = ∫ Ω 𝑓 (x′) 𝐺 �x − x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� 𝑑x′ is applied to the Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, an overbar denotes the spatial filtering, Ω is the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐺 and ¯Δ are the filter kernel and filter width, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The governing equations for the LES can be correspondingly derived as (Sagaut 2006) 𝜕 ¯𝑢𝑖 𝜕𝑥𝑖 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4) 𝜕 ¯𝑢𝑖 𝜕𝑡 + 𝜕 � ¯𝑢𝑖 ¯𝑢 𝑗 � 𝜕𝑥 𝑗 = − 𝜕 ¯𝑝 𝜕𝑥𝑖 − 𝜕𝜏𝑖 𝑗 𝜕𝑥 𝑗 + 𝜈 𝜕2 ¯𝑢𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + ¯F𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) Here, the unclosed SGS stress tensor 𝜏𝑖 𝑗 = 𝑢𝑖𝑢 𝑗 − ¯𝑢𝑖 ¯𝑢 𝑗 cannot be directly calculated using the resolved variables ¯𝑢𝑖, and additional SGS stress modeling is required to make the LES equations solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Conventional subgrid-scale models for LES The SGS models aim to establish the approximate constitutive equation for SGS unclosed terms using the known resolved variables, and reconstruct the nonlinear interactions between the resolved large scales and unsolved small scales as accurately as possible (Moser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Johnson 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The explicit SGS models consist of the functional and structural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The functional modeling adopts the eddy-viscosity forms to mimic the forward kinetic energy transfer from the resolved large scales to the residual small scales, while the structural models can accurately recover the unclosed SGS stress by the hypothesis of scale similarity or using the truncated series expansions with high a priori accuracy (Sagaut 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' One of the most widely-used functional models is the Smagorinsky model (Smagorinsky 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Lilly 1967), expressed as 𝜏𝐴 𝑖 𝑗 = 𝜏𝑖 𝑗 − 𝛿𝑖 𝑗 3 𝜏𝑘𝑘 = −2𝐶2 𝑆 ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) where 𝛿𝑖 𝑗 denotes the Kronecker delta operator, ¯𝑆𝑖 𝑗 = 1 2 �𝜕 ¯𝑢𝑖/𝜕𝑥 𝑗 + 𝜕 ¯𝑢 𝑗/𝜕𝑥𝑖 � is the filtered strain-rate tensor and | ¯𝑆| = (2 ¯𝑆𝑖 𝑗 ¯𝑆𝑖 𝑗)1/2 represents the characteristic filtered strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The superscript “A” represents the trace-free anisotropic part of the arbitrary variables, namely, (•) 𝐴 𝑖 𝑗 = (•)𝑖 𝑗 − (•)𝑘𝑘𝛿𝑖 𝑗/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The isotropic SGS stress 𝜏𝑘𝑘 is absorbed into the pressure term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶2 𝑆 is the Smagorinsky coefficient and can be determined empirically or by a theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The most common approach is based on the least-squares dynamic procedure using the Germano identity, giving rise to the dynamic Smagorinsky model (DSM), whose coefficient is given by (Germano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Lilly 1992) 𝐶2 𝑆 = ⟨𝐿 𝐴 𝑖 𝑗M𝑖 𝑗⟩ ⟨M𝑘𝑙M𝑘𝑙⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) where the Leonard stress 𝐿𝑖 𝑗 = � ¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗, 𝐿 𝐴 𝑖 𝑗 = 𝐿𝑖 𝑗 − 1 3𝛿𝑖 𝑗𝐿𝑘𝑘 and M𝑖 𝑗 = ˜𝛼𝑖 𝑗 − 𝛽𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, a tilde stands for the test filtering operation at the double-filtering scale ˜Δ = 2 ¯Δ, the variables 𝛼𝑖 𝑗 = 2 ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗 and 𝛽𝑖 𝑗 = 2 ˜Δ2| ˜¯𝑆| ˜¯𝑆𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The scale-similarity model 𝜏𝑖 𝑗 = � ¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗 is a typical structural model and can correctly reconstruct the SGS stress with high a priori accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, these structural models often exhibit insufficient dissipation and numerical instability in the a posteriori testings of LES due to the underestimation of the forward kinetic energy cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dynamic mixed model (DMM) combines the scale-similarity model with the dissipative 7 Smagorinsky term, and is given by (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2008) 𝜏𝑖 𝑗 = 𝐶1 ¯Δ2 �� ¯𝑆 �� ¯𝑆𝑖 𝑗 + 𝐶2 � � ¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) Similar to the DSM model, model coefficients of the DMM model 𝐶1 and 𝐶2 are dynamically determined by the least-squares algorithm using the Germano identity, expressed respectively as (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020) 𝐶1 = � 𝑁2 𝑖 𝑗 � � 𝐿𝑖 𝑗 𝑀𝑖 𝑗 � − � 𝑀𝑖 𝑗𝑁𝑖 𝑗 � � 𝐿𝑖 𝑗𝑁𝑖 𝑗 � � 𝑁2 𝑖 𝑗 � � 𝑀2 𝑖 𝑗 � − � 𝑀𝑖 𝑗𝑁𝑖 𝑗 �2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4) 𝐶2 = � 𝑀2 𝑖 𝑗 � � 𝐿𝑖 𝑗𝑁𝑖 𝑗 � − � 𝑀𝑖 𝑗𝑁𝑖 𝑗 � � 𝐿𝑖 𝑗 𝑀𝑖 𝑗 � � 𝑁2 𝑖 𝑗 � � 𝑀2 𝑖 𝑗 � − � 𝑀𝑖 𝑗𝑁𝑖 𝑗 �2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) where 𝑀𝑖 𝑗 = 𝐻1,𝑖 𝑗 − ˜ℎ1,𝑖 𝑗, and 𝑁𝑖 𝑗 = 𝐻2,𝑖 𝑗 − ˜ℎ2,𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, ℎ1,𝑖 𝑗 = −2 ¯Δ2 �� ¯𝑆 �� ¯𝑆𝑖 𝑗, ℎ2,𝑖 𝑗 = � ¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗, 𝐻1,𝑖 𝑗 = −2 ˜Δ2 ��� ˜¯𝑆 ��� ˜¯𝑆𝑖 𝑗, and 𝐻2,𝑖 𝑗 = � ˜¯𝑢𝑖 ˜¯𝑢 𝑗 − ˆ˜¯𝑢𝑖 ˆ˜¯𝑢 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The hat stands for the test filtering at scale ˆΔ = 4 ¯Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The unfiltered variables can be accurately recovered by the resolved filtered field using the iterative approximate deconvolution procedure, namely (Stolz & Adams 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001) 𝑢∗ 𝑖 = 𝐴𝐷 𝑁 ( ¯𝑢𝑖) = 𝑁 ∑︁ 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ ¯𝑢𝑖, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6) where the asterisk represents the approximately unfiltered variables, 𝐴𝐷 𝑁 is the abbreviation of the 𝑁-th order approximate deconvolution, 𝐼 is the identity, and the symbol “⊗” stands for the spatial convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For any two functions 𝑓 and 𝑔, 𝑓 ⊗ 𝑔 = ∫ +∞ −∞ 𝑓 (x′) 𝑔 (x − x′) 𝑑x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The unclosed SGS stress then can be recovered with the scale-similarity form by the approximate deconvolution method (ADM), given by (Bardina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1980) 𝜏𝑖 𝑗 = 𝑢∗ 𝑖 𝑢∗ 𝑗 − ¯𝑢∗ 𝑖 ¯𝑢∗ 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) The number of iterations for the ADM model is recommended to be 𝑁 =3 ∼ 5 (Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The accuracy of the ADM model becomes higher, while the numerical stability drops, as the number of iterations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Hence, 𝑁 = 5 is selected in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to maintain the numerical stability of the a posteriori testings of LES [𝜕 ¯𝑢𝑖/𝜕𝑡 = ¯𝑅𝑖 ( ¯𝑢𝑖, 𝑡)], Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2001) and Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2004) introduced a secondary filtering relaxation term [𝜕 ¯𝑢𝑖/𝜕𝑡 = ¯𝑅𝑖 ( ¯𝑢𝑖, 𝑡)+ ¯𝑆𝑖 ( ¯𝑢𝑖)], yielding ¯𝑆𝑖 ( ¯𝑢𝑖) = −𝜒 � 𝐼 − 𝐺 ⊗ 𝑁 ∑︁ 𝑛=1 (𝐼 − 𝐺)𝑛−1 � ⊗ ¯𝑢𝑖, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8) where 𝜒 is the empirical regularization coefficient, which is approximately insensitive to the LES results in previous studies, and we choose 𝜒 = 0 and 1 for comparisons in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Adjoint-based variational optimal mixed models (VOMM) The mixed model is composed of the structural parts and the dissipative functional terms, and its general form can be written as (Sagaut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2000) 𝜏𝑖 𝑗 �𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� = 𝑁 ∑︁ 𝑛=1 𝐶𝑛𝑇 (𝑛) 𝑖 𝑗 � ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ�, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) 8 where 𝑇 (𝑛) 𝑖 𝑗 � ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� represents the 𝑛-th basis stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) denotes the corre- sponding model coefficient and 𝑁 is the number of basis stress tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The model coefficients are generally respectively determined by the multivariate least-squares algorithm proposed by Germano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (1991) and Lilly (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Many previous studies have shown that the dynamic mixed models give rise to an excessive dissipation of energy in the transitional regions and dissipation underestimation if the filter scales are sufficiently large, especially in situations of grid anisotropy (Meneveau & Katz 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Moser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In recent years, data-driven based high-accuracy SGS models are successively proposed (Kutz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Duraisamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2019a) proposed an artificial-neural-network-based mixed model which accurately recovers the unclosed SGS terms by estimating mixed model coefficients with local flow characteristics as inputs of the machine-learning strategy, yielding better predictions of LES statistics than the classical dynamic mixed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The input features of the data-driven closure models are crucial for the accuracy of SGS models (Gamahara & Hattori 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Park & Choi 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Incorporating the accurate structural parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', filtered velocity gradients at the neighboring stencil turn out to improve the performance of data-driven SGS models effectively (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019b, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Moreover, the spatial flow structures at scales between ¯Δ/2 and 2 ¯Δ are found to be essential for the SGS modeling of LES at the filter scale ¯Δ (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The strategy of the blind deconvolution with the artificial neural network was proposed to recover the unknown original unfiltered variables from the known filtered quantities with high accuracy (Maulik & San 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Maulik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A deconvolutional-artificial-neural-network (DANN) framework was further proposed to accurate reconstruct the SGS unclosed terms both in a priori and a posteriori analyses of isotropic turbulence (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020, 2021a), and successfully applied to the chemically reacting compressible turbulence (Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It was demonstrated that the DANN models embed the properties of symmetry and realizability conditions, which preserve the physical reliability of the DANN framework (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to enhance the interpretability of black-box machine-learning SGS models, a semi-explicit ANN-based spatial gradient model and constant-coefficient spatial gradient models are successively proposed by the elaborate Taylor expansions of velocity gradients in the neighboring stencil locations (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The machine-learning-based SGS models trained by high-fidelity simulation data can be regarded as the structural models with high a priori accuracy, requiring additional indispensable dissipation to account for the spatial discretization effect and ensure the numerical stability in the a posteriori studies of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition to the machine-learning-assisted SGS models, some a priori information about statistics of turbulence acquired from experimental data or DNS results like energy spectra, structure functions, and probability density functions of physical quantities can be used to determine or correct the model coefficients of SGS models to improve the model accuracy greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These a priori knowledge of turbulent statistical quantities can be dynamically assimilated into the closure models via the data-assimilation based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Among these data-assimilation techniques, adjoint-based variational methods adopt the optimal control strategy to efficiently calculate all the gradients of cost functionals for the model coefficients by solving the forward governing equations and the backward adjoint equations (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Then, the model coefficients of SGS models are iteratively updated using the gradient-based optimization algorithm until the optimal values are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The cost functionals measure the discrepancies of statistical quantities in turbulence between the LES results and measurements from the experimental or DNS data, which can greatly alleviate the impact of chaotic field observations on the performance of data assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In this work, we resort to the state-of-art adjoint-based data-assimilation approaches to establish a general optimal SGS framework to determine model parameters adaptively for various grid resolutions of LES in different turbulence scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Adjoint LES equations and gradient evaluations with the mixed model We optimize the model coefficients of the SGS closure model to minimize the statistical discrepancies between the LES calculations and the reference values acquired from the experi- mental or DNS data, which can be defined as the minimal optimization problem constrained by the governing equations (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The constrained optimization problem for the turbulent closure modeling is expressed as min 𝐶𝑛 J � 𝜙 ( ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , 𝜙 � ¯𝑢ref 𝑖 �� , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑅0 ( ¯𝑢𝑖) = 𝜕 ¯𝑢𝑖 𝜕𝑥𝑖 = 0, 𝑅𝑖 ( ¯𝑢𝑖, ¯𝑝) = 𝜕 ¯𝑢𝑖 𝜕𝑡 + 𝜕( ¯𝑢𝑖 ¯𝑢𝑗) 𝜕𝑥𝑗 + 𝜕 ¯𝑝 𝜕𝑥𝑖 − 𝜈 𝜕2 ¯𝑢𝑖 𝜕𝑥𝑗𝜕𝑥𝑗 − F 𝑖 + 𝜕𝜏𝑖 𝑗 𝜕𝑥𝑗 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) where J � 𝜙 ( ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , 𝜙 � ¯𝑢ref 𝑖 �� = 𝑇∫ 0 ∫ Ω 𝐽 � 𝜙 ( ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛, x, 𝑡) , 𝜙 � ¯𝑢ref 𝑖 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' x, 𝑡�� 𝑑x𝑑𝑡 denotes the total cost functions, 𝐽 � 𝜙 ( ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛, x, 𝑡) , 𝜙 � ¯𝑢ref 𝑖 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' x, 𝑡�� is the discrepancy of statistical quantities 𝜙 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' kinetic energy spectra, structure functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=') between the LES results ¯𝑢𝑖 and reference values ¯𝑢ref 𝑖 (experimental or DNS data) at a certain state (𝐶𝑛, x, 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) denotes model coefficients of the SGS mixed model 𝜏𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛) 𝑖 𝑗 , and 𝑡 ∈ [0,𝑇] is the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, “s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='t.” stands for the abbreviation of “subject to”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑅0 and 𝑅𝑖 (𝑖 = 1, 2, 3) represent the LES continuity equation and momentum equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Lagrangian functional L is introduced to take the dynamics of LES variables ¯v = [ ¯𝑝, ¯𝑢1, ¯𝑢2, ¯𝑢3]𝑇 into account and convert the constrained optimization (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) into the un- constrained optimization problem, namely (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2006) min 𝐶𝑛 L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , where L = J � 𝜙 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , 𝜙 � ¯vref�� − 3 ∑︁ 𝑘=0 𝑇 ∫ 0 ∫ Ω 𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) · ¯𝑣† 𝑘𝑑x𝑑𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) Here, ¯v† = � ¯𝑝†, ¯𝑢† 1, ¯𝑢† 2, ¯𝑢† 3 �𝑇 are the adjoint LES variables of ¯v, where ¯𝑝† and ¯𝑢† 𝑖 are the adjoint pressure and adjoint velocity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the sake of brevity, the inner product of time and space is defined by ⟨f, g⟩x,𝑡 = 𝑇∫ 0 ∫ Ω f (x, 𝑡) · g (x, 𝑡) 𝑑x𝑑𝑡, where f (x, 𝑡) and g (x, 𝑡) denote the arbitrary physical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Lagrangian functional L can be simplified as L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) = J (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛)− 3� 𝑘=0 � 𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , ¯v†� x,𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The sensitivity of the Lagrangian functional L can be derived by 𝛿L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) = 𝛿J (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) − 3 ∑︁ 𝑘=0 � 𝑅𝑘 (𝛿¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , ¯v†� x,𝑡 − 3 ∑︁ 𝑘=0 � 𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝛿𝐶𝑛) , ¯v†� x,𝑡, = 𝛿J (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝜕¯v 𝛿¯v, ¯v† � x,𝑡 − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝜕𝐶𝑛 𝛿𝐶𝑛, ¯v† � x,𝑡 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4) where 𝜕𝑅𝑘/𝜕¯v and 𝜕𝑅𝑘/𝜕𝐶𝑛 are the tangent operators of the governing equations 𝑅𝑘 (𝑘 = 0, 1, 2, 3) for the variables ¯v and parameters 𝐶𝑛 with the perturbation field 𝛿¯v = ¯v (𝐶𝑛 + 𝛿𝐶𝑛) − ¯v (𝐶𝑛) , 𝑛 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 is the sensitivity of the cost functional J and calculated as the Gâteaux-Fréchet derivative (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001) 10 of J at 𝐶𝑛 in the direction 𝛿𝐶𝑛, namely 𝛿J (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝛿𝐶𝑛) = lim 𝜀→0 𝑑 𝑑𝜀 J (¯v (𝐶𝑛 + 𝜀𝛿𝐶𝑛)) = � 𝜕𝐽 𝜕¯v, 𝛿¯v � x,𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) The adjoint identity (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001) can be obtained via the integral by part, given by � R (¯v) , ¯v†� x,𝑡 = � ¯v, R† � ¯v†�� x,𝑡 + � ¯F, ¯v†� 𝑡 �� Γ + �¯v, ¯v†� x ��𝑇 0 = � ¯v, R† � ¯v†�� x,𝑡 + 𝐵𝑇, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6) where the partial differential equations R (¯v) = 𝜕¯v/𝜕𝑡 + 𝜕 ¯F/𝜕x = 0 with the associated adjoint operator R† �¯v†�, ¯F denotes the fluxes and Γ is the boundary of the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝐵𝑇 = � ¯F, ¯v†� 𝑡 �� Γ + �¯v, ¯v†� x ��𝑇 0 represents the boundary and temporal integral terms, which determines the boundary and terminal conditions of the adjoint equations to give 𝐵𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ⟨f, g⟩𝑡 = 𝑇∫ 0 f (x, 𝑡) · g (x, 𝑡) 𝑑𝑡 and ⟨f, g⟩x = ∫ Ω f (x, 𝑡) · g (x, 𝑡) 𝑑x denote the temporal and spatial inner products, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 can be expressed with the adjoint identity, namely (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Delport et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2009, 2011) � 𝜕𝑅𝑘 𝜕¯v · 𝛿¯v, ¯v† � x,𝑡 = � 𝛿¯v, � 𝜕𝑅𝑘 𝜕¯v �† ¯v† � x,𝑡 + 𝐵𝑇, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) where (𝜕𝑅𝑘/𝜕¯v)† is the adjoint operator of the LES tangent Jacobian tensor 𝜕𝑅𝑘/𝜕¯v, (𝑘 = 0, 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Substitute the Fréchet derivative 𝛿J (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) and the adjoint identity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) into the sensitivity of the Lagrangian functional L (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4), and we get 𝛿L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) = � 𝜕𝐽 𝜕¯v − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 𝜕¯v �† ¯v†, 𝛿¯v � x,𝑡 − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝜕𝐶𝑛 𝛿𝐶𝑛, ¯v† � x,𝑡 − 𝐵𝑇, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8) To avoid calculating the perturbation field 𝛿¯v in the first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8, the inner product should be equal to 0 and the corresponding adjoint LES equations can be derived by 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 𝜕¯v �† ¯v† − 𝜕𝐽 𝜕¯v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='9) Substitute the specific forms of the LES equations 𝑅𝑘 (𝑘 = 0, 1, 2, 3) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2), and the adjoint LES equations can be written as 𝜕 ¯𝑢† 𝑖 𝜕𝑥𝑖 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10) 𝜕 ¯𝑢† 𝑖 𝜕𝑡 + � 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 � ¯𝑢 𝑗 + 𝜕 ¯𝑝† 𝜕𝑥𝑖 + 𝜈 𝜕2 ¯𝑢† 𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + 𝜕𝜏† 𝑖 𝑗 𝜕𝑥 𝑗 + 𝜕𝐽 𝜕 ¯𝑢𝑖 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11) where 𝜏† 𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛),† 𝑖 𝑗 denotes the adjoint SGS mixed model and 𝑇 (𝑛),† 𝑖 𝑗 is the 𝑛-th adjoint basis stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The detailed derivation of the adjoint LES equations can refer to the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The terminal conditions of the adjoint LES equations is determined by the last term of adjoint identity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6), namely �¯v†, 𝛿¯v � x ��𝑇 0 = �¯v† (𝑇) , 𝛿¯v (𝑇) � x − �¯v† (0) , 𝛿¯v (0) � x = �¯v† (𝑇) , 𝛿¯v (𝑇) � x, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='12) where 𝛿¯v (0) = 0, since the unperturbed initial LES field is exactly given by the filtered DNS (fDNS) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The terminal conditions ¯v† (𝑇) = � ¯𝑢† 𝑖 (𝑇) , ¯𝑝† (𝑇) �𝑇 = 0 make the temporal integral 11 terms �� 𝛿¯v, ¯v†� x �𝑇 0 equal to zero and the calculation of the terminal perturbation 𝛿¯v (𝑇) is obviated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The terminal conditions ( ¯𝑢† 𝑖 (𝑇) = 0, ¯𝑝† (𝑇) = 0) and boundary conditions of the adjoint LES equations are identified by setting 𝐵𝑇 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The sensitivity of the Lagrangian functional L can be further expressed as 𝛿L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) = − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝜕𝐶𝑛 𝛿𝐶𝑛, ¯v† � x,𝑡 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='13) where 𝜕𝑅0/𝜕𝐶𝑛 = 0, and 𝜕𝑅𝑖/𝜕𝐶𝑛 = 𝜕 𝜕𝐶𝑛 � 𝜕𝜏𝑖 𝑗 𝜕𝑥𝑗 � = 𝜕𝑇 (𝑛) 𝑖 𝑗 /𝜕𝑥 𝑗 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) denotes the 𝑛-th SGS basis force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Once the LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) temporally advances forward in the time horizon 𝑡 ∈ [0,𝑇] and the adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11) are integrated backward with zero terminal conditions, the gradients of Lagrangian functional for the SGS model coefficients can be calculated efficiently by 𝜕L 𝜕𝐶𝑛 = 𝛿L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝛿𝐶𝑛 = − � 𝜕𝑇 (𝑛) 𝑖 𝑗 𝜕𝑥 𝑗 , ¯𝑢† 𝑖 � x,𝑡 , (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='14) The adjoint-based gradient evaluations are independent of the parameter perturbations 𝛿𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁), which are very efficient compared to the finite difference algorithm and forward sensitivity analysis with at least 𝑁 parameter perturbations and 𝑁 + 1 LES equation calculations for each optimization iteration (Chandramouli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sirignano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' MacArt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Energy budget analysis of the adjoint LES equations Before proceeding to the introduction of the variational optimal mixed models, it is essential to analyze the energy budget of the adjoint LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint LES kinetic energy ( ¯E† = ¯𝑢† 𝑖 ¯𝑢† 𝑖 /2) equation is derived through multiplying the adjoint velocity ¯𝑢† 𝑖 on both sides of the adjoint LES momentum equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11), namely 𝜕 ¯E† 𝜕𝑡 + 𝜕 ¯P𝑗 𝜕𝑥 𝑗 = ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 + ¯𝐷† − ¯Π† − ¯𝐽†, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='15) where ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 denotes the adjoint energy production term due to the shear strain rate ¯𝑆𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, ¯P𝑗 is the adjoint spatial transport flux, ¯𝐷 is the adjoint viscous dissipation term, ¯Π† is the adjoint variable of the SGS energy flux ¯Π = −𝜏𝑖 𝑗 ¯𝑆𝑖 𝑗 and ¯𝐽† is the energy injected from the discrepancy between LES results and reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These terms are respectively defined by ¯P𝑗 = ¯E† ¯𝑢 𝑗 + � ¯𝑝† + ¯𝑢𝑖 ¯𝑢† 𝑖 � ¯𝑢† 𝑗 + � 𝜈 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 + 𝜏† 𝑖 𝑗 � ¯𝑢† 𝑖 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='16) ¯𝐷 = 𝜈 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='17) ¯Π† = −𝜏† 𝑖 𝑗 ¯𝑆† 𝑖 𝑗, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='18) ¯𝐽† = ¯𝑢† 𝑖 𝜕𝐽 𝜕 ¯𝑢𝑖 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='19) 12 where ¯𝑆† 𝑖 𝑗 = � 𝜕 ¯𝑢† 𝑖 /𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗/𝜕𝑥𝑖 � /2 represents the adjoint strain-rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The backward evolution of the adjoint volume-averaged kinetic energy can be written as − 𝜕 � ¯E†� 𝜕𝑡 = − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � − � ¯𝐷†� + � ¯Π†� + � ¯𝐽†� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='20) where � ¯𝐷†� is pure dissipation term that drains out the adjoint energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' � ¯Π†� denotes the adjoint SGS energy transport term which represents the forward adjoint energy transfer from large scales to unsolved residual scales if � ¯Π†� > 0, otherwise stands for the adjoint SGS energy backscatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The accurate reconstruction of � ¯Π†� is crucial for the SGS modeling of LES and gradient evaluations with respect to the SGS model coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' � ¯𝐽†� is the loss-induced adjoint energy injection term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' � ¯𝐷†� is the viscous dissipation which enhances the numerical stability of the adjoint LES field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' � ¯𝐽†� is the adjoint energy production due to the discrepancy between LES evaluation and reference data, which dominates the accuracy of the sensitivity calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The large-scale strain-rate tensor ¯𝑆𝑖 𝑗 can be decomposed into its principal components using the eigendecomposition approach, such that (Wang & Gao 2013) ¯𝑆𝑖 𝑗=𝜆1𝑞(1) 𝑖 𝑞(1) 𝑗 + 𝜆2𝑞(2) 𝑖 𝑞(2) 𝑗 + 𝜆3𝑞(3) 𝑖 𝑞(3) 𝑗 = 3 ∑︁ 𝑘=1 𝜆𝑘𝑞(𝑘) 𝑖 𝑞(𝑘) 𝑗 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='21) where 𝜆1, 𝜆2 and 𝜆3 are the eigenvalues of the shear strain rate, with 𝑞(1) 𝑖 , 𝑞(2) 𝑖 and 𝑞(3) 𝑖 being the associated eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here,𝜆1+𝜆2+𝜆3 = 0 for the trace-free strain rate ¯𝑆𝑖 𝑗 in the incompressible turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Hence, the quadratic term − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='20 is further expressed as − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � = − 3 ∑︁ 𝑘=1 � 𝜆𝑘 � 𝑞(𝑘) 𝑖 ¯𝑢† 𝑖 � � 𝑞(𝑘) 𝑗 ¯𝑢† 𝑗 �� = − 3 ∑︁ 𝑘=1 � 𝜆𝑘 � 𝑞(𝑘) 𝑖 ¯𝑢† 𝑖 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='22) The sign of the eigenvalues 𝜆𝑘, (𝑘 = 1, 2, 3) determines the contribution of the adjoint energy from the quadratic term − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � is productive or dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The quadratic terms with negative eigenvalues of the shear strain rate produce the positive adjoint energy production, while those with positive eigenvalues drain out the adjoint energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In previous studies of chaotic adjoint methods, the adjoint-based gradients are found to grow exponentially with time and finally numerically diverge in a long time horizon for the chaotic flows (Wang & Gao 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Ashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garai & Murman 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The terms � ¯𝐷†� , � ¯Π†� and � ¯𝐽†� in the volume-averaged adjoint energy equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='20) are less likely to cause the exponential growth of the adjoint energy, since the adjoint energy term � ¯E†� does not appears explicitly in these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It can be further shown that the quadratic term − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � plays the dominant role in the exponential growth of the adjoint variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We apply the Cauchy-Schwarz inequality to the inner product terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='22, such that (Talnikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2017) � 𝑞(𝑘) 𝑖 ¯𝑢† 𝑖 �2 ⩽ � 𝑞(𝑘) 𝑖 𝑞(𝑘) 𝑖 � � ¯𝑢† 𝑖 ¯𝑢† 𝑖 � = 2 ���q(𝑘)��� E † (𝑘 = 1, 2, 3) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='23) where “∥·∥” denotes the L2 norm of the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the quadratic terms with negative eigenvalues (adjoint energy production), the evolution of the adjoint energy can be approximated using the leading principal vectors as − 𝜕 � E †� 𝜕𝑡 ≈ 2|𝜆|∞∥q∥∞ � E †� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='24) 13 where |𝜆|∞ = max Ω {−𝜆1, −𝜆2, −𝜆3} denotes the magnitude of the leading negative eigenvalue in the entire domain Ω and ∥q∥∞ represents the corresponding eigenvector magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint energy is then calculated by the backward time interval, namely � E †� (𝑡) ≈ � E †� (𝑇) exp [2|𝜆|∞∥q∥∞ (𝑇 − 𝑡)] , 𝑡 ∈ [0,𝑇] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='25) The quadratic term − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � with negative eigenvalues makes the adjoint energy grow exponentially over time and numerically unstable if it cannot be suppressed by the adjoint dissipation in a long time horizon 𝑡 ∈ [0,𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to stabilize the adjoint equations during every iteration, an additional symmetric tensor ¯𝑆𝑎 𝑖 𝑗 (Ashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garai & Murman 2021) is introduced to maintain the numerical stability of the adjoint momentum (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11), and the stabilized adjoint momentum equations are then expressed as 𝜕 ¯𝑢† 𝑖 𝜕𝑡 + � 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 � ¯𝑢 𝑗 + ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗 + 𝜕 ¯𝑝† 𝜕𝑥𝑖 + 𝜈 𝜕2 ¯𝑢† 𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + 𝜕𝜏† 𝑖 𝑗 𝜕𝑥 𝑗 + 𝜕𝐽 𝜕 ¯𝑢𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26) Consequently, the stabilized adjoint kinetic energy equation is written by 𝜕E † 𝜕𝑡 + 𝜕P 𝑗 𝜕𝑥 𝑗 = ¯𝑢† 𝑖 � ¯𝑆𝑖 𝑗 − ¯𝑆𝑎 𝑖 𝑗 � ¯𝑢† 𝑗 + ¯𝐷† − ¯Π† − ¯𝐽†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='27) Here, the quadratic term ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 < 0 � − ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 > 0 � is responsible for the exponential growth of the adjoint energy, and the minimal artificial symmetric tensor is added to keep the adjoint variables numerically stable in advancing backward of the adjoint LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The artificial symmetric tensor ¯𝑆𝑎 𝑖 𝑗 can be optimized by the suboptimal minimization problem (Ashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garai & Murman 2021), such that min ¯𝑆𝑎 𝑖 𝑗 1 2 ¯𝑆𝑎 𝑖 𝑗 ¯𝑆𝑎 𝑖 𝑗, 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯𝑢† 𝑖 � ¯𝑆𝑖 𝑗 − ¯𝑆𝑎 𝑖 𝑗 � ¯𝑢† 𝑗 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='28) We use the sequential quadratic programming (SQP) approach (Boggs & Tolle 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Chung & Freund 2022) to efficiently solve the suboptimal problem, and the augmented Lagrangian functional L is applied to the constrained minimization problem, namely L = 1 2 ¯𝑆𝑎 𝑖 𝑗 ¯𝑆𝑎 𝑖 𝑗 + 𝜆 � ¯𝑢† 𝑖 � ¯𝑆𝑖 𝑗 − ¯𝑆𝑎 𝑖 𝑗 � ¯𝑢† 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='29) where 𝜆 is the Lagrangian multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Karush–Kuhn–Tucker (KKT) optimal conditions (Kuhn & Tucker 1951;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Blonigan & Wang 2018) are obtained by taking the derivatives of the cost functional with respect to the augmented optimal variables ( ¯𝑆𝑎 𝑖 𝑗 and 𝜆), derived by 𝜕L 𝜕 ¯𝑆𝑎 𝑖 𝑗 = ¯𝑆𝑎 𝑖 𝑗 − 𝜆 � ¯𝑢† 𝑖 ¯𝑢† 𝑗 � = 0 ⇒ ¯𝑆𝑎 𝑖 𝑗 = 𝜆 � ¯𝑢† 𝑖 ¯𝑢† 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='30) 𝜕L 𝜕𝜆 = ¯𝑢† 𝑖 � ¯𝑆𝑖 𝑗 − ¯𝑆𝑎 𝑖 𝑗 � ¯𝑢† 𝑗 = 0 ⇒ ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 = ¯𝑢† 𝑖 ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='31) By multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='30 by ¯𝑢† 𝑖 from the left and right by ¯𝑢† 𝑗 , and then substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='31, the Lagrangian multiplier 𝜆 is calculated by 𝜆 = ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � ¯𝑢† 𝑘 ¯𝑢† 𝑘 �2 = ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 4E †2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Initial SGS parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='SGS parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='SGS model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Forward LES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='equations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='LES statistics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Loss function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Stop criterion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Adjoint SGS model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Stop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='L-BFGS gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Gradient Calculations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Stop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Reference statistics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Initial velocity field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='L-BFGS gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Terminal condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Loss sensitivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Backward adjoint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='LES equations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Backward adjoint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='LES equations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Stop criterion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 1: Schematic diagram of the adjoint-based variational optimal mixed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The minimal artificial symmetric tensor ¯𝑆𝑎 𝑖 𝑗 can be obtained by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='32 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='30, yielding ¯𝑆𝑎 𝑖 𝑗 = ��� ��� ¯𝑢† 𝑚 ¯𝑆𝑚𝑛 ¯𝑢† 𝑛 4E †2 ¯𝑢† 𝑖 ¯𝑢† 𝑗, if ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 < 0, 0 , if ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='33) The artificial momentum term ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗 is thus additionally calculated in the stabilized adjoint momentum equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26), namely ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗 = �� �� ¯𝑢† 𝑚 ¯𝑆𝑚𝑛 ¯𝑢† 𝑛 2E † ¯𝑢† 𝑖 , if ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 < 0, 0 , if ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='34) The minimal stabilization term ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗 can efficiently maintain the numerical stability of LES adjoint variables in the long-term chaotic turbulent calculations as much as possible, without deteriorating the correct evaluations of the adjoint-based gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Adjoint-based variational optimal mixed models (VOMM) In this research, we select the mixed model comprised of the Smagorinsky dissipative term (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) and approximate deconvolution model (ADM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6) in the scale-similarity form, expressed as (Sagaut 2006) 𝜏𝑖 𝑗 = 𝐶1 � ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗 � + 𝐶2 � 𝑢∗ 𝑖 𝑢∗ 𝑗 − 𝑢∗ 𝑖 𝑢∗ 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='35) where 𝑢∗ 𝑖 denotes the approximate unfiltered velocity recovered by the iterative van Cittert procedure (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In previous studies (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020), we have conducted error analyses to validate that deconvolutional-type SGS models with scale-similarity form (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) perform better than those with theconventionaldirect-modelingform(𝜏𝑖 𝑗 = 𝑢∗ 𝑖 𝑢∗ 𝑗− ¯𝑢𝑖 ¯𝑢 𝑗),satisfyingtheproperties of symmetry and realizability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The model coefficients 𝐶1 and 𝐶2 are optimally identified by minimizing the discrepancy between statistical quantities calculated by the LES results and those measured by the filtered DNS (fDNS) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The selected statistics should be able to sufficiently quantify the multiscale transport behaviours of turbulence, meanwhile facilitating the practical measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS stress 𝜏𝑖 𝑗 and SGS force 𝜕𝜏𝑖 𝑗/𝜕𝑥 𝑗 are intermediate variables, 15 𝑅𝑒𝜆 𝐸𝑘 𝑘max𝜂 𝜂/ℎDNS 𝐿𝐼 /𝜂 𝜆/𝜂 𝑢rms 𝜔rms 𝜀 252 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='01 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='30 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='73 Table 1: One-point statistics for the DNS of forced homogeneous isotropic turbulence with grid resolution of 10243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and their statistics are relatively difficult to be obtained through the actual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the statistics of velocity are more convenient to measure and the velocity spectrum clearly quantifies the turbulent kinetic energy distributions at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS modeling is especially concerned with the accurate reconstruction of small scales near the filter width, therefore we select the dissipation spectrum as the optimization statistical quantities 𝜙 ( ¯𝑢𝑖) to increase the weights of small scales, namely (Pope 2000) 𝜙 ( ¯𝑢𝑖, 𝑘, 𝑡) = 𝐷 (𝑘, 𝑡) = ∫ k 𝜈𝑘2 ¯𝑣∗ 𝑖 (k, 𝑡) ¯𝑣𝑖 (k, 𝑡) 𝛿 (|k| − 𝑘) 𝑑k, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='36) where 𝛿 (·) denotes the Dirac delta function and the star symbol represents complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑘 and k stand for the wavenumber magnitude and wavenumber vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, ¯𝑣 𝑗 (𝜅, 𝑡) = F � ¯𝑢 𝑗 (x, 𝑡) � = � k ¯𝑢 𝑗 (x, 𝑡) 𝑒−𝑖k·x is the 𝑗-th velocity component in Fourier space, where F {·} represents the 3D Fourier transform, and 𝑖 is the imaginary unit with 𝑖2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The optimization problem constrained by the governing equations for the SGS parameters 𝐶1 and 𝐶2 is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2, where the cost functional for the dissipation spectrum 𝐷 (𝑘, 𝑡) is given by J � 𝜙, 𝜙fDNS� = 𝑇 ∫ 0 𝑘max ∑︁ 𝑘=1 𝐽 � 𝐷 (𝑘, 𝑡) , 𝐷fDNS (𝑘, 𝑡) � 𝑑𝑡, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='37) where 𝑘max = 𝑁LES/3 is the effective maximum wavenumber, 𝑁LES is the number of LES grids, and the discrepancy function 𝐽 � 𝐷 (𝑘, 𝑡) , 𝐷fDNS (𝑘, 𝑡) � = � 𝐷 (𝑘, 𝑡) − 𝐷fDNS (𝑘, 𝑡) �2 takes the L2 norm of the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The gradients of the loss function with respect to the model coefficients 𝐶1 and 𝐶2 are evaluated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='14, where the adjoint variables ¯𝑢† 𝑖 are calculated by backward advancing the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26) with zero terminal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The sensitivity term 𝜕𝐽/𝜕 ¯𝑢𝑖 is calculated by the chain rule, namely 𝜕𝐽 𝜕 ¯𝑢𝑖 = 𝜕𝐽 𝜕𝐷 𝜕𝐷 𝜕 ¯𝑢𝑖 = 2 � 𝐷 − 𝐷fDNS� F−1 � 2𝜈𝑘2 ¯𝑣𝑖 (k, 𝑡) 𝛿 (|k| − 𝑘) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='38) where F−1 {·} denotes the 3D inverse Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In the stabilized adjoint momentum equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26), the adjoint SGS stress is given by 𝜏† 𝑖 𝑗 = 𝐶1𝑇 (1),† 𝑖 𝑗 + 𝐶2𝑇 (2),† 𝑖 𝑗 , where the associated adjoint basis stress tensors 𝑇 (1),† 𝑖 𝑗 and 𝑇 (2),† 𝑖 𝑗 are expressed in detail as 𝑇 (1),† 𝑖 𝑗 = − ¯Δ2 � �� ¯𝑆 �� ¯𝑆† 𝑖 𝑗 + 2 ¯𝑆𝑘𝑙 ¯𝑆† 𝑘𝑙 �� ¯𝑆 �� ¯𝑆𝑖 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='39) 𝑇 (2),† 𝑖 𝑗 = 𝑁 ∑︁ 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='40) 16 (𝑎) �� � �� � �� � �� � � �� �� �� �� �� �� �� �� �� � �� � ���� � ���� � � � ��� DNS ��� ���� � � � �� (𝑏) �� � �� � �� � �� � � �� �� �� �� �� �� �� �� �� �� �� �� ���� � ���� � � � ��� DNS ��� ���� � � � �� Figure 2: Velocity and dissipation spectra of DNS and filtered DNS in forced homogeneous isotropic turbulence with grid resolution of 10243: (𝑎) velocity spectra, and (𝑏) dissipation spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Diamond represent the cutoff wavenumber 𝑘𝑐=16 ( ¯Δ = 32ℎDNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' where 𝑁 = 5 denotes the number of iterations for the AD procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The detailed derivation of the adjoint SGS stress tensors for the VOMM model can refer to the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To our knowledge, few previous works have studied the mixed SGS models and given the detailed derivations of the adjoint SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Once the gradients of the cost functional for the model coefficients are obtained by successively solving the forward LES equations and backward stabilized adjoint LES equations, a gradient- based iterative optimization procedure can be established, namely (Liu & Nocedal 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Badreddine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2014) 𝐶 (𝑘+1) 𝑛 = 𝐶 (𝑘) 𝑛 + 𝛾(𝑘)𝑑 (𝑘) 𝑛 , (𝑛 = 1, 2, · · · , 𝑁) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) where 𝐶 (𝑘) 𝑛 is the 𝑛-th model coefficient during the 𝑘-th gradient-based optimal iteration, 𝑑 (𝑘) 𝑛 denotes the updated direction of the 𝑛-th model coefficient and 𝛾(𝑘) represents the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We use a popular quasi-Newton method named limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm to update the directions 𝑑 (𝑘) 𝑛 (Liu & Nocedal 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The step size 𝛾(𝑘) is calculated by the backtracking-Armijo line search method in the L-BFGS algorithm (Armijo 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In summary, the diagram of the VOMM model is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1, and the calculation steps are listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (1) We first select the pure structural ADM model without the dissipative Smagorinsky term as the initial SGS model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='35) with model coefficients 𝐶 (0) 1 = 0 and 𝐶 (0) 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2) The LES transient statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the dissipation spectrum shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='36) is then evaluated by forward calculating the LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) initialized by the filtered DNS velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The statistical discrepancy (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='37) between the LES statistics and the a priori measurable benchmark data (fDNS data) is measured to evaluate the performance of the SGS model with current parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (3) Afterwards, the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26) are integrated back- ward with zero terminal conditions, driven by the loss sensitivity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='38) and corresponding adjoint SGS model (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='39 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradients of augmented functional with respect to the model coefficients (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='14) are sequentially evaluated using the adjoint variables and the SGS basis forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4) The L-BFGS gradient-based optimization algorithm (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) is adopted to iteratively update the SGS model parameters by repeating the above calculations until the stopping criteria are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 17 � �� �� �� �� �� �� �� �� �� ��� ���������� � ��� ��� ��� ��� ��� ��� ��� ��� ��� � J�J0 � � � ��� DNS ��� � �� � � �� � ��� � �� � � �� � ��� � �� � � ��� � Figure 3: The evolution of the normalized cost function in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' FGR LES Resolution 𝐶 (0) 1 𝐶 (0) 2 𝐶opt 1 𝐶opt 2 1 323 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0529 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='229 2 643 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='027 4 1283 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0030 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='000 Table 2: The initial and optimal parameters of the VOMM model for LES computations with the filter width ¯Δ = 32ℎDNS in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The stop criteria for the VOMM model for the optimization iterations are summarized as follows: (a) the number of iterations reaches the maximum number of iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) the ratio of the current loss to the initial loss is smaller than a given error threshold 𝜖0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝜖0 = 1%) , namely, J (𝑘)/J (0) ⩽ 𝜖0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) the difference of model coefficients between two successive iterations is negligible, namely, ���𝐶 (𝑘+1) 𝑛 − 𝐶 (𝑘) 𝑛 ��� / ���𝐶 (0) 𝑛 ��� ⩽ 𝜖0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Eventually, the optimal parameters of the VOMM model are automatically obtained after reaching the given stopping optimization criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A posteriori studies of the VOMM models In order to examine the performance of the proposed VOMM model, the a posteriori evaluations are respectively carried out for the forced, decaying homogeneous isotropic turbulence and temporally evolving turbulent mixing layer in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The results of the filtered direct numerical simulation (DNS) are the benchmark for the performance evaluations of the large-eddy simulation (LES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We first introduce the detailed settings of DNS for these three turbulent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The DNS data are then explicitly filtered by the commonly-used Gaussian filter, which is expressed 18 Model(FGR=1,𝑁 = 323) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='066 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='584 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='273 Model(FGR=2,𝑁 = 643) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='870 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='418 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='594 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='285 Model(FGR=4,𝑁 = 1283) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='512 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='517 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='588 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='240 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='645 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='321 Table 3: The average computational cost of SGS stress modeling 𝜏𝑖 𝑗 for LES computations with the filter width ¯Δ = 32ℎDNS in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' as (Pope 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sagaut 2006) 𝐺 �r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� = � 6 𝜋 ¯Δ2 �1/2 exp � −6r2 ¯Δ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) The filter scale ¯Δ = 32ℎDNS is selected for both the forced and decaying homogeneous isotropic turbulence, while ¯Δ = 8ℎDNS for the temporally evolving turbulent mixing layer, where ℎDNS denotes the grid spacing of DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Three conventional SGS models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', the dynamic Smagorinsky model (DSM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1), the dynamic mixed model (DMM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) and the approximate deconvolution model with standard secondary filtering regularization (ADM, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8) are adopted to compare against the VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The consistent instantaneous snapshots of the filtered DNS data are used to initialize the LES calculations for different SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Both the turbulent statistics and transient contours are evaluated and compared with different SGS models for the a posteriori testings of the three canonical turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Forced homogeneous isotropic turbulence We perform the direct numerical simulation of forced incompressible isotropic turbulence using the uniform grid resolution 𝑁 = 10243 in a cubic box of (2𝜋)3 with periodic boundary conditions (ℎDNS = 2𝜋/1024) (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020a,c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pseudo-spectral method is used for the spatial discretization of the governing equations (Canuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Peyret 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The nonlinear advection terms are fully dealiased by the two-thirds dealiasing rule (Canuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A second-order two-step Adams-Bashforth explicit scheme is used for time integration (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The kinematic viscosity is chosen as 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='001, and large-scale forcing is applied to the two lowest wavenumber shells to maintain the turbulence in statistical equilibrium, giving rise to the Taylor Reynolds number Re𝜆 ≈ 250 (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The detailed one-point statistics of DNS data for the forced isotropic turbulence are summarized in Table 1 (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝑘max = 2𝜋 3ℎDNS denotes the largest effective wavenumber after the fully dealiasing, and 𝜔rms = √︁ ⟨𝜔𝑖𝜔𝑖⟩ represents the root-mean-square value of the vorticity magnitude, where 𝜔 = ∇ × u stands for the vorticity which is the curl of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Kolmogorov length ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='scale 𝜂 and the integral length scale 𝐿𝐼 stand for the smallest resolved scale and the largest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 4: Velocity spectra for different SGS models in the a posteriori analysis of forced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS: (a) log-log for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) semi-log for FGR=1, 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) log-log for FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (d) semi-log for FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (e) log-log for FGR=4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 1283;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (f) semi-log for FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' characteristic scale of turbulence, and are defined respectively by 𝜂 = � 𝜈3 𝜀 �1/4 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) 𝐿𝐼 = 3𝜋 2(𝑢rms)2 ∫ +∞ 0 𝐸 (𝑘) 𝑘 𝑑𝑘, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) where 𝜀 is the spatial average dissipation rate of kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The total turbulent kinetic energy 𝐸𝑘 = ⟨𝑢𝑖𝑢𝑖⟩ /2 = ∫ +∞ 0 𝐸 (𝑘) 𝑑𝑘, and 𝐸 (𝑘) represents the velocity spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The resolution parameters 𝑘max𝜂 ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1 and 𝜂/ℎDNS ⩾ 1 indicate that the grid resolution is sufficient to capture ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 5: Second-order structure functions of the filtered velocity for LES in the a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='posteriori analysis of forced homogeneous isotropic turbulence with the same filter scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the smallest turbulent eddy scales and ensure the convergence of turbulent kinetic energy at all scales (Ishihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2007, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to alleviate the impact of initial conditions, the forced homogeneous isotropic turbulence is run for a long period after the flow gradually reaches a statistically steady state (more than 50 large-eddy turnover times 𝜏 = 𝐿𝐼 /𝑢rms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We select data of the last ten large-eddy turnover times as a benchmark for LES comparisons (total forty flow-field snapshots of DNS data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In this paper, the Gaussian filter (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) is used as the explicit filter to calculate the filtered physical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Theselected filter width ¯Δ = 32ℎDNS and the correspondingcutoff wavenumber is 𝑘𝑐 = 𝜋/ ¯Δ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The velocity and dissipation spectra of the DNS and filtered DNS at ¯Δ = 32ℎDNS are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The filtered velocity spectrum nearly overlaps with the DNS data in a Kolmogorov scaling law of 𝑘−5/3 at the low wavenumber region, while it drops significantly at the region larger than the truncated wavenumber 𝑘𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Overall 12% of the turbulent kinetic energy is filtered out in the residual velocity field at the filter scale ¯Δ = 32ℎDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the filtered dissipation spectrum gradually grows with the power of law scaling 𝑘1/3 at the low-wavenumber inertial region, and drops sharply where the cutoff wavenumber exceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The small scales near the truncated wavenumbers are essential for the reconstruction of the filtered dissipation spectrum and also very important for the residual SGS modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, these small scales account for a very small proportion of the turbulent kinetic energy, almost several orders of magnitude smaller than the large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Thus, the dissipation spectrum instead of the kinetic energy spectrum is chosen as the optimization objective function of the proposed VOMM model in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori testings of LES are essential to validate the practical performance of the SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' LES calculations use the same kinematic viscosity (𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='001) with the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The filter width is fixed to ¯Δ = 32ℎDNS and the impact of the spatial discretization errors on the SGS models is investigated by changing the grid resolution of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Three different filter-to-grid ratios FGR= ¯Δ/ℎLES=1, 2 and 4 are chosen to study the influence of spatial discretization on the SGS modeling, and the corresponding grid points of LES are 𝑁 = 323, 643 and 1283, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The proposed VOMM model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='35) is compared against the classical SGS models, including the dynamic Smagorinsky model (DSM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1), the dynamic mixed model (DMM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) and the standard approximate deconvolution model with secondary filtering regularization (ADM, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The relaxation factors of ADM model 𝜒=0 and 1 are chosen for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ratios of the time steps for LES and DNS are Δ𝑡LES/Δ𝑡DNS = {10, 10, 5} for different grids (FGR=1, 2 and 4 with 𝑁 = 323, 643 and 1283).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Among the filtered DNS data of the ten large- eddy turnover periods, the data of the first two large-eddy turnover times are used for the adjoint optimization of the VOMM model (only the dissipation spectrum is used, stored once every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1𝜏, twenty sets in total), and the remaining data of the last eight large-eddy turnover times are used for the a posteriori accuracy validation of the LES models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 6: Fourth-order structure functions of the filtered velocity for LES in the a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='posteriori analysis of forced homogeneous isotropic turbulence with the same filter scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' At the adjoint-based optimization stage of the VOMM model, the calculations of the adjoint equations are consistent with the primary LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We adopt the same pseudo-spectral numerical scheme to spatially discrete the stabilized adjoint momentum equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A second-order two-step Adams-Bashforth explicit scheme is applied for the time backward integration with zero terminal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Since the large-scale forcing is assumed to be nearly independent of the filtered velocity, the large-scale forcing term does not appear in the adjoint momentum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' During the adjoint optimization stage (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1) of the VOMM model, the pure structural ADM model without the dissipative Smagorinsky term is selected as the initial SGS model with model coefficients 𝐶 (0) 1 = 0 and 𝐶 (0) 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The LES forward evolution is initialized by the filtered DNS velocity field and the dissipation spectrum is calculated when the filtered DNS data are available (every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The statistical discrepancy of the dissipation spectrum between the LES and fDNS data is evaluated and recorded as the cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradients of the cost functional with respect to the model coefficients are calculated through backward integrating the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26) with zero terminal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS model coefficients are then iteratively updated by the gradient-based L-BFGS optimization algorithm (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) until reaching the stopping criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figure 3 shows the evolution of the cost function normalized by the initial discrepancy during the adjoint-based optimization in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The loss functions (prediction errors of dissipation spectra between LES and fDNS data) for all three different filter-to-grid ratio cases (FGR=1,2 and 4) gradually converge and become stationary within less than twenty iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The error is significantly reduced by nearly an order of magnitude for the cases of FGR=1 and 2 within about ten iterations, and is drastically reduced to 20% of the initial state at FGR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results indicate that the adjoint-based L-BFGS gradient optimization is very efficient and effectively obtains the optimal model coefficients within several iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The optimal parameters of the VOMM model are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The magnitude of the eddy- viscosity coefficient (( ���𝐶opt 1 ���) ) dramatically reduces from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0529 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='003 with the increasing of FGR and LES resolutions, while the coefficient of the ADM part (𝐶opt 2 ) gradually approaches unity, which is identical to the theoretical value derived from the Taylor series expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Once the optimal model coefficients are obtained, we further examine the a posteriori performance of the VOMM model using the filtered DNS data of the last eight large-eddy turnover periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Table 3 gives the average computational cost for the SGS stress modeling at the same filter width ¯Δ = 32ℎDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For all three different grid resolutions, the computation time of the VOMM model is only about 30% of that of the DMM model, without significantly increasing the computational cost in comparison to the ADM models (𝜒 = 0 and 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='The velocity spectra of different SGS models with the filter scale ¯Δ = 32ℎDNS in comparison to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑏) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 7: Sixth-order structure functions of the filtered velocity for LES in the a posteriori ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='analysis of forced homogeneous isotropic turbulence with the same filter scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' those of the DNS and filtered DNS (fDNS) data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The velocity spectrum of DNS data exhibits a sufficiently long inertial range with a typical 𝑘−5/3 scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The spectrum of fDNS almost overlaps with that of DNS at the low-wavenumber region, but is obviously lower than that of DNS near the truncated wavenumber since the small-scale kinetic energy at high wavenumbers is filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' LES only solves the large-scale variables with the filtered Navier-Stokes equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5), leaving the effect of residual small scales to be approximately reconstructed by the SGS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Therefore the statistics of an ideal LES would overlap with that of the fDNS data as closely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' When the grid resolution of LES is sufficiently coarse and the grid spacing of LES is equal to the filter scale (FGR=1, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figs 4a and 4b), the spatial discretization error is significant and deteriorates the accuracy of the SGS stress modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' LES calculations with traditional SGS models are very difficult to obtain accurate predictions of the turbulent kinetic energy cascade at FGR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The velocity spectra predicted by the ADM models with 𝜒 = 0 and 1 exhibit numerical unstable, and kinetic energy at high wavenumbers is obviously overestimated due to the insufficient dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DSM and DMM models also have dramatic overestimations at high-wavenumber regions, with predictions even larger than that of the DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, VOMM model predicts the velocity spectra most accurately among these SGS models whose results nearly coincide with that of fDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the cases of fine grid resolutions (FGR=2 and 4), the pure ADM model (𝜒 = 0) is still numerically unstable since the pure structural model itself cannot produce sufficient SGS dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ADM model with the standard secondary-filtering regularization (𝜒 = 1) exhibits excessively dissipative, and the small-scale kinetic energy at high wavenumbers is extremely exhausted and much lower than that of fDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The predictions of DSM and DMM models illustrate the obviously tilted distribution, where kinetic energy at low wavenumbers is accumulated, while that near the truncated wavenumber is diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dynamic least-square procedure for both DSM and DMM models would overestimate the eddy-viscosity coefficient for the cases of fine grid resolutions (FGR=2 and 4), and small-scale flow structures near the truncated wavenumbers are exhausted by the excessive dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The turbulent kinetic energy is transferred from large scales to small scales through the forward energy cascade process of the nonlinear advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The lack of the sufficient flow structures near the cutoff wavenumber leads to the energy accumulation in the intermediate wavenumber region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model is superior to the other SGS models and can accurately predict the velocity spectra at all different grid resolutions of LES, with the predictions very close to the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to further examine the reconstruction of multiscale properties of turbulence by the SGS models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' we calculate the longitudinal structure functions of the filtered velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' namely ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' �� �� � � � �� �� �� �� �� �� �� �� �� �� � �� � ��� � � �� � � � � � ��� DNS ��� � � ���� ��� ��� ����� � �� ����� � �� ���� (𝑑) �� �� �� �� �� � � � � � � � � � � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' �� �� � � � �� �� �� �� �� �� �� �� �� �� � �� � ��� � � �� � � � � � ��� DNS ��� � � ���� ��� ��� ����� � �� ����� � �� ���� Figure 8: PDFs of the normalized velocity increments 𝛿r ¯𝑢/ ¯𝑢rms for LES at grid resolution of 323 in the a posteriori analysis of forced homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS: (a) r = ¯Δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) r = 2 ¯Δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) r = 3 ¯Δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (d) r = 4 ¯Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2018, 2019a) ¯𝑆𝑛(𝑟) = ����� 𝛿𝑟 ¯𝑢 ¯𝑢rms ���� 𝑛� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4) where 𝑛 represents the order of structure function and 𝛿𝑟 ¯𝑢 = [¯u (x + r) − ¯u (x)] · ˆr denotes the longitudinal velocity increment at the separation r with the unit distance vector ˆr = r/|r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figures 5, 6 and 7 respectively compare the second-order, fourth-order and sixth-order structure functions of the filtered velocity for different SGS models with the filtered DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For all three grid resolutions of LES (FGR=1, 2 and 4), all SGS models predict the lower-order structure functions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5) much better than the higher-order structure functions (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Besides, the predictions of structure functions are improved greatly with the increasing of the grid resolution, and those of all SGS models almost coincide with each other at large separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ADM models (both 𝜒 = 0 and 1) give the worst predictions and obviously overestimate the structure function at small distances r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DSM and DMM models also predict the structure functions greater than the fDNS data at small separations but underestimate the structure functions at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model can accurately reconstruct the structure functions with different orders at both small and large separations, almost overlapping with those of the filtered DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We then evaluate the probability density functions (PDFs) of the filtered velocity increments to measure the spatial correlations of turbulence, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 8, where the velocity increments 𝛿𝑟 ¯𝑢/ ¯𝑢rms are normalized by the root-mean-square value of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The cases of fine grid resolutions (FGR=2 and 4) are very similar to that of FGR=1 and not shown in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The PDFs of the velocity increments exhibit approximately symmetrical distribution, relatively 24 (a) fDNS (b) DMM (c) ADM (𝜒=0) (d) ADM (𝜒=1) (e) DSM (f) VOMM Figure 9: Contours of the normalized vorticity ¯𝜔/ ¯𝜔rms fDNS at an arbitrary 𝑥1-𝑥2 plane at 𝑡/𝜏 ≈ 4 for LES at a grid resolution of 643 (FGR=2) in forced homogeneous isotropic turbulence with the filter width ¯Δ = 32ℎDNS: (a) fDNS, (b) DMM, (c) ADM(𝜒=0), (d) ADM(𝜒=1), (e) DMM, and (f) VOMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' rms 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 C1/2πrms 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 C1/2π25 � �� �� �� �� �� �� �� �� �� ��� ���������� � ��� ��� ��� ��� ��� ��� ��� ��� ��� � J�J0 � � � ��� DNS ��� � �� � � �� � ��� � �� � � �� � ��� � �� � � ��� � Figure 10: The evolution of the normalized cost function in decaying homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' FGR LES Resolution 𝐶 (0) 1 𝐶 (0) 2 𝐶opt 1 𝐶opt 2 1 323 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0398 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='150 2 643 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0094 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='326 4 1283 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='101 Table 4: The initial and optimal parameters of the VOMM model for LES computations with the filter width ¯Δ = 32ℎDNS in decaying homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' concentrated at small distances while gradually becoming wider as the distance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The PDFs predicted by the ADM, DSM and DMM models are significantly wider than those of the fDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In comparison with these traditional SGS models, the VOMM model gives the most accurate prediction of the velocity increments for different distances, which are in reasonable agreement with the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We finally examine the reconstruction of instantaneous spatial flow structures by plotting the contours of the normalized vorticity magnitude as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The vorticity contours are consistently extracted on an arbitrary 𝑥1-𝑥2 plane for the isotropic turbulence at the same time with approximately four large-eddy turnover periods ( 𝑡/𝜏 ≈ 4) at a grid resolution of 643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It is noteworthy that the exact point-to-point correlations are difficult to achieve under the long- term forecasting of LES due to the chaotic nature of the turbulence and extreme sensitivity to perturbations (Pope 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022c, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model overpredicts some unrealistic small-scale structures, which are obviously different from the band-like or strip-like spatial structures of the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The DSM, DMM and ADM (𝜒 = 1) models only predict the large-scale vorticity structures and some small scales are excessively dissipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to these traditional SGS models, the VOMM model predicts the vortex structures very similar to the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 26 Model(FGR=1,𝑁 = 323) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='070 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='590 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='269 Model(FGR=2,𝑁 = 643) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='026 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='589 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='553 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='317 Model(FGR=4,𝑁 = 1283) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='026 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='287 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='521 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='531 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='393 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='586 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='330 Table 5: The average computational cost of SGS stress modeling 𝜏𝑖 𝑗 for LES computations with the filter width ¯Δ = 32ℎDNS in decaying homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 11: Temporal evolutions of the turbulent kinetic energy 𝐸𝑘 for LES in the a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='posteriori analysis of decaying homogeneous isotropic turbulence with the same filter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='scale ¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Decaying homogeneous isotropic turbulence In order to investigate the impact of turbulent unsteady evolution on SGS stress modeling, the numerical simulation of decaying homogeneous isotropic turbulence in a cubic box of (2𝜋)3 with periodic boundary conditions is investigated in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The numerical simulation method is consistent with the forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We spatially discretize the governing equations using the pseudo-spectral method with the two-thirds dealiasing rule at a uniform grid resolution of 𝑁 = 10243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The temporal discretization scheme adopts the second- order two-step Adams-Bashforth explicit method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The statistically steady isotropic turbulence data of the forced isotropic turbulence (detailed statistics see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1) is used as the initial field for DNS decaying turbulence without the large-scale forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The kinematic viscosity is set to 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='001 and the initial Taylor Reynolds number is Re𝜆 ≈ 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We calculate the DNS data of decaying turbulence for about six large-eddy turnover times (𝜏 = 𝐿𝐼 /𝑢rms), the first two of which are used for the adjoint-based optimization to determine the model coefficients of VOMM model (only the dissipation spectrum is used, stored once every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1𝜏, twenty sets in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori studies of LES adopt the consistent kinematic viscosity (𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='001) with the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Gaussian filter (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) is selected as the explicit filter with the given filter width ¯Δ = 32ℎDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Similar to the forced isotropic turbulence, three different filter-to-grid ratios FGR= ¯Δ/ℎLES=1,2 and 4 are chosen to investigate the impact of the spatial discretization on the SGS stress modeling with the corresponding grid resolutions of LES 𝑁 = 323, 643 and 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 12: Temporal evolutions of the average dissipation rate ¯𝜀 for LES in the a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='posteriori analysis of decaying homogeneous isotropic turbulence with the same filter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='scale ¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' adjoint-based optimization of the VOMM model (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1) is first performed to determine the optimal model coefficients using the dissipation spectra as the cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model without the Smagorinsky part is used as the initial SGS model with parameters 𝐶 (0) 1 = 0 and 𝐶 (0) 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradients of the cost functional for the model coefficients are evaluated by successively forward solving the LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) and backward integrating the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The gradient-based L-BFGS optimization algorithm (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) is used for iteratively updating the SGS model parameters until reaching the stopping criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The evolution of the cost function normalized by the initial loss during the adjoint-based optimization for the decaying isotropic turbulence is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The loss functions for all three cases of different grid resolutions (FGR=1,2 and 4) drop rapidly at the beginning and gradually reach a plateau within approximately twenty iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The prediction errors of the optimization objective are considerably reduced to 10% of the initial state for both FGR=1 and 2, and substantially decreased to about 20% of the original value at FGR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradient optimization can quickly obtain the optimal model parameters within a limited number of iterations (less than 100 optimization iterations, namely, 200 LES evaluations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Table 4 gives the optimal parameters of the VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The magnitude of the dissipative Smagorinsky coefficient ( ���𝐶opt 1 ���) significantly drops from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0398 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='002 as the LES resolution increases, which is slightly lower than that in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the coefficient of the structural part (𝐶opt 2 ) is asymptotically close to unity as the grid spacing of LES becomes smaller, similar to the results of forced isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori performance of the VOMM model is further validated after determining the optimal SGS model coefficients by the adjoint-based gradient optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We compare the proposed VOMM model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='35) with the classical SGS models including the DSM model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1), DMM model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) and the ADM model regularized by the standard secondary- filtering technique (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The time steps of LES are given as Δ𝑡LES/Δ𝑡DNS = {10, 10, 5} for different grid resolutions (FGR=1, 2 and 4 with 𝑁 = 323, 643 and 1283).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The average computational costs for the SGS stress modeling with different grid resolutions using different SGS models at the same filter scale ¯Δ = 32ℎDNS are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The computation time of the VOMM model only accounts for approximately 30% of the time of DMM model and slightly increases in computational cost compared to the ADM models with 𝜒 = 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figures 11 and 12 respectively compare the temporal evolutions of the turbulent kinetic energy and the resolved dissipation rate ( ¯𝜀 = 2𝜈 � ¯𝑆𝑖 𝑗 ¯𝑆𝑖 𝑗 � ) of different SGS models with the filtered DNS (fDNS) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The turbulent kinetic energy gradually decays from the initial statistically steady state over time, since there are no additional forcing driving the dissipative turbulent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' All the classical SGS models (DSM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DMM and ADM models) clearly overestimate the kinetic energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� � ��� ��� ���� ��� ��� ����� � �� ����� � �� ���� ( 𝑓 ) �� � �� � �� � �� �� �� �� �� �� �� �� �� �� �� � �� � � � ��� � � � � � ��� DNS ��� � �� !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� � ��� ��� ���� ��� ��� ����� � �� ����� � �� ���� Figure 13: Velocity spectra for different SGS models in the a posteriori analysis of decaying homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS at 𝑡/𝜏 ≈ 2 and 4: (a) FGR=1, 𝑁 = 323 at 𝑡/𝜏 ≈ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=1, 𝑁 = 323 at 𝑡/𝜏 ≈ 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) FGR=2, 𝑁 = 643 at 𝑡/𝜏 ≈ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (d) FGR=2, 𝑁 = 643 at 𝑡/𝜏 ≈ 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (e) FGR=4, 𝑁 = 1283 at 𝑡/𝜏 ≈ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (f) FGR=4, 𝑁 = 1283 at 𝑡/𝜏 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' throughout the time, which differs significantly from the benchmark fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model gives reasonable predictions of the turbulent kinetic energy, which is the closest to the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The average dissipation rate displays a decline trend with time, similar to that of the turbulent kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, all conventional SGS models wrongly predict the non- monotonic tendency of the average dissipation rate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the case of sufficiently coarse grid resolution of LES (FGR=1 with 𝑁 = 323), DSM, DMM and ADM models overpredict the dissipation rate with an erroneous temporal evolution that first increases and then decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' When the grid resolution of LES becomes fine (FGR=2 and 4 with 𝑁 = 643 and 1283), DSM and DMM models obviously underestimate the dissipative rate at the early stage of decaying turbulence (𝑡/𝜏 ⩽ 3), then DMM model gradually becomes closer to the fDNS data while DSM model 29 (𝑎) � ��� � ��� � ��� � � �� � � #"$ ��� � ��� ��� ��� ��� ��� ��� ��� � � �� � � � � � ��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� � �� %�� � ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) � ��� � ��� � ��� � � �� � � #"$ ��� � ��� ��� ��� ��� ��� ��� ��� � � �� � � � � � ��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� � �� %�� � ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑐) � ��� � ��� � ��� � � �� � � #"$ ��� � ��� ��� ��� ��� ��� ��� ��� � � ��� � � � � � ��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� � �� %�� � ��� ��� ��� ��� ����� � �� ����� � �� ���� Figure 14: PDFs of the normalized vorticity ¯𝜔/ ¯𝜔rms fDNS for LES in the a posteriori analysis of decaying homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS at 𝑡/𝜏 ≈ 4: (a) FGR=1, 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' overestimates the dissipation rate with the decaying of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model ( 𝜒 = 0 ) always gives the overestimations of the dissipation rate for all three different grid resolutions of LES, even though the pure ADM model can accurately predict the turbulent kinetic energy at a sufficiently high grid resolution (FGR=4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results demonstrate that the pure structural ADM model without any dissipative terms might not accurately predict all physical quantities of LES (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', the average dissipation rate), even if the grid resolution is high enough compared to the filter scale (FGR=4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ADM model with standard secondary-filtering regularization (𝜒 = 1) provides excessive dissipation similar to the DSM model with mispredictions of first underestimating and then overestimating the average dissipation rate over time at FGR=2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In comparison to these classical SGS models, the VOMM model accurately predicts the temporal evolutions of average dissipation rate for all three different grid resolutions, which agrees fairly well with the benchmark filtered DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The transient velocity spectra of different SGS models at the filter width ¯Δ = 32ℎDNS with two different time instants 𝑡/𝜏 ≈ 2 and 4 are further illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The velocity spectra exhibit an overall decrease, and the kinetic energy at all wavenumbers declines with the decaying of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' All the classical SGS models (DSM, DMM and ADM models) overpredict the kinetic energy at high wavenumbers for the coarse grid-resolution case (FGR=1 with 𝑁 = 323 ) and the excessive kinetic energy stacked at small scales leads to the numerical instability of LES, which gradually intensifies with the evolution of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The conventional SGS models provide insufficient model dissipation to balance the discretization errors and the small-scale kinetic energy cannot be effectively dissipated in time at FGR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the fine grid-resolution cases (FGR=2 and 4 with 𝑁 = 643 and 1283), the dissipation of the traditional SGS models (DSM, DMM models, and ADM model with 𝜒 = 1) is too strong to diminish most small-scale flow structures near the truncated wavenumber, which hinders the normal transmission of turbulent kinetic energy cascades from large scales to small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Therefore, the kinetic energy of classical SGS models accumulates in the region of intermediate wavenumbers, leading to the overestimations of the turbulent kinetic energy with time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 11) at FGR=2 and 4 with 𝑁 = 643 and 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' LES using the pure ADM model with 𝜒 = 0 is always numerically unstable and lacks necessary SGS dissipation to drain out the small-scale kinetic energy for all different grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to these classical SGS models, the VOMM model can accurately reconstruct the kinetic energy cascade with the predictions that nearly coincide with those of fDNS at all three different grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Furthermore, we compare the PDFs of the normalized vorticity magnitude at the dimensionless time 𝑡/𝜏 ≈ 4 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The vorticity is normalized by the root-mean-square values of the vorticity calculated by the fDNS data for comparisons of different grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM models with 𝜒 = 0 gives the worst prediction of the vorticity with erroneous peaks 30 (a) fDNS (b) DMM (c) ADM (𝜒=0) (d) ADM (𝜒=1) (e) DSM (f) VOMM Figure 15: Contours of the normalized vorticity ¯𝜔/ ¯𝜔rms fDNS at an arbitrary 𝑥1-𝑥2 plane at 𝑡/𝜏 ≈ 4 for LES at a grid resolution of 643 (FGR=2) in decaying homogeneous isotropic turbulence with the filter width ¯Δ = 32ℎDNS: (a) fDNS, (b) DMM, (c) ADM(𝜒=0), (d) ADM(𝜒=1), (e) DMM, and (f) VOMM.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 C1/2πrms 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 C1/2π31 (𝑎) (𝑏) π Figure 16: Diagram of the temporally evolving mixing layer with the mean velocity profile: (a) schematic of the mixing layer, (b) mean streamwise velocity profile ⟨𝑢1⟩ along the normal (𝑥2) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' of PDFs significantly different from the fDNS data for all three grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The secondary filtering technique (𝜒 = 1) of the ADM model cannot improve the prediction of vorticity very well, whose estimations are still obviously different from the benchmark fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DSM and DMM models underestimate the PDF of vorticity and have wrong predictions of the PDF peak at the coarse grid-resolution case (FGR=1 with 𝑁 = 323 ), while greatly improving the predictions of PDFs with the increasing of the grid resolution (FGR=2 and 4 with 𝑁 = 643 and 1283).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model outperforms these classical SGS models at all three different grid resolutions, which gives a reasonably good prediction for both the locations and the peaks of the PDFs of the vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The reconstruction of transient spatial vorticity structures are finally demonstrated by the contours of the normalized vorticity magnitude shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The instantaneous snapshots are selected on an arbitrary 𝑥1-𝑥2 slice at the consistent time instant 𝑡/𝜏 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model predicts the excessive stochastic small-scale structures, which significantly differ from the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The other SGS models can predict the large-scale vorticity structures quite well, but the VOMM model reconstruct the spatial vortex structures very similar to the benchmark fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model can accurately recover more flow structures and the temporal evolution of the vortex with suitable SGS dissipation and accurate structural modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Temporally evolving turbulent mixing layer The turbulent mixing layer is one of the cardinal flows in the fluid-mechanics community, which is widely applied to the investigation of turbulent combustion, chemical reaction mixing process, and fundamental studies of flow instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The turbulent mixing layer involves the unsteady shear process of vortex shedding and transition from laminar to turbulent flows, which are remarkably suitable for investigating the impact of non-uniform turbulent shear and mixing on the SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The temporally evolving turbulent mixing layer characterized by the Kelvin–Helmholtz instability induced by the initial velocity difference is considered in this paper (Vreman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The free-shear mixing layer is governed by the same Navier-Stokes equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) without the forcing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figure 16 illustrates the diagram of the flow configuration for the temporally evolving turbulent mixing layer with the initial hyperbolic tangent streamwise velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The numerical simulation of mixing layer is performed in a cuboid domain with lengths 𝐿1×𝐿2×𝐿3 = 8𝜋×8𝜋×4𝜋 at the uniform grid resolution of 𝑁1 × 𝑁2 × 𝑁3 = 512 × 512 × 256 where 𝑥1 ∈ [−𝐿1/2, 𝐿1/2], 𝑥2 ∈ [−𝐿2/2, 𝐿2/2] and 𝑥3 ∈ [−𝐿3/2, 𝐿3/2] denote the streamwise, transverse and spanwise directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To enable a periodic configuration in the normal direction, the initial 32 𝑁1 × 𝑁2 × 𝑁3 𝐿1 × 𝐿2 × 𝐿3 𝜈∞ 𝑅𝑒𝜃 𝛿0 𝜃 Δ𝑈 Δ𝑑/ℎDNS ℎDNS Δ𝑡DNS 512 × 512 × 256 8𝜋 × 8𝜋 × 4𝜋 5 × 10−4 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='08 2 8 𝜋/64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='002 Table 6: Numerical parameters for the DNS of the temporally evolving mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' mean streamwise velocity (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 16b) is given by (Sharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022a) ⟨𝑢1⟩ = Δ𝑈 2 � tanh � 𝑥2 2𝛿0 𝜃 � − tanh � 𝑥2 + 𝐿2/2 2𝛿0 𝜃 � − tanh � 𝑥2 − 𝐿2/2 2𝛿0 𝜃 �� , for − 𝐿2 2 ⩽ 𝑥2 ⩽ 𝐿2 2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) where Δ𝑈 = 2 is the velocity difference between two equal and opposite free streams across the shear layer, 𝛿0 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='08 denotes the initial momentum thickness, and ⟨·⟩ stands for a spatial average over all the homogeneous directions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑥1 and 𝑥3 directions for the mixing layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The initial mean transverse and spanwise velocities are both set to zero, namely, ⟨𝑢2⟩ = ⟨𝑢3⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Since the initial mean velocity field is periodic in all three directions, the triply periodic boundary conditions are adopted and the pseudo-spectral method with the two-thirds dealiasing rule is used for the spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' An explicit two-step Adam-Bashforth scheme is selected as the time-advancing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to effectively suppress the influence of the top and bottom boundaries on the central mixing layer, two numerical diffusion buffer zones are applied near the vertical edges of domain (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The thickness of the buffer layer is set to 15𝛿0 𝜃 in the paper, which is sufficiently large and has a negligible effect on the calculations of mixing layer (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The digital filter method is used to generate the spatially-correlated initial perturbation imposed on the mean velocities with the digital filter width Δ𝑑 = ¯Δ = 8ℎDNS consistent to the filter scale of LES (Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The initial Reynolds stress distribution (𝑅𝑖 𝑗 = � 𝑢′ 𝑖𝑢′ 𝑗 � where 𝑢′ 𝑖 = 𝑢𝑖 − ⟨𝑢𝑖⟩ represents the fluctuated velocity) of the digital filter method is assumed as a vertical distribution of 𝑅𝑖 𝑗 = 𝐴 � 1 − ⟨𝑢1⟩2� 𝐼𝑖 𝑗 with the identity 𝐼𝑖 𝑗 and peak amplitude 𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='025Δ𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The kinematic viscosity of shear layer is set to 𝜈∞ = 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The momentum thickness quantifies the range of turbulence region in the mixing layer, which is defined by (Rogers & Moser 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019) 𝛿𝜃 = 𝐿2/4 ∫ −𝐿2/4 � 1 4 − � ⟨ ¯𝑢1⟩ Δ𝑈 �2� 𝑑𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6) Correspondingly, the Reynolds number based on the momentum thickness 𝑅𝑒𝜃 is expressed as 𝑅𝑒𝜃 = Δ𝑈𝛿𝜃 𝜈∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) Here, the initial momentum thickness Reynolds number is 𝑅𝑒0 𝜃 = 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The detailed numerical parameters of DNS for the temporally evolving mixing layer is summarized in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We calculate the DNS of the mixing layer for total of eight hundred time units (𝑡/𝜏𝜃 = 800) normalized by 𝜏𝜃 = 𝛿0 𝜃/Δ𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to reduce the impact of initial random disturbances on the temporal development of the shear layer, six numerical experiments with different random initializations are performed, one of which is adopted for the parameter optimization of the VOMM model, while the remaining five are used to evaluate the ensemble-averaged physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori studies of LES are conducted using the explicit Gaussian filter (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) with 33 � �� �� �� �� �� �� �� �� �� ��� ���������� � ��� ��� ��� ��� ��� ��� ��� ��� ��� � J�J0 � � � �� DNS ��� � �� � � �� � � �� ��� � �� � � ��� � � �� Figure 17: The evolution of the normalized cost function in temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' FGR LES Resolution 𝐶 (0) 1 𝐶 (0) 2 𝐶opt 1 𝐶opt 2 1 642 × 32 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0637 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='188 2 1282 × 64 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0126 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='000 Table 7: The initial and optimal parameters of the VOMM model for LES computations with the filter width ¯Δ = 8ℎDNS in temporally evolving mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the given filter scale ¯Δ = 8ℎDNS and initialized by the same instantaneous velocity field of the filtered DNS at 𝑡/𝜏𝜃 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Two different filter-to-grid ratios FGR= ¯Δ/ℎLES=1 and 2 are selected to study the influence of the spatial resolution or discretization error on the SGS stress modeling with the corresponding grid resolutions of LES: 𝑁 = 642 × 32 and 1282 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The results from the previous two turbulence problems (forcing and decaying homogenous isotropic turbulence) indicate that the statistics of turbulence are very close and similar when the grid resolution is sufficiently fine (FGR=2 and 4) and the discretization error is considered negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the statistics of LES with a relatively coarse grid resolution (FGR=1) are distinctly different from those of LES with satisfactory grid resolutions (FGR=2 and 4), since the spatial discretization error of FGR=1 is considerably significant and dominates the SGS modelling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Therefore, the a posteriori testings of LES at both FGR=1 and 2 are essential for performance evaluations of the SGS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dissipation spectrum of the filtered DNS is consistently used as the objective function to optimize the model parameters of the VOMM model during the period (assess every 𝑡/𝜏𝜃 = 10 with total thirty-six groups at 50 ⩽ 𝑡/𝜏𝜃 ⩽ 400) of the adjoint-based optimization (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model without the dissipative term is adopted as the initial SGS model with coefficients 𝐶 (0) 1 = 0 and 𝐶 (0) 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We calculate the adjoint-based gradients of the cost functional for the model parameters by backward integrating the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS model coefficients are iteratively updated by the L-BFGS optimization method (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) until the stopping criterion is ultimately satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figure 17 gives the optimization 34 Model(FGR=1,𝑁 = 642 × 32) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='646 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='362 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='590 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='232 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='330 Model(FGR=2,𝑁 = 1282 × 64) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='756 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='370 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='465 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='460 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='908 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='590 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='300 Table 8: The average computational cost of SGS stress modeling 𝜏𝑖 𝑗 for LES computations with the filter width ¯Δ = 8ℎDNS in temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' process of the cost function during the adjoint-based optimization for the temporally evolving mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The loss functions for both FGR=1 and 2 drop dramatically and reach a steady plateau within less than ten iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The cost function of FGR=1 shows a more distinct reduction with approximately 8% of the initial level than that of FGR=2 decreasing to the 10% of original value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The optimal parameters of VOMM model are quickly obtained by the effective gradient- based optimization within a limited number of iterations (around 10 optimization evaluations, namely, 20 LES calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Table 7 summarizes the optimal parameters of the VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The parameter magnitude of the dissipative Smagorinsky term ( ���𝐶opt 1 ���) obviously decreases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0637 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0126 when the FGR increases from 1 to 2, while the ADM coefficient (𝐶opt 2 ) generally tends towards unity, similar to the cases of isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We then examine the a posteriori performance of the proposed VOMM model once the SGS model coefficients are determined by the adjoint-based gradient optimization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to demonstrate the generality of the optimal model parameters that are insensitive to the initial perturbations, ensemble-averaged quantities are evaluated by five numerical experiments with different initial random disturbances from the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The time steps of LES are set as Δ𝑡LES/Δ𝑡DNS = {10, 5} to guarantee the consistent CFL number for different grid resolutions (FGR=1 and 2 with 𝑁 = 642 × 32 and 1282 × 64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model is compared with the conventional SGS models (DSM, DMM and ADM models), and the average modeling costs for different SGS models are listed in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model evaluates efficiently with about 30% computational cost of the DMM model which is similar to those of the ADM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figure 18 illustrates the temporal evolutions of the momentum thickness 𝛿𝜃 in LES calculations of different SGS models compared to the benchmark fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' At the case of coarse grid resolution (FGR=1 with 𝑁 = 642 × 32), all conventional SGS models underpredict the momentum thickness at the early stage of shear layer development (𝑡/𝜏𝜃 ⩽ 300) but give obvious overestimations in the linear growth region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the fine-grid-resolution case (FGR=2 with 𝑁 = 1282 × 64), DMM and ADM (𝜒=1) models can capture the growth rate of momentum thickness well at the beginning of temporal development, but still overpredict the thickness with the developing of shear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The prediction of the pure ADM model with 𝜒 = 0 is irregular and nonlinear all the time without an apparent linear self-similar region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The DSM model at different grid resolutions gives the clearly tilted temporal evolutions, where the momentum thickness is underestimated at the beginning of transition region and overpredicted in the region of linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the predictions of the VOMM model always coincide well with those of fDNS, and they accurately capture the temporal growth rate in the linear region at both grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Furthermore, the evolutions of the turbulent kinetic energy in the streamwise and spanwise 35 (𝑎) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ��� ��� ��� ��� ��� � � ��� � �� � � �� � ��� � � � �" DNS !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ��� ��� ����� � �� ����� � �� ���� (𝑏) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ��� ��� ��� ��� ��� � � ��� � �� � � ��� � ��� � � � �" DNS !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ��� ��� ����� � �� ����� � �� ���� Figure 18: Temporal evolutions of the momentum thickness 𝛿𝜃 for LES in the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter scale ¯Δ = 8ℎDNS: (a) FGR=1, 𝑁 = 642 × 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 1282 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (𝑎) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ����� ���� ����� ���� � "� ��� � �� � � �� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ����� ���� ����� ���� � "� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� ��� ��� ����� � �� ����� � �� ���� Figure 19: Temporal evolutions of the streamwise turbulent kinetic energy 𝐸𝑘1 for LES in the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter scale ¯Δ = 8ℎDNS: (a) FGR=1, 𝑁 = 642 × 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 1282 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' directions are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 19 and 20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The comparisons of transverse turbulent kinetic energy for different SGS models are very similar to those in the spanwise direction, not shown in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The turbulent kinetic energy of DNS in different directions gradually increases with the developing of the shear layer, since the initial perturbated velocity field is approximately laminar and steadily transitions to turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The temporal development of the streamwise kinetic energy can be approximately regarded as a linear growth with time, which is distinctly different from that of spanwise kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' All classical SGS models predict both streamwise and spanwise kinetic energy much larger than the benchmark fDNS results at both grid resolutions of LES, except that the pure ADM model gives underestimations of kinetic energy in the fine-grid-resolution case (FGR=2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to these traditional models, the VOMM model accurately predicts the kinetic energy at different grid resolutions in both streamwise and spanwise directions, and is the closest to the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The profiles of the resolved Reynolds shear stress component ¯𝑅12 = � ¯𝑢′ 1 ¯𝑢′ 2 � at time instants 𝑡/𝜏𝜃 ≈ 500 and 800 are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 21, which is the dominant Reynolds stress term due to the intense mixing along the streamwise and normal directions (Vreman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The normal distribution of the Reynolds stress is a second-order statistic of turbulence which has high requirements for the accuracy of SGS modeling of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ADM models underpredict the Reynolds stress, while DSM and DMM models give obvious overestimations at 36 (𝑎) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ����� ���� ����� � "� ��� � �� � � �� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ����� ���� ����� � "� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� ��� ��� ����� � �� ����� � �� ���� Figure 20: Temporal evolutions of the spanwise turbulent kinetic energy 𝐸𝑘3 for LES in the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter scale ¯Δ = 8ℎDNS: (a) FGR=1, 𝑁 = 642 × 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 1282 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (𝑎) ���� ���� ���� � ��� ��� ��� x 2 /4π ����� � ���� ���� ���� ���� ���� ���� � � � �� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � "�� � � ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) ���� ���� ���� � ��� ��� ��� x 2 /4π ����� � ���� ���� ���� ���� ���� ���� � � � �� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � "�� � � ��� ��� ��� ��� ����� � �� ����� � �� ���� Figure 21: The transient profile of the resolved Reynolds shear stress ¯𝑅12 = � ¯𝑢′ 1 ¯𝑢′ 2 � along the cross-stream direction for LES in the a posteriori analysis of temporally evolving turbulent mixing layer with filter scale ¯Δ = 8ℎDNS at grid resolution of 𝑁 = 1282 × 64: (a) 𝑡/𝜏𝜃 ≈ 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) 𝑡/𝜏𝜃 ≈ 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to these classical SGS models, the VOMM model gives the prediction closest to the fDNS results, and accurately recovers the transient profiles of Reynolds stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We further compare the velocity spectra of different SGS models with the DNS and filtered DNS data at time instants 𝑡/𝜏𝜃 ≈ 500 and 800, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The spectra of DNS at 𝑡/𝜏𝜃 ≈ 500 and 800 are very similar since the instantaneous velocity fields at different moments are both at the self-similar stage of mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the coarse grid-resolution case at FGR=1 with 𝑁 = 642 × 32, the conventional SGS models (DSM, DMM and ADM models) always give the overestimations of the small-scale kinetic energy at high wavenumbers, and the excess kinetic energy accumulates at small scales and exacerbates the numerical instability of LES over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS dissipation provided by these conventional SGS models is insufficient to stabilize the numerical perturbations induced by the spatial discretization errors, which cannot effectively drain out the small-scale kinetic energy in time at FGR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the case of fine grid resolution at FGR=2 with 𝑁 = 1282×64, the pure ADM model is still numerically unstable, whose prediction distinctly deviates from the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' And the velocity spectra predicted by the other conventional SGS models (DSM, DMM and ADM with 𝜒=0) diminish at high-wavenumber regions and accumulate in the region of intermediate wavenumbers, since these traditional SGS models are too dissipative at the fine grid-resolution case to recover the effect of small-scale flow structures near the cutoff wavenumber, giving rise to the blockage of the kinetic energy cascade from large scales to small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the kinetic energy cascade can be correctly constructed with high accuracy by 37 (𝑎) �� � � � �� � �� � �� �� " �� �� �� �� �� �� �� �� �� �� �� �� ��"� ��� � �� � � �� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � #�� � � ��� ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) �� � � � �� � �� � �� �� " �� �� �� �� �� �� �� �� �� �� �� �� ��"� ��� � �� � � �� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � #�� � � ��� ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑐) �� � � � �� � �� � �� �� " �� �� �� �� �� �� �� �� �� �� �� �� ��"� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � #�� � � ��� ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑑) �� � � � �� � �� � �� �� " �� �� �� �� �� �� �� �� �� �� �� �� ��"� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � #�� � � ��� ��� ��� ��� ��� ����� � �� ����� � �� ���� Figure 22: Velocity spectra for different SGS models in the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter scale ¯Δ = 8ℎDNS at 𝑡/𝜏𝜃 ≈ 500 and 800: (a) FGR=1, 𝑁 = 642 × 32 at 𝑡/𝜏𝜃 ≈ 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=1, 𝑁 = 642 × 32 at 𝑡/𝜏𝜃 ≈ 800;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) FGR=2, 𝑁 = 1282 × 64 at 𝑡/𝜏𝜃 ≈ 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (d) FGR=2, 𝑁 = 1282 × 64 at 𝑡/𝜏𝜃 ≈ 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the VOMM model, and the predictions are always in reasonable agreement with those of fDNS at different grid resolutions and time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The reconstruction of vortex structures is finally compared with different SGS models by displaying the iso-surface of the Q-criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Q-criterion is a useful visualization tool for observing vortex structures in turbulent flows, and is the second invariant of velocity gradient tensor, namely (Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Dubief & Delcayre 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019) 𝑄 = 1 2 � ¯Ω𝑖 𝑗 ¯Ω𝑖 𝑗 − ¯𝑆𝑖 𝑗 ¯𝑆𝑖 𝑗 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8) where ¯Ω𝑖 𝑗 = 1 2 �𝜕 ¯𝑢𝑖/𝜕𝑥 𝑗 − 𝜕 ¯𝑢 𝑗/𝜕𝑥𝑖 � represents the rotation-rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The instantaneous iso- surface of Q at 𝑡/𝜏𝜃 ≈ 500 is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 23 during the self-similar stage of the mixing layer for Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 colored by the streamwise velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Q iso-surface of fDNS contains a large number of elaborate vortex structures near the middle 𝑥1-𝑥3 plane of the shear layer, including the rib-like vortices, hairpin vortices and complex helical vortices, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DSM, DMM and ADM (𝜒=1) models exhibit an excessive dissipation that only large-scale rib-like vortex structures remain, while the pure ADM model with 𝜒=0 suffers from numerical instability of LES and overpredicts many nonphysical small-scale structures caused by numerical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model can accurately reconstruct much more vortex structures, highlighting its advantage in improving the accuracy of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 38 (a) fDNS (b) DMM (c) ADM (𝜒=0) (d) ADM (𝜒=1) (e) DSM (f) VOMM Figure 23: The iso-surface of the Q-criterion at 𝑄=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 colored by the streamwise velocity at 𝑡/𝜏𝜃 ≈ 500 in the a posteriori analysis of temporally evolving turbulent mixing layer with filter scale ¯Δ = 8ℎDNS at grid resolution of 𝑁 = 1282 × 64: (a) fDNS, (b) DMM, (c) ADM(𝜒=0), (d) ADM(𝜒=1), (e) DMM, and (f) VOMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' u1 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': 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variational optimal mixed model (VOMM) is developed for the large-eddy simulation of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We first derive the original adjoint LES equations with the general SGS model, and then carry out the energy budget analysis of adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These detailed derivations demonstrate that the quadratic term with negative eigenvalues of the shear strain rate is responsible for the exponential temporal growth of the adjoint-based gradients, giving rise to the numerical divergence in a long time horizon for the chaotic turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' This issue might greatly limits the application of the adjoint-based variational methods and optimal control strategy in turbulence problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' An additional stabilization term is introduced to maintain the numerical stability of the adjoint LES equations and is efficiently calculated by the sequential quadratic programming (SQP) approach, without degrading the accuracy of gradient evaluations for the SGS model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Subsequently, the stabilized adjoint LES equations are correspondingly formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The approximate deconvolution model (ADM) in the scale-similarity form and the dissipative Smagorinsky term are selected as the basis tensors of the proposed VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The parameters of the VOMM model are optimally identified by minimizing the statistical discrepancies between dissipation spectra of the LES and those of the benchmark filtered DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradients of cost functional for model coefficients are efficiently evaluated by successively forward solving the LES equations and backward integrating the stabilized adjoint LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The gradient-based L-BFGS optimization algorithm is adopted for iteratively updating the VOMM model parameters until the optimal values are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Three turbulent flow scenarios including the forced homogeneous isotropic turbulence, de- caying homogeneous isotropic turbulence and temporally evolving turbulent mixing layer are investigated to examine the a posteriori performance of the VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure structural ADM model without the dissipative Smagorinsky term is selected as the initial SGS model for the parameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The loss functions of the dissipation spectra can dramatically converge and reach the optimal state of only about 10% of the initial value within less than twenty iterations (about forty LES evaluations) during the adjoint-based gradient optimization at different grid resolutions for these three types of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results indicate that the adjoint-based gradient optimization is an effective tool to obtain the optimal parameters of VOMM model within only a few iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Meanwhile, the computational efficiency of the proposed method is independent of the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Once the optimal SGS model coefficients are determined by the adjoint-based gradient optimization, the a posteriori accuracy of the VOMM model is further tested in comparison with the classical SGS models, including the dynamic Smagorinsky model (DSM), dynamic mixed model (DMM), the pure ADM model and ADM model with the standard secondary- filtering regularization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The various statistics of turbulence and the instantaneous flow structures are comprehensively compared for LES calculations of different SGS models with the benchmark filtered DNS data at different grid resolutions of three turbulent flow scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In the cases of forced and decaying homogeneous isotropic turbulence, the filter scale is fixed to ¯Δ = 32ℎDNS and the impact of the spatial discretization errors on the SGS modeling is studied by changing the grid resolution of LES with three different filter-to-grid ratios FGR=1, 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori performance of the proposed VOMM model is systematically evaluated by comparison to the conventional SGS models (DSM, DMM and ADM models) in terms of the velocity spectra, structure functions with different orders, PDFs of the velocity increments and vorticity, temporal evolutions of the turbulent kinetic energy and average dissipation rate, as well as the instantaneous vorticity contours at different grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model always exhibits numerical instability due to the insufficient sufficient SGS dissipation for all grid- resolution cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dynamic models and standard regularized ADM model underpredict the 40 model dissipation in the case of coarse grid resolution (FGR=1), with the excess kinetic energy accumulated at small scales leading to the numerical instability of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS dissipation imposed by these classical SGS models is insufficient to suppress the numerical perturbations dominated by the spatial discretization, and it cannot effectively drain out the small-scale kinetic energy in time at FGR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the traditional SGS models are too dissipative that most small- scale flow structures near the truncated wavenumber are diminished, giving rise to the blockage of the kinetic energy cascade from large scales to small scales at situations of satisfactory grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model can correctly reconstruct the kinetic energy cascade and the evolution of dissipation rate with high accuracy, which is essential for the isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the VOMM model accurately predicts various flow statistics and transient spatial flow structures, which are always in reasonable agreement with the benchmark filtered DNS results at different grid resolutions and times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In the context of the temporally evolving turbulent mixing layer, the unsteady evolution of the shear layer from the initial perturbed velocity field gradually transitions to fully developed turbulence is challenging for the SGS modeling of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model can accurately reconstruct the temporal evolutions of characteristic physical quantities of the mixing layer, including the momentum thickness, turbulent kinetic energy in different directions and transient velocity spectra at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The corresponding predictions of VOMM are closest to the filtered DNS results and superior to these conventional SGS models (DSM, DMM and ADM models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The profiles of Reynolds shear stress at the self-similar stage of the shear layer are critical for the development of mixing layer, and all conventional SGS models are not able to accurately predict the vertical distributions with significant deviations from the benchmark fDNS result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model predicts the Reynolds stress fairly well at different time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Besides, it can be clearly observed from the iso-surface of Q-criterion that the VOMM model accurately recovers the diverse spatial vortex structures very similar to the benchmark fDNS data in comparison to the classical SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Furthermore, for the cases of three turbulent flow scenarios with different grid resolutions, the computational cost of the proposed VOMM model is only about 30% the time of the DMM model, which is very efficient and competitive compared to the classical SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results suggest that the proposed VOMM model has high a posteriori accuracy and computational efficiency by assimilating the a priori knowledge of turbulence statistics, and can be a promising tool to develop advanced SGS models in the LES of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Eventually, it is noteworthy that fine-tuning a small number of model parameters of some traditional SGS models can significantly improve the a posteriori accuracy of LES using the proposed adjoint-based optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the predictions of LES in complex turbulent flows using the VOMM model might be dramatically accurate as the number of model coefficients increases, while the computational cost of the adjoint-based approach hardly varies with to the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Although the high-fidelity turbulence statistics is provided by DNS data in the current study, the experimental measurements can also be assimilated using the same optimization procedure to increase the accuracy of LES modeling for a particular type of complex turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In future work, we would further apply the VOMM model with the existing optimal parameters to more complex turbulent flows and generalize to turbulence with different filter scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' This work was supported by the National Natural Science Foundation of China (NSFC Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 91952104, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 92052301, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 12172161, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 12161141017), by the National Numerical Windtunnel Project (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' NNW2019ZT1-A04), by the NSFC Basic Science Center Program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 11988102), by the Shenzhen Science and Technology Program (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' KQTD20180411143441009), by Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 41 GML2019ZD0103), and by Department of Science and Technology of Guangdong Province (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019B21203001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' This work was also supported by Center for Computational Science and Engineering of Southern University of Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Declaration of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Derivation of the adjoint large-eddy simulation equations The large-eddy simulation (LES) equations are expressed as (Pope 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sagaut 2006) 𝑅0 ( ¯𝑢𝑖) = 𝜕 ¯𝑢𝑖 𝜕𝑥𝑖 = 0, (A 1) 𝑅𝑖 ( ¯𝑢𝑖, ¯𝑝) = 𝜕 ¯𝑢𝑖 𝜕𝑡 + 𝜕 � ¯𝑢𝑖 ¯𝑢 𝑗 � 𝜕𝑥 𝑗 + 𝜕 ¯𝑝 𝜕𝑥𝑖 − 𝜈 𝜕2 ¯𝑢𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 − F 𝑖 + 𝜕𝜏𝑖 𝑗 𝜕𝑥 𝑗 = 0, (A 2) where an overbar denotes the filtered variables with filter scale ¯Δ, ¯𝑢𝑖 and ¯𝑝 denote the filtered velocity and pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝜈 is the kinematic viscosity, and ¯F𝑖 represents the large-scale forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The unclosed SGS stress 𝜏𝑖 𝑗 = 𝑢𝑖𝑢 𝑗 − ¯𝑢𝑖 ¯𝑢 𝑗 is modeled by the 𝑁-parameter mixed model 𝜏𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛) 𝑖 𝑗 � ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� with the basis stress tensors 𝑇 (𝑛) 𝑖 𝑗 and model coefficients 𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The sensitivities of the governing equations for the LES variables ¯v = [ ¯𝑝, ¯𝑢1, ¯𝑢2, ¯𝑢3]𝑇 are given by 𝛿𝑅𝑘 = 𝜕𝑅𝑘 𝜕¯v · 𝛿¯v = � 𝜕𝛿 ¯𝑢𝑖 𝜕𝑥𝑖 𝜕𝛿 ¯𝑢𝑖 𝜕𝑡 + 𝜕( ¯𝑢𝑗 𝛿 ¯𝑢𝑖) 𝜕𝑥𝑗 + 𝜕( ¯𝑢𝑖 𝛿 ¯𝑢𝑗) 𝜕𝑥𝑗 + 𝜕𝛿 ¯𝑝 𝜕𝑥𝑖 − 𝜈 𝜕2 𝛿 ¯𝑢𝑖 𝜕𝑥𝑗𝜕𝑥𝑗 + 𝜕𝛿𝜏𝑖 𝑗 𝜕𝑥𝑗 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (A 3) The adjoint LES equations are derived by the adjoint identity acting on the adjoint variables ¯v† = � ¯𝑝†, ¯𝑢† 1, ¯𝑢† 2, ¯𝑢† 3 �𝑇 , namely � 𝜕𝑅𝑘 𝜕¯v · 𝛿¯v, ¯v† � x,𝑡 = � 𝛿¯v, � 𝜕𝑅𝑘 𝜕¯v �† ¯v† � x,𝑡 + 𝐵𝑇, (A 4) where 𝐵𝑇 denotes the boundary and temporal integral terms, and 𝐵𝑇 = 0 can identify the boundary and terminal conditions of the adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The corresponding adjoint LES equations can be expressed as 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 𝜕¯v �† ¯v† − 𝜕𝐽 𝜕¯v = 0, (A 5) where 𝜕𝐽/𝜕¯v = � 0, 𝜕𝐽 𝜕 ¯𝑢1 , 𝜕𝐽 𝜕 ¯𝑢2 , 𝜕𝐽 𝜕 ¯𝑢3 �𝑇 denotesthesensitivityofthecostfunctional 𝐽 � ¯𝑢𝑖, ¯𝑢ref 𝑖 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛, x, 𝑡� which quantifies the discrepancy between ¯𝑢𝑖 and the reference data ¯𝑢ref 𝑖 in the LES calculations under the given parameters 𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) at a certain space-time state (x, 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, the terms (𝜕𝑅𝑘/𝜕¯v)† · ¯v† (𝑘 = 0, 1, 2, 3) are derived by multiplying the perturbation LES equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A 3) with the adjoint LES variables ¯v†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and then integrating by parts to rearrange all of the 42 differential operators without 𝛿¯v onto the adjoint variables ¯v† ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' yielding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ¯𝑝† + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕( ¯𝑢𝑗 𝛿 ¯𝑢𝑖) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕( ¯𝑢𝑖 𝛿 ¯𝑢𝑗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ 𝜕𝛿 ¯𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 − 𝜈 𝜕2 𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗𝜕𝑥𝑗 + 𝜕𝛿𝜏𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑝 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑡 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢 𝑗 + 𝜈 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕2 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗𝜕𝑥𝑗 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝜏𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='− ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕2𝜏𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢𝑖𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑖+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='terminal condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ 𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ¯𝑢 𝑗 + 𝜈 𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝜏𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑖 − 𝜈 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ 𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿 ¯𝑝 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑝† + ¯𝑢 𝑗 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='boundary condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (A 6) The adjoint LES equations are written in detail as 𝜕 ¯𝑢† 𝑖 𝜕𝑥𝑖 = 0, (A 7) 𝜕 ¯𝑢† 𝑖 𝜕𝑡 + � 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 � ¯𝑢 𝑗 + 𝜈 𝜕2 ¯𝑢† 𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + 𝜕 𝜕𝑥 𝑗 � ¯𝑢† 𝑘 𝜕𝜏𝑗𝑘 𝜕 ¯𝑢𝑖 � − ¯𝑢† 𝑘 𝜕2𝜏𝑗𝑘 𝜕 ¯𝑢𝑖𝜕𝑥 𝑗 + 𝜕𝐽 𝜕 ¯𝑢𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (A 8) It is worth noting that the adjoint SGS term ¯𝑢† 𝑘 𝜕2𝜏𝑗𝑘 𝜕 ¯𝑢𝑖𝜕𝑥𝑗 can lead to the non-conservation of the adjoint momentum and deteriorate the evaluation of the adjoint-based gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To our knowledge, few previous studies have addressed this critical issues that make the LES adjoint field prone to numerical instability and eventual divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to maintain the momentum conservation in the adjoint equations, we remove ¯𝑢† 𝑘 𝜕2𝜏𝑗𝑘 𝜕 ¯𝑢𝑖𝜕𝑥𝑗 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A 8, and the conservative adjoint LES equations are obtained as 𝜕 ¯𝑢† 𝑖 𝜕𝑥𝑖 = 0, (A 9) 𝜕 ¯𝑢† 𝑖 𝜕𝑡 + � 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 � ¯𝑢 𝑗 + 𝜈 𝜕2 ¯𝑢† 𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + 𝜕𝜏† 𝑖 𝑗 𝜕𝑥 𝑗 + 𝜕𝐽 𝜕 ¯𝑢𝑖 = 0, (A 10) where 𝜏† 𝑖 𝑗 = ¯𝑢† 𝑘 𝜕𝜏𝑗𝑘 𝜕 ¯𝑢𝑖 is the adjoint SGS stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' If the unclosed SGS terms is modeled by the 𝑁-parameter mixed model 𝜏𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛) 𝑖 𝑗 � ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� with the basis stress tensors 𝑇 (𝑛) 𝑖 𝑗 and model coefficients 𝐶𝑛, the adjoint SGS stresses are correspondingly represented as 𝜏† 𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛),† 𝑖 𝑗 with the associated adjoint basis stress tensors 𝑇 (𝑛),† 𝑖 𝑗 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Derivation of the adjoint SGS stress for the VOMM model The present variational optimal mixed model (VOMM) combines the approximate deconvolu- tion model (ADM) in the scale-similarity form with the dissipative Smagorinsky part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' expressed as 𝜏𝑖 𝑗 = 𝐶1𝑇 (1) 𝑖 𝑗 + 𝐶2𝑇 (2) 𝑖 𝑗 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' with 𝑇 (1) 𝑖 𝑗 = ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑇 (2) 𝑖 𝑗 = 𝑢∗ 𝑖 𝑢∗ 𝑗 − 𝑢∗ 𝑖 𝑢∗ 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 1) where 𝑢∗ 𝑖 = 𝑁� 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ ¯𝑢𝑖 stands for the 𝑖-th approximate unfiltered velocity component recovered by the iterative van Cittert procedure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 is the number of iterations for the AD procedure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐼 is the identity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and the symbol “⊗” is the spatial convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝐶1 and 𝐶2 are SGS 43 model coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The variation of the first basis SGS tensor 𝑇 (1) 𝑖 𝑗 with respect to the velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' is derived by 𝛿𝑇 (1) 𝑖 𝑗 = ¯Δ2 � | ¯𝑆|𝛿 ¯𝑆𝑖 𝑗 + �𝛿| ¯𝑆|� ¯𝑆𝑖 𝑗 � = ¯Δ2 � | ¯𝑆| 𝜕 ¯𝑆𝑖 𝑗 𝜕 ¯𝑢𝑘 + 𝜕| ¯𝑆| 𝜕 ¯𝑢𝑘 ¯𝑆𝑖 𝑗 � 𝛿 ¯𝑢𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 2) where the derivatives of the shear strain-rate tensor and characteristic strain rate for the velocity are further written as 𝜕 ¯𝑆𝑖 𝑗 𝜕 ¯𝑢𝑘 = 1 2 𝜕 𝜕 ¯𝑢𝑘 � 𝜕 ¯𝑢𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢 𝑗 𝜕𝑥𝑖 � = 1 2 � 𝜕𝛿𝑖𝑘 𝜕𝑥 𝑗 + 𝜕𝛿 𝑗𝑘 𝜕𝑥𝑖 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 3) and 𝜕| ¯𝑆| 𝜕 ¯𝑢𝑘 = 𝜕| ¯𝑆| 𝜕 ¯𝑆𝑖 𝑗 𝜕 ¯𝑆𝑖 𝑗 𝜕 ¯𝑢𝑘 = ¯𝑆𝑖 𝑗 | ¯𝑆| � 𝜕𝛿𝑖𝑘 𝜕𝑥 𝑗 + 𝜕𝛿 𝑗𝑘 𝜕𝑥𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(B 4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='The inner product between the variation of the first basis SGS force and the adjoint velocity is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='derived by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='= − ¯Δ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='| ¯𝑆| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿𝑖𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 + 𝜕𝛿 𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿𝑚𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑛 + 𝜕𝛿𝑛𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 ¯𝑆𝑚𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='| ¯𝑆| ¯𝑆𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑘 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='= − ¯Δ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='| ¯𝑆| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� 2 ¯𝑆𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='| ¯𝑆| ¯𝑆𝑚𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑛 + 𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑘 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 5) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the adjoint strain-rate tensor ¯𝑆† 𝑖 𝑗 = � 𝜕 ¯𝑢† 𝑖 /𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗/𝜕𝑥𝑖 � /2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and the inner product term can be further expressed as ¯𝑢† 𝑖 𝜕𝛿𝑇 (1) 𝑖 𝑗 𝜕𝑥 𝑗 = � 𝜕 𝜕𝑥 𝑗 � − ¯Δ2 � | ¯𝑆| ¯𝑆† 𝑖 𝑗 + 2 ¯𝑆𝑘𝑙 ¯𝑆† 𝑘𝑙 | ¯𝑆| ¯𝑆𝑖 𝑗 ��� 𝛿 ¯𝑢𝑖 + 𝜕 𝜕𝑥 𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (1) 𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 6) Thus, the adjoint basis stress tensor 𝑇 (1),† 𝑖 𝑗 is given by 𝑇 (1),† 𝑖 𝑗 = − ¯Δ2 � | ¯𝑆| ¯𝑆† 𝑖 𝑗 + 2 ¯𝑆𝑘𝑙 ¯𝑆† 𝑘𝑙 | ¯𝑆| ¯𝑆𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 7) The common filter function 𝐺 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' top-hat, Gaussian and spectral filters) is symmetric spatial filter, and is self-adjoint, namely (Vreman 2004) ⟨𝐺 ⊗ 𝑓 , 𝑔⟩x = ⟨ 𝑓 , 𝐺 ⊗ 𝑔⟩x, (B 8) where 𝑓 (x) and 𝑔 (x) are arbitrary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The 𝐺𝑛 filter with spatially filtering 𝑛 times (𝐺𝑛 = 𝐺 ⊗ 𝐺 ⊗ · · · ⊗ 𝐺) also satisfies the self-adjoint property proved by the mathematical induction method, expressed as ⟨𝐺𝑛 ⊗ 𝑓 , 𝑔⟩x = � 𝐺 ⊗ 𝐺𝑛−1 ⊗ 𝑓 , 𝑔 � x = � 𝐺𝑛−1 ⊗ 𝑓 , 𝐺 ⊗ 𝑔 � x = · · · = ⟨ 𝑓 , 𝐺𝑛 ⊗ 𝑔⟩x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 9) The (𝐼 − 𝐺) filter is also a symmetric filter, and the approximate deconvolution procedure 𝐻 = 𝑁� 𝑛=1 (𝐼 − 𝐺)𝑛−1 is thus the self-adjoint filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The second basis SGS tensor 𝑇 (2) 𝑖 𝑗 can be described using the AD abbreviated notation, namely 𝑇 (2) 𝑖 𝑗 = 𝑢∗ 𝑖 𝑢∗ 𝑗 − 𝑢∗ 𝑖 𝑢∗ 𝑗 = 𝐺 ⊗ � (𝐻 ⊗ ¯𝑢𝑖) �𝐻 ⊗ ¯𝑢 𝑗 �� − [𝐺 ⊗ (𝐻 ⊗ ¯𝑢𝑖)] � 𝐺 ⊗ �𝐻 ⊗ ¯𝑢 𝑗 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 10) 44 The variation of the second basis SGS tensor 𝑇 (2) 𝑖 𝑗 with respect to the velocity, expressed as 𝛿𝑇 (2) 𝑖 𝑗 = 𝐺 ⊗ � (𝐻 ⊗ 𝛿 ¯𝑢𝑖) 𝑢∗ 𝑗 � +𝐺 ⊗ � 𝑢∗ 𝑖 �𝐻 ⊗ 𝛿 ¯𝑢 𝑗 �� −[𝐺 ⊗ (𝐻 ⊗ 𝛿 ¯𝑢𝑖)] 𝑢∗ 𝑗 −𝑢∗ 𝑖 � 𝐺 ⊗ �𝐻 ⊗ 𝛿 ¯𝑢 𝑗 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 11) The inner product between the variation of the second basis SGS force and the adjoint velocity is given by 𝜕𝛿𝑇 (2) 𝑖 𝑗 𝜕𝑥 𝑗 ¯𝑢† 𝑖 = − 𝜕 ¯𝑢† 𝑖 𝜕𝑥𝑗 𝛿𝑇 (2) 𝑖 𝑗 + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � = −2 ¯𝑆† 𝑖 𝑗 � 𝐺 ⊗ � (𝐻 ⊗ 𝛿 ¯𝑢𝑖) 𝑢∗ 𝑗 �� + 2 ¯𝑆† 𝑖 𝑗 [𝐺 ⊗ (𝐻 ⊗ 𝛿 ¯𝑢𝑖)] 𝑢∗ 𝑗 + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 12) The inner product term can be further simplified by the self-adjoint property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' such that 𝜕𝛿𝑇 (2) 𝑖 𝑗 𝜕𝑥𝑗 ¯𝑢† 𝑖 = −2 � 𝐺 ⊗ ¯𝑆† 𝑖 𝑗 � � (𝐻 ⊗ 𝛿 ¯𝑢𝑖) 𝑢∗ 𝑗 � + 2 � 𝐺 ⊗ � ¯𝑆† 𝑖 𝑗𝑢∗ 𝑗 �� (𝐻 ⊗ 𝛿 ¯𝑢𝑖) + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � = 𝐻 ⊗ � −2 ¯𝑆† 𝑖 𝑗𝑢∗ 𝑗 + 2 ¯𝑆† 𝑖 𝑗𝑢∗ 𝑗 � 𝛿 ¯𝑢𝑖 + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � = � 𝜕 𝜕𝑥𝑗 � 𝐻 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 �� + 𝐻 ⊗ � 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 𝑢∗ 𝑗 − 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 𝑢∗ 𝑗 �� 𝛿 ¯𝑢𝑖 + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 13) It is quite notable that the second adjoint SGS term makes the non-conservation of the adjoint momentum, therefore we discard the second adjoint SGS term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Thus, the second adjoint basis stress tensor 𝑇 (2),† 𝑖 𝑗 can be written as 𝑇 (2),† 𝑖 𝑗 = 𝐻 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 � = 𝑁 ∑︁ 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 14) In summary, the adjoint SGS stress of the proposed VOMM model is represented by 𝜏† 𝑖 𝑗 = 𝐶1𝑇 (1),† 𝑖 𝑗 + 𝐶2𝑇 (2),† 𝑖 𝑗 , (B 15) where the adjoint basis stress tensors are 𝑇 (1),† 𝑖 𝑗 = − ¯Δ2 � | ¯𝑆| ¯𝑆† 𝑖 𝑗 + 2 ¯𝑆𝑘𝑙 ¯𝑆† 𝑘𝑙 | ¯𝑆| ¯𝑆𝑖 𝑗 � and 𝑇 (2),† 𝑖 𝑗 = 𝑁� 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} 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release the work +under a Creative Commons +Attribution 4.0 International +License (CC BY 4.0). +Summary +pyssam is a Python library for creating statistical shape and appearance models (SSAMs) +for biological (and other) shapes such as bones, lungs or other organs. A point cloud best +describing the anatomical ‘landmarks’ of the organ are required from each sample in a small +population as an input. Additional information such as landmark gray-value can be included to +incorporate joint correlations of shape and ‘appearance’ into the model. Our library performs +alignment and scaling of the input data and creates a SSAM based on covariance across the +population. The output SSAM can be used to parameterise and quantify shape change across +a population. pyssam is a small and low dependency codebase with examples included as +Jupyter notebooks for several common SSAM computations. The given examples can easily be +extended to alternative datasets, and also alternative tasks such as medical image segmentation +by incorporating a SSAM as a constraint for segmented organs. +Statement of need +Statistical shape (and appearance) models (SSAMs) have drawn significant interest in biomed- +ical engineering and computer vision research due to their ability to automatically deduce a +linear parameterisation of shape covariances across a small population of training data (Baka +et al., 2011; Cootes et al., 1995; Heimann & Meinzer, 2009; Väänänen et al., 2015). The +classic statistical shape model (SSM) approach uses a point cloud of landmarks which are +in correspondence across several instances of a shape. The covariances of how the shape +changes across the training population are computed, and principal component analysis (PCA) +is used to parameterise the different modes of shape variation (Cootes et al., 1995). This +approach paved the way for automatic algorithms which could significantly aid medical image +segmentation (similar to an atlas) (Irving et al., 2011), characterise how the organ shape varies +over a population as a diagnostic tool (Osanlouy et al., 2020), or even reconstruct a full 3D +structure from a sparser imaging modality such as planar X-ray images (Baka et al., 2011; +Väänänen et al., 2015). +We have found that available open-source toolkits such as Statismo and Scalismo (Lüthi et +al., 2012) suffer from an exhaustive number of dependencies and are difficult to adapt to +new tasks, datasets and I/O datatypes. ShapeWorks (Cates et al., 2017) is another strongly +developed library for statistical shape modelling, but it uses an alternative method of extracting +landmarks (a so-called particle-based method) which is less broadly used and more complex +than a landmark-based system (where landmarks can be defined in any desired way for different +anatomical shapes). Additionally, as the machine learning ecosystem has strong foundations in +Python, building statistical models in C++, Scala or other languages reduces compatibility +with the majority of modern machine learning developments (Bhalodia et al., 2018). We +therefore implemented a lightweight Python framework for SSAMs which is easily adaptable +with few dependencies, making it suitable for integrating as part of a broader codebase, as +well as installing and running on high-performance computing clusters where users do not have +root access to install many dependencies. We provide Jupyter notebooks on readthedocs and +Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software, +TBD, TBD. https://doi.org/TBD +1 +arXiv:2301.04416v1 [q-bio.QM] 11 Jan 2023 + +two example datasets that allow users new to coding or SSAMs to learn how these models +work in an interactive way to ease access when learning a new research topic and library. +Overview +The main modelling classes are built on the abstract base class StatisticalModelBase, +which has several methods for pre-processing data and performing PCA (Figure 1). There +are also several global variables that are inherited which are related to principal components, +component variances and model parameters. The classes for SSM and SAM pre-process the data +(align to zero mean and standard deviation of one) and can compute the population mean +shape/appearance. Finally, the SSAM class for shape and appearance modelling inherits all +of these, but also imports the SSM and SAM methods to pre-process shape and appearance +features separately, before they are merged into one dataset for modelling. +StatisticalModelBase +SSM +StatisticalModelBase +SAM +SSAM +StatisticalModelBase +StatisticalModelBase +SSM +SAM +Figure 1: Schematic overview of the codebase. Each modelling class is abstracted from the Statis +ticalModelBase class and contains several inherited variables such as model weights and principal +components. The SSAM class inherits from StatisticalModelBase, but also uses pre-processing +pipelines from SSM and SAM. +Examples +Here we present two example applications of pyssam. The first example examines shape +variations in a toy dataset created for this study, which has a tree structure. Tree structures +appear often in biology, including the lung airways and vascular system. Toy datasets such as +these are a simple means to visualise and interpret the modelling and code framework. We then +provide a more complex example which considers the left lower lobe of human lungs obtained +from CT data (Tang et al., 2019). This example considers shape and appearance, where the +appearance is the gray-value at the landmark location on an X-ray projection (obtained with +the AppearanceFromXray helper class). +Statistical shape modelling toy dataset +To understand the shape modelling process, we have provided a dataset class called Tree +which creates a number of tree shapes which are randomly computed based on global minimum +and maximum values for angle and branch length ratio (between parent and child). Tree +parameters are shown in Figure 2a. Tree nodes are converted to a numpy array and used to +initialise pyssam.SSM. At initialisation of the SSM class, the landmarks are aligned, scaled to +unit standard deviation and stacked into a matrix of shape (Nf, 3NL) where Nf is the number +of features (samples in our training dataset) and NL is the number of landmarks (each with +a x, y, z coordinates). All y coordinates in this case are zero, meaning the data is actually +2D but we preserve a 3D coordinate system for simplicity in generalising the code to more +Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software, +TBD, TBD. https://doi.org/TBD +2 + +common 3D applications. The code below shows how we can simply obtain a SSM from a set +of landmarks. +from glob import glob +import numpy as np +import pyssam +tree_class = pyssam.datasets.Tree(num_extra_ends=1) +landmark_coordinates = np.array( +[tree_class.make_tree_landmarks() for i in range(0, num_samples)] +) +ssm_obj = pyssam.SSM(landmark_coordinates) +ssm_obj.create_pca_model(ssm_obj.landmarks_scale) +mean_shape_columnvector = ssm_obj.compute_dataset_mean() +L1 +L2 +θ +0 +10 +20 +30 +40 +Number of components +50 +60 +70 +80 +90 +100 +Variance [%] +(a) +(b) +Figure 2: Overview of tree dataset population. Panels show (a) a visualisation of 100 tree samples, +and (b) cumulative variance versus the number of PCA components constructed by the statistical +shape model. Inset of (a) shows a legend describing the morphological parameters varied to create +the tree dataset. These parameters include the initial branch length, L1, the branch length ratio +LR = L2/L1, and branching angle θ. +Shape and appearance modelling of lung shape and chest X-ray images +In the following example, we show a real application where 3D landmark for the left lower +lung lobe are projected onto digitally reconstructed X-rays (Väänänen et al., 2015) and the +gray-value is used to obtain appearance. Example landmark data was obtained using an +automatic algorithm (Ferrarini et al., 2007). Appearance information is extracted from the +X-ray images using AppearanceFromXray (part of pyssam.utils). We use landmarks, +X-ray images as well as origin and pixel spacing information for the X-ray images to extract +appearance as follows +appearance_xr = pyssam.AppearanceFromXray( +IMAGE_DATASET, IMAGE_ORIGIN, IMAGE_SPACING +) +appearance_values = appearance_xr.all_landmark_density( +landmarks_coordinates +) +Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software, +TBD, TBD. https://doi.org/TBD +3 + +The SSAM can then be trained in a similar way as the SSM in subsection with the following +code snippet: +ssam_obj = pyssam.SSAM(landmark_coordinates, appearance_values) +ssam_obj.create_pca_model(ssam_obj.shape_appearance_columns) +mean_shape_appearance_columnvector = ssam_obj.compute_dataset_mean() +The shape and appearance modes can then be computed based on the model parameters +(ssam.model_parameters). The computed model parameters (eigenvectors and eigenvalues +of the covariance matrix) can be used to morph the shape and appearance using ssam.morph +_model (part of StatisticalModelBase in Figure 1) by +x ≈ ¯x + Φ · b +(1) +where x is a new array containing shape and appearance, ¯x is the training dataset mean +shape and appearance, Φ is the model principal components (eigenvectors of the training data +covariance matrix), b is the model parameters, which is an array of weights unique to each +data sample. The model parameter a mode m should be within [−3 +� +σ2 +m, 3 +� +σ2 +m], where +σ2 +m is the explained variance of m (mth largest eigenvalue of the covariance matrix) (Cootes +et al., 1995). +Each mode of shape and appearance variation is visualised, as shown for a representative mode +in Figure 3. This shows how lung shape influences the gray-value of lung pixels on the X-ray +image. In this case, the change in shape and appearance are mainly due to how the lung +interacts with adjacent structures such as the heart, rib cage and diaphragm. +Figure 3: First mode of SSAM variation for lung lobe dataset. Panels show shape and appearance +morphed using ssam.morph_model method and varying the model parameters (ssam.model_parame +ters), from -2, 0 (mean shape) and 2. +Acknowledgement +JW was funded by a 2019 PhD Scholarship from the Carnegie-Trust for the Universities of +Scotland. +Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software, +TBD, TBD. https://doi.org/TBD +4 + +References +Baka, N., Kaptein, B. L., Bruijne, M. de, Walsum, T. van, Giphart, J., Niessen, W. J., & +Lelieveldt, B. P. (2011). 2D–3D shape reconstruction of the distal femur from stereo X-ray +imaging using statistical shape models. Medical Image Analysis, 15(6), 840–850. +Bhalodia, R., Elhabian, S. Y., Kavan, L., & Whitaker, R. T. (2018). DeepSSM: A deep +learning framework for statistical shape modeling from raw images. International Workshop +on Shape in Medical Imaging, 244–257. +Cates, J., Elhabian, S., & Whitaker, R. (2017). Shapeworks: Particle-based shape corre- +spondence and visualization software. In Statistical shape and deformation analysis (pp. +257–298). Elsevier. +Cootes, T. F., Taylor, C. J., Cooper, D. H., & Graham, J. (1995). Active shape models-their +training and application. Computer Vision and Image Understanding, 61(1), 38–59. +Ferrarini, L., Olofsen, H., Palm, W. M., Van Buchem, M. A., Reiber, J. H., & Admiraal-Behloul, +F. (2007). GAMEs: Growing and adaptive meshes for fully automatic shape modeling and +analysis. Medical Image Analysis, 11(3), 302–314. +Heimann, T., & Meinzer, H.-P. (2009). +Statistical shape models for 3D medical image +segmentation: A review. Medical Image Analysis, 13(4), 543–563. +Irving, B., Goussard, P., Gie, R., Todd-Pokropek, A., & Taylor, P. (2011). Segmentation of +obstructed airway branches in CT using airway topology and statistical shape analysis. 2011 +IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 447–451. +Lüthi, M., Blanc, R., Albrecht, T., Gass, T., Goksel, O., Büchler, P., Kistler, M., Bousleiman, +H., Reyes, M., Cattin, P., & others. (2012). Statismo-a framework for PCA based statistical +models. The Insight Journal, 2012, 1–18. +Osanlouy, M., Clark, A. R., Kumar, H., King, C., Wilsher, M. L., Milne, D. G., Whyte, K., +Hoffman, E. A., & Tawhai, M. H. (2020). Lung and fissure shape is associated with age in +healthy never-smoking adults aged 20–90 years. Scientific Reports, 10(1), 1–13. +Tang, H., Zhang, C., & Xie, X. (2019). Automatic pulmonary lobe segmentation using deep +learning. arXiv Preprint arXiv:1903.09879. +Väänänen, S. P., Grassi, L., Flivik, G., Jurvelin, J. S., & Isaksson, H. (2015). Generation of +3D shape, density, cortical thickness and finite element mesh of proximal femur from a +DXA image. Medical Image Analysis, 24(1), 125–134. +Williams et al. (2023). pyssam – a Python library for statistical modelling of biomedical shape and appearance. Journal of Open Source Software, +TBD, TBD. https://doi.org/TBD +5 + diff --git a/59E3T4oBgHgl3EQfQwmQ/content/tmp_files/load_file.txt b/59E3T4oBgHgl3EQfQwmQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d6536484499806d0889b6f8c440d937d24b92483 --- /dev/null +++ b/59E3T4oBgHgl3EQfQwmQ/content/tmp_files/load_file.txt @@ -0,0 +1,236 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf,len=235 +page_content='pyssam – a Python library for statistical modelling of biomedical shape and appearance Josh Williams1, Ali Ozel1, and Uwe Wolfram1 1 School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, UK DOI: TBD Software Repository Archive Editor: Pending Editor Reviewers: Pending Reviewers Submitted: N/A Published: N/A License Authors of papers retain copyright and release the work under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='0 International License (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Summary pyssam is a Python library for creating statistical shape and appearance models (SSAMs) for biological (and other) shapes such as bones, lungs or other organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' A point cloud best describing the anatomical ‘landmarks’ of the organ are required from each sample in a small population as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Additional information such as landmark gray-value can be included to incorporate joint correlations of shape and ‘appearance’ into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Our library performs alignment and scaling of the input data and creates a SSAM based on covariance across the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The output SSAM can be used to parameterise and quantify shape change across a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' pyssam is a small and low dependency codebase with examples included as Jupyter notebooks for several common SSAM computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The given examples can easily be extended to alternative datasets, and also alternative tasks such as medical image segmentation by incorporating a SSAM as a constraint for segmented organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Statement of need Statistical shape (and appearance) models (SSAMs) have drawn significant interest in biomed- ical engineering and computer vision research due to their ability to automatically deduce a linear parameterisation of shape covariances across a small population of training data (Baka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Cootes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Heimann & Meinzer, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Väänänen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The classic statistical shape model (SSM) approach uses a point cloud of landmarks which are in correspondence across several instances of a shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The covariances of how the shape changes across the training population are computed, and principal component analysis (PCA) is used to parameterise the different modes of shape variation (Cootes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' This approach paved the way for automatic algorithms which could significantly aid medical image segmentation (similar to an atlas) (Irving et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2011), characterise how the organ shape varies over a population as a diagnostic tool (Osanlouy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2020), or even reconstruct a full 3D structure from a sparser imaging modality such as planar X-ray images (Baka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Väänänen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' We have found that available open-source toolkits such as Statismo and Scalismo (Lüthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2012) suffer from an exhaustive number of dependencies and are difficult to adapt to new tasks, datasets and I/O datatypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' ShapeWorks (Cates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2017) is another strongly developed library for statistical shape modelling, but it uses an alternative method of extracting landmarks (a so-called particle-based method) which is less broadly used and more complex than a landmark-based system (where landmarks can be defined in any desired way for different anatomical shapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Additionally, as the machine learning ecosystem has strong foundations in Python, building statistical models in C++, Scala or other languages reduces compatibility with the majority of modern machine learning developments (Bhalodia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' We therefore implemented a lightweight Python framework for SSAMs which is easily adaptable with few dependencies, making it suitable for integrating as part of a broader codebase, as well as installing and running on high-performance computing clusters where users do not have root access to install many dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' We provide Jupyter notebooks on readthedocs and Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' pyssam – a Python library for statistical modelling of biomedical shape and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Journal of Open Source Software, TBD, TBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='org/TBD 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='04416v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='QM] 11 Jan 2023 two example datasets that allow users new to coding or SSAMs to learn how these models work in an interactive way to ease access when learning a new research topic and library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Overview The main modelling classes are built on the abstract base class StatisticalModelBase, which has several methods for pre-processing data and performing PCA (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' There are also several global variables that are inherited which are related to principal components, component variances and model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The classes for SSM and SAM pre-process the data (align to zero mean and standard deviation of one) and can compute the population mean shape/appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Finally, the SSAM class for shape and appearance modelling inherits all of these, but also imports the SSM and SAM methods to pre-process shape and appearance features separately, before they are merged into one dataset for modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' StatisticalModelBase SSM StatisticalModelBase SAM SSAM StatisticalModelBase StatisticalModelBase SSM SAM Figure 1: Schematic overview of the codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Each modelling class is abstracted from the Statis ticalModelBase class and contains several inherited variables such as model weights and principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The SSAM class inherits from StatisticalModelBase, but also uses pre-processing pipelines from SSM and SAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Examples Here we present two example applications of pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The first example examines shape variations in a toy dataset created for this study, which has a tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Tree structures appear often in biology, including the lung airways and vascular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Toy datasets such as these are a simple means to visualise and interpret the modelling and code framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' We then provide a more complex example which considers the left lower lobe of human lungs obtained from CT data (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' This example considers shape and appearance, where the appearance is the gray-value at the landmark location on an X-ray projection (obtained with the AppearanceFromXray helper class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Statistical shape modelling toy dataset To understand the shape modelling process, we have provided a dataset class called Tree which creates a number of tree shapes which are randomly computed based on global minimum and maximum values for angle and branch length ratio (between parent and child).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Tree parameters are shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Tree nodes are converted to a numpy array and used to initialise pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' At initialisation of the SSM class, the landmarks are aligned, scaled to unit standard deviation and stacked into a matrix of shape (Nf, 3NL) where Nf is the number of features (samples in our training dataset) and NL is the number of landmarks (each with a x, y, z coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' All y coordinates in this case are zero, meaning the data is actually 2D but we preserve a 3D coordinate system for simplicity in generalising the code to more Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' pyssam – a Python library for statistical modelling of biomedical shape and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Journal of Open Source Software, TBD, TBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='org/TBD 2 common 3D applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The code below shows how we can simply obtain a SSM from a set of landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' from glob import glob import numpy as np import pyssam tree_class = pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='Tree(num_extra_ends=1) landmark_coordinates = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='array( [tree_class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='make_tree_landmarks() for i in range(0, num_samples)] ) ssm_obj = pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='SSM(landmark_coordinates) ssm_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='create_pca_model(ssm_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='landmarks_scale) mean_shape_columnvector = ssm_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='compute_dataset_mean() L1 L2 θ 0 10 20 30 40 Number of components 50 60 70 80 90 100 Variance [%] (a) (b) Figure 2: Overview of tree dataset population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Panels show (a) a visualisation of 100 tree samples, and (b) cumulative variance versus the number of PCA components constructed by the statistical shape model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Inset of (a) shows a legend describing the morphological parameters varied to create the tree dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' These parameters include the initial branch length, L1, the branch length ratio LR = L2/L1, and branching angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Shape and appearance modelling of lung shape and chest X-ray images In the following example, we show a real application where 3D landmark for the left lower lung lobe are projected onto digitally reconstructed X-rays (Väänänen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2015) and the gray-value is used to obtain appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Example landmark data was obtained using an automatic algorithm (Ferrarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Appearance information is extracted from the X-ray images using AppearanceFromXray (part of pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='utils).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' We use landmarks, X-ray images as well as origin and pixel spacing information for the X-ray images to extract appearance as follows appearance_xr = pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='AppearanceFromXray( IMAGE_DATASET, IMAGE_ORIGIN, IMAGE_SPACING ) appearance_values = appearance_xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='all_landmark_density( landmarks_coordinates ) Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' pyssam – a Python library for statistical modelling of biomedical shape and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Journal of Open Source Software, TBD, TBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='org/TBD 3 The SSAM can then be trained in a similar way as the SSM in subsection with the following code snippet: ssam_obj = pyssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='SSAM(landmark_coordinates, appearance_values) ssam_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='create_pca_model(ssam_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='shape_appearance_columns) mean_shape_appearance_columnvector = ssam_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='compute_dataset_mean() The shape and appearance modes can then be computed based on the model parameters (ssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='model_parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The computed model parameters (eigenvectors and eigenvalues of the covariance matrix) can be used to morph the shape and appearance using ssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='morph _model (part of StatisticalModelBase in Figure 1) by x ≈ ¯x + Φ · b (1) where x is a new array containing shape and appearance, ¯x is the training dataset mean shape and appearance, Φ is the model principal components (eigenvectors of the training data covariance matrix), b is the model parameters, which is an array of weights unique to each data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' The model parameter a mode m should be within [−3 � σ2 m, 3 � σ2 m], where σ2 m is the explained variance of m (mth largest eigenvalue of the covariance matrix) (Cootes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=', 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Each mode of shape and appearance variation is visualised, as shown for a representative mode in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' This shows how lung shape influences the gray-value of lung pixels on the X-ray image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' In this case, the change in shape and appearance are mainly due to how the lung interacts with adjacent structures such as the heart, rib cage and diaphragm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Figure 3: First mode of SSAM variation for lung lobe dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Panels show shape and appearance morphed using ssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='morph_model method and varying the model parameters (ssam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='model_parame ters), from -2, 0 (mean shape) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Acknowledgement JW was funded by a 2019 PhD Scholarship from the Carnegie-Trust for the Universities of Scotland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' pyssam – a Python library for statistical modelling of biomedical shape and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Journal of Open Source Software, TBD, TBD.' metadata={'source': 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shape, density, cortical thickness and finite element mesh of proximal femur from a DXA image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Medical Image Analysis, 24(1), 125–134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' pyssam – a Python library for statistical modelling of biomedical shape and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' Journal of Open Source Software, TBD, TBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} +page_content='org/TBD 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfQwmQ/content/2301.04416v1.pdf'} diff --git a/8dFLT4oBgHgl3EQftC_m/content/tmp_files/2301.12150v1.pdf.txt b/8dFLT4oBgHgl3EQftC_m/content/tmp_files/2301.12150v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..10191bb7e6c387d202d5783c3b9817183ca66c3f --- /dev/null +++ b/8dFLT4oBgHgl3EQftC_m/content/tmp_files/2301.12150v1.pdf.txt @@ -0,0 +1,540 @@ +Wrapping pathways of anisotropic dumbbell +particles by giant unilamellar vesicles +Ali Azadbakht,†,∥ Billie Meadowcroft,‡,¶,∥ Thijs Varkevisser,†,§,∥ Anđela Šarić,‡ and +Daniela J. Kraft∗,† +†Soft Matter Physics, Huygens-Kamerlingh Onnes Laboratory, Leiden University, PO Box +9504, 2300 RA Leiden, the Netherlands +‡Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria +¶Department of Physics and Astronomy, Institute for the Physics of Living Systems, +University College London, London WC1E 6BT, United Kingdom +§Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, Science +Park 904, 1098 XH Amsterdam, Netherlands +∥These authors contributed equally to this work. +E-mail: kraft@physics.leidenuniv.nl +Abstract +Endocytosis is a key cellular process involved in the uptake of nutrients, pathogens +or the diagnosis and therapy of diseases. Most studies have focused on spherical ob- +jects, whereas biologically relevant shapes can be highly anisotropic. In this letter, we +use an experimental model system based on Giant Unilamellar Vesicles (GUVs) and +dumbbell-shaped colloidal particles to mimic and investigate the first stage of the pas- +sive endocytic process: engulfment of an anisotropic object by the membrane. Our +model has specific ligand-receptor interactions realized by mobile receptors on the vesi- +cles and immobile ligands on the particles. Through a series of experiments, theory +1 +arXiv:2301.12150v1 [cond-mat.soft] 28 Jan 2023 + +and molecular dynamics simulations, we quantify the wrapping process of anisotropic +dumbbells by GUVs and identify distinct stages of the wrapping pathway. We find that +the strong curvature variation in the neck of the dumbbell as well as membrane tension +are crucial in determining both the speed of wrapping and the final states. +The engulfment of objects through the cell membrane is critical for endocytic processes +such as phagocytosis1–3 and receptor-mediated endocytosis. The latter is often exploited by +viruses for cell entry and proliferation4 and key to nanomedical applications such as drug +delivery and imaging.5 To single out receptor-mediated effects from active mechanisms in- +volved in the engulfment,6 simplified passive model systems can be employed, which recently +led to a conclusive understanding of the wrapping of spherical objects.7,8 However, biological +objects such as bacteria and viruses4,9,10 as well as nanoparticles relevant for applications +in nanomedicine but also nanotoxicology11 often posses non-spherical shapes. Moreover, in +vitro experiments with nanoparticles and simulations have shown that the size and shape +influence their likelihood to be taken up by endocytosis.6,12–17 +The wrapping pathways of spheres at sufficiently low membrane tensions have been shown +to be a continuous transition from attached to fully wrapped, occurring either spontaneously +or after activation.7,8,18 In contrast, anisotropic particles such as ellipsoids and rods, are +expected to reorient during the wrapping process or become trapped in metastable states +due to their varying curvature.19–27 The aspect ratio of these particles as well as the degree of +rounding of their tip were the key parameters affecting the wrapping orientation with respect +to the membrane and their metastable and stable states.24,27 Despite the extensive work in +theory and simulations and exciting observations on shape-dependence in phagocytosis,28 no +experimental work has investigated the passive wrapping process of anisotropic particles by +lipid membranes and tested these predictions yet. +In this letter, we employ an experimental model system based on Giant Unilamellar +Vesicles (GUVs) and colloidal dumbbell particles to investigate the wrapping of micrometre- +sized anisotropic objects by lipid membranes. Our model system is designed to have mobile +2 + +ligands on the vesicles and immobile receptors on the particles mimicking receptor-mediated +endocytotic systems.18,29,30 We quantify the wrapping pathways of anisotropic dumbbells +by lipid membranes and test if their initial orientation affects the final states. Molecular +dynamics simulations of the same system corroborate our experimental data, allowing us to +inspect the dynamics of the process that was inaccessible to experiment. We find that the +strong curvature variation in the neck of the dumbbell as well as membrane tension and not +their initial orientation are crucial in both determining the speed of wrapping and the final +states. +We investigate the wrapping process of anisotropic objects by a lipid membrane using a +model system consisting of GUVs and colloidal particles, (see Fig. 1a). We chose the simplest +object that features anisotropy: a dumbbell shaped colloidal particle that consists of two +equal sized spheres. +The colloid dumbbells were obtained from aggregating polystyrene +spheres with diameter ds=0.98± 0.03 µm31 by briefly lowering the pH to 5.3 and then +quenching the process by increasing the pH to 8.6.32 This process yielded 5-10% dimers with +a long axis of 1.96 ± 0.06 µm and a short axis of 0.98 ± 0.03 µm. GUVs were prepared by +electroswelling from 97.5% w/w 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC). +To realize strong ligand-receptor mediated binding we doped the GUVs with 2% w/w 1,2- +dioleoyl-sn-glycero-3-phosphoethanolamine-N-[biotin-2000] (DOPE-PEG2000-Biotin) and the +dumbbells with 2.2×103/µm2 NeutrAvidin following,31 see Fig. 1b and c and see particle +functionalization and quantification of binding affinity in Supporting Information. We sup- +press electrostatic interactions by working in 50 mM Phosphate Buffered Saline, and achieve +colloidal stability by coating the dumbbells with polyethyleneglycol (PEG5000). Imaging of +the position and orientation of the dumbbells and membranes in three dimensions was made +possible by dying the colloids with BODIPY, represented by a green color throughout the +manuscript, as well as including 0.5% w/w 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine- +N-(lissamine rhodamine B sulfonyl) (DOPE-Rhodamine) into the GUVs, represented by a +magenta color. See Fig. 1c. Confocal stacks and image sequences were acquired with an +3 + +inverted Nikon TI-e microscope, equipped with a 60x (NA 1.2) objective and A1-R scan +head. 2D image sequences were taken at 59 fps, which enables tracking of the dumbbells in +real time. Experimental details are described in the Supporting Information. +To initiate the wrapping process, we used optical tweezers to bring dumbbell particles in +contact with the GUV. They subsequently diffused on the GUV surface before suddenly and +quickly becoming wrapped, a process that took between a few seconds and a few minutes +depending on membrane tension, see Figure 1e and Movie S1. To capture the wrapping pro- +cess with high speed, we adjusted the focal height during acquisition of the image sequence. +After wrapping, the dumbbell continued to diffuse on the inside of the vesicle. +We quantify the wrapping process of a dumbbell by measuring the angle θ between the +major axis of the dumbbell and surface normal of the GUV and distance d of the dumbbell +with respect to the undistorted surface of the GUV, see Figure 1d. We inferred the 3D +position of the dumbbell from the position of its lobes with respect to the GUV. To improve +the accuracy of tracking, particles were tracked only when their center of mass was between +-0.8R basal +slip (29 ± 5 MPa), pyramidal slip (203 ± 7 MPa) and tensile twin nucleation +(above 148 MPa), while the CRSS for prismatic slip only increases up to 105 ± 4 +MPa. The changes in the CRSS for slip and tensile twinning in Mg-Y-Ca alloys +expectedly modify the dominant deformation mechanisms in polycrystals. In particular, +tensile twinning is replaced by prismatic slip during compressive deformation +along the a-axis. The reduction of twinning (which generally induces strong anisotropy +in the plastic deformation in textured alloys), and the activation of prismatic slip +(which provides an additional plastic deformation mechanism with limited hardening) +were responsible for the large tensile ductility of the alloy. + +Keywords: Mg-Y-Ca alloys; micropillar compression; critical resolved shear stress; +plastic anisotropy; tension-compression asymmetry; tensile ductility. + +1. Introduction +Pure Mg and Mg alloys generally present poor ductility and formability, especially +at room temperature (Huang et al., 2022; Sun et al., 2019; Tang et al., 2022; Yaghoobi +et al., 2022). As a result, forming of rolled sheets and extruded bars becomes difficult +and limits the application of wrought Mg alloys in different industrial sectors (Li and +Fang, 2022). Thus, understanding the origin of the lack of ductility and formability is +of paramount importance to develop new Mg alloys that overcome these limitations. +The poor ductility of Mg alloys is primarily traced to its low-symmetry hexagonal +closed packed (HCP) lattice structure, which results in very large differences in the +critical resolved shear stress (CRSS) between basal and non-basal slip systems as well +as in the easy activation of tensile twinning (Lee et al., 2018). Plastic deformation in +pure Mg is initially accommodated by basal slip, which only provides two +independent slip systems (Partridge, 1967). This process leads to the development of a +strong basal texture during rolling and extrusion. Moreover, plastic deformation along +the c-axis (which is necessary to activate five independent slip systems to fulfil the von- + + +3 +Mises criterion for homogeneous plastic deformation) is absorbed by tensile twinning, +which is triggered at much lower CRSS than that necessary to produce pyramidal +slip (Graff et al., 2007; Sukedai and Yokoyama, 2010). However, the plastic strain +associated with tensile twinning is very limited (at most 7%), moreover, tensile twining +is a polar mechanism that only occurs when the stress along the c-axis of the crystal is +tensile (Mayama et al., 2011). This leads to a large buildup of stresses to activate +pyramidal slip in grains that are not suitably oriented for twinning and/or that cannot +accommodate more plastic deformation by twinning (Obara et al., 1973; Reed-Hill and +Robertson, 1957). The stress concentrations in these grains facilitate the nucleation of +cracks and limit the ductility (Zhang et al., 2022). Moreover, huge differences in the +flow stress and the strain hardening rate between tension and compression appear in +textured microstructures, which also lead to fracture during bending and forming +operations (Agnew and Duygulu, 2005; Basu et al., 2021). +The strategies to improve ductility and formability of Mg alloys have been directed +towards promoting the activation of multiple slip, including non-basal and non- +basal slip, and to suppress deformation twinning. Multiple slip leads to more +homogeneous plastic deformation and limits texture development during rolling and +extrusion while twinning promotes plastic anisotropy in textured microstructures +(Ahmad et al., 2019; G. Liu et al., 2017; Zhang et al., 2016a). For instance, precipitation +hardening in Mg-Zn alloys leads to large enhancements in the CRSS for basal (Alizadeh +and LLorca, 2020; Chun and Byrne, 1969; Wang and Stanford, 2015) and pyramidal +slip (Alizadeh et al., 2021) and, thus, to an important reduction in the pyramidal-to- +basal CRSS ratio. Nevertheless, the large increase in flow stress inherently decreases +the ductility due to the strong accumulation of geometrically necessary dislocations +around the precipitates (Rosalie et al., 2012). In addition, precipitates also increase the +CRSS for twin growth but do not affect the CRSS for twin nucleation (Wang et al., +2019b). As the latter is normally higher than the former, the presence of precipitates do +not contribute to hinder the development of twinning. The only difference induced by +the precipitates is a larger number of smaller twins, as compared to the precipitate-free +condition (Stanford et al., 2012). Thus, precipitate is not very efficient to enhance the + + +4 +ductility of Mg alloys (Fu et al., 2019; Jain et al., 2010). +Strategies based on solid solution hardening have been more successful to improve +the ductility of Mg alloys if the alloying elements are properly chosen. For instance, +Sandlöbes et al. (Sandlöbes et al., 2013, 2012, 2011) reported that the addition of 3 wt. % +Y led to Mg alloys with a tensile ductility > 25 %, which was associated with the +presence of a large density of pyramidal dislocations in the deformed sample. +This behavior was mainly attributed to a reduction in the ratio between the CRSS of the +< c+a > pyramidal slip and < a > basal slip, which was ~3.2 according to in situ high +energy X-ray diffraction tests (Huang et al., 2018; Wang et al., 2018) and ~2.8-4.8 from +micropillar compression tests (Wu et al., 2020). Large ductility and formability are not +achieved, however, by the addition of other elements in solid solution (such as Al or Zn) +because the pyramidal-to-basal CRSS ratio in these alloys are > 10 (Li et al., 2021a; +Wang et al., 2020). Zhu et al., (2019) found that the addition of 0.47 wt. % of Ca in +solid solution enhanced the activity of prismatic and pyramidal I dislocations as +well as the cross-slip between basal and non-basal slip planes, improving the tensile +ductility to ~18 % in a Mg-0.47 Ca (wt. %) alloy. And several authors reported a large +improvement in the ductility of binary Mg-Zn and Mg-Al alloys through the addition +of small amount of Ca (Hofstetter et al., 2015; Sandlöbes et al., 2017; Wang et al., +2021b). This behavior was supported by our recent micropillar compression tests that +showed that the addition of Ca to Mg-Zn alloys reduced the pyramidal-to-basal CRSS +ratio values, that were similar to those found in Mg-Y alloys (Wang et al., 2021a). +Finally, Wu et al., (2018) showed that the presence of Y and Ca reduces the energy for +cross-slip/double cross-slip of pyramidal dislocations, leading to new dislocation +loops which accommodate plastic deformation. In contrast, the cross-slip is inhibited in +pure Mg (or in Mg-Al and Mg-Zn alloys) (Wu et al., 2018), by the favorable +dissociation of edge pyramidal dislocation segments into sessile segments in the +basal plane. +Regarding the effect of solid solution on tensile twinning, several investigations +reported an increase in the CRSS for twin nucleation and growth with the addition of +Al (Wang et al., 2020), Zn (Li et al., 2021a), Y (Li et al., 2021b) as well as Ca to Mg- + + +5 +Zn alloys (Wang et al., 2021a). However, the CRSS for twin nucleation and growth +were lower than that for pyramidal slip in the corresponding alloy, thus, tensile +twinning was still preferred over pyramidal slip to accommodate plastic deformation in +grains suitable oriented for twinning. In addition, the addition of 4Y (wt. %) could +significantly suppress the tensile twinning (with CRSS larger than 113 MPa) and +promote the dislocations (with CRSS around 106 MPa) (Wu et al., 2020). The +results summarized above point to the beneficial effects of Y and Ca in solid solution +to reduce the plastic anisotropy of Mg. Thus, the co-addition of Ca and Y is expected +to promote the homogeneous deformation and improve the plastic deformability of Mg +alloys, taking advantages of the significant suppression effect of Y on the tensile +twinning, the promotion effect of Ca on the non-basal slips, simultaneously the +positive effect of Ca and Y on the activation of the slips. Ca enhances the +activation of prismatic and pyramidal slip while Y has similar effects on +slip. Moreover, experimental results on the tensile behavior of an extruded Mg – 2.4 +wt. % Y – 0.3 wt. % Ca (Zhou et al., 2013) showed a very large tensile elongation +(~37 %) but there is not information available in the literature -to the authors’ +knowledge- on the concurrent effects of Y and Ca in solid solution on the dominant +deformation mechanisms and this is the main objective of this investigation. Thus, the +CRSS for different slip systems and twinning was determined in a Mg-Y-Ca alloy from +micropillar compression tests in single crystals with different orientations. The +deformation mechanisms were ascertained from slip trace analysis in the scanning +electron microscope (SEM), transmission electron microscopy (TEM) observations of +the dislocations as well as transmission Kikuchi diffraction (TKD). This information +was used to rationalize the excellent ductility of Mg-Y-Ca and to provide guidelines to +design novel Mg alloys with improved ductility and formability. + +2. Materials and experimental techniques +2.1 Materials +The Mg-Y-Ca alloy was prepared from pure Mg (99.99 wt. %), Mg-30 Ca (wt. %) +and Mg-30 Y (wt. %) master alloys in a resistance furnace under a protective + + +6 +atmosphere of CO2 and SF6. The actual chemical composition of the ingot, obtained by +inductively coupled plasma atomic emission spectroscopy, was Mg-5Y-0.08Ca (wt. %). +The cast alloy was solution treated at 400 ℃ for 12 h, followed by extrusion at 300 ℃ +with an extrusion ratio of ~ 18:1. Afterwards, parallelepipedal samples of 10×10×5 mm3 +were cut from the extruded specimens and homogenized at 550 ℃ for 20 days within +quartz capsules filled with Ar to induce grain growth. +2.2 Experimental techniques +Tensile and compressive tests were carried out along the extrusion direction in +polycrystalline specimens at crosshead speed of 0.5 mm/min, using a universal testing +machine (Z100-TEW) at room temperature. The dimensions of the gage section of the +dog-bone tensile specimens were 18×3.4×1.4 mm3 (length × width × thickness), while +cylindrical specimens of 8 mm in diameter and 12 mm in length were used in the +compression tests. Deformation was measured with an extensometer and 3 specimens +were tested in each condition. +The crystallographic orientation of the grains in the sample was characterized via +electron back-scattered diffraction (EBSD) in a Tescan Mira-3 SEM with an Oxford +Instruments Nordlys EBSD detector at an accelerating voltage of 20 kV. The surface of +the sample was mechanically ground using abrasive SiC papers with a grit size of 1200, +2000, 3000, 5000 and 7000. Subsequently, the sample surface was electropolished in +an ethanol solution with 10 (vol. %) perchloric acid at -30 ℃ and 30 V for 90 s to +remove the surface damage induced by grinding and reveal the grain boundaries. The +EBSD data were analyzed using the Channel 5 software and the Oxford Instruments +AZtec Nanoanalysis software package v6.0 along with AZtec Crystal. Several grains +whose orientations were appropriate to active different deformation modes were +selected to mill the micropillars. +Micropillars of 5 × 5 μm2 square cross and an aspect ratio 2:1 were milled from +the selected grains using a FEI Helios G4 UX Focused Ion Beam (FIB)/SEM dual beam +microscope operated at 30 kV. These dimensions are known to minimize size effects +during mechanical deformation while the time and effort to mill each micropillar are +reasonable (Wang et al., 2021a). An initial ion current of 9.3 nA was used to remove the + + +7 +surrounding material and it was reduced to 2.5 nA when the beam was getting closer to +the actual dimensions of the micropillar. A final ion current of 80 pA was used in the +final polishing step to minimize the surface damage due to Ga+ ion-implantation. The +final taper of the micropillars was < 1.5°. +Micropillar compression tests were performed in ex situ using a Hysitron +Triboindenter TI950 system though a diamond flat punch of 10 μm in diameter. All the +tests were conducted under displacement control up to a maximum strain of 10 % at a +nominal strain rate of 10-3 s-1. The experimental displacement was corrected to account +for the elastic deflection of the matrix material beneath the micropillars following the +Sneddon correction (Sneddon, 1965). To this end, the elastic modulus of each grain was +determined via the nanoindentation method with a Berkovich tip in the same grain +where the micropillar was milled. More details about micropillar manufacturing and the +compression set-up can be found in (Sneddon, 1965; Wang et al., 2021a). +The engineering stress-strain curves were obtained from the load and the corrected +elastic deflection of the micropillar using the initial cross-sectional area and the height +of the micropillars measured in the SEM. The yield stress, σy , was determined from +the loss of linearity in the stress-strain curve following the methodology described in +(Alizadeh and LLorca, 2020; Maaß et al., 2009). From this information, the CRSS of +the active slip system was determined as +CRSS = SF × σy (1) +where SF is the Schmid factor of the corresponding slip system, computed from the +crystallographic orientation of each crystal (Table 1). +The slip traces on the top and lateral surfaces of the deformed micropillars were +characterized in a Tescan Mira-3 SEM to ascertain the active slip planes. The active slip +plane and direction were identified from the micropillar orientation using VESTA +software (Momma and Izumi, 2008). Moreover, TEM and TKD were used to determine +the dislocation activity and the orientation of the micropillar after deformation. To this +end, a thin lamella was lifted-out along the loading direction from the deformed pillars +and thinned to < 100 nm in thickness using FIB. The TKD maps were collected in a + + +8 +Tescan Mira-3 SEM at 30 kV with a step size of 20 nm. The TEM observations were +carried out using a Talos F200X G2 microscope operated at 200 kV. The two-beam +condition was applied to obtain dislocation contrast. Moreover, the “g·b” visibility +criterion was used to identify the types of dislocation, i.e., the dislocation is in contrast +when g!⃗ · b!⃗ ≠ 0, where g!⃗ is the diffraction vector and b!⃗ the Burgers vector. +2.3 First-principles calculations +In order to study the influence of Y and/or Ca atoms on the deformation +mechanisms in Mg alloys, the generalized stacking fault energy (GSFE) curves of +different slip systems were calculated via the first-principles calculations using the +Vienna Ab initio Simulation Package (VASP) (Kresse and Furthmüller, 1996). The +exchange-correlation function was described using the generalized gradient +approximation (GGA) with the Perdew-Burke-Ernzerholf functional (PBE), based on +the projector augmented wave (PAW) (Blöchl, 1994) method. +A supercell with 12-layers containing 48 atoms was defined for different slip +systems, as indicated in Fig. 1. Each supercell was separated by 15 Å vacuum to +eliminate the influence of the periodic boundary conditions. The formation energy was +initially calculated for different positions of the solute atoms and the configurations +with lower formation energy was selected as the most stable ones (Yuasa et al., 2014). +In the binary Mg47N1 (N = Y, Ca) alloys, the most stable configuration was found when +one Mg atom at the center site of the stacking fault plane was substituted by a solute +atom X. In the ternary Mg46N1X1 (N = Y, and X = Ca) alloy, the most stable +configuration was found when one Mg atom at the center site of the stacking fault plane +was substituted by a Ca atom. Then, one of the eleven nearest Mg atoms from the Ca +atom was substituted by one Y atom, as shown in Fig. S1 in the supplementary material. +The exact position of the Y atom was determined from the formation energy (Ding et +al., 2019; Dong et al., 2018). The formation energies for every occupancy of the Y atom +are listed in Table S1 in the supplementary material. +The conventional direct crystal slip methods were employed to obtain the GSFE +curves of different slip systems The perfect supercell was cut into two free parts and + + +9 +one part was displaced with respect to the other one along the slip direction. The atomic +positions were relaxed only along the direction perpendicular to the stacking fault plane +(Wang et al., 2020). A residual force threshold of 0.01 eV/Å was performed in all +geometric relaxations until the electronic energy converged to less than 10-5 eV/cell. +The Brillouin zone for the GSFE of the basal slip system, the prismatic slip system, and +the pyramidal slip system was set as 8×8×1, 10×6×1, and 6×10×1, respectively, with +an energy cutoff of 480 eV (Dong et al., 2018; Wang et al., 2013). + + +Fig. 1. Schematic illustration of the models to calculate the GSFE for (a) basal slip (b) +prismatic slip, and (c) pyramidal Ⅰ slip. The most stable positions of Y and Ca atoms +determined by the lowest formation energy are marked by blue and purple atoms, +respectively. Stacking fault planes are noted by the dotted lines. + +3. Results +3.1 Mechanical behavior of polycrystals +The inverse pole figure (IPF) map of the as-extruded Mg-Y-Ca alloy along the +(a) basal slip +(b) prismatic slip +(c) pyramidalⅠslip +[11!00] +[0001] +[112!0] +[101!1] +[112!0] +[112!3] +105° +[112!0] +[11!00] +[0001] +Mg +Ca +Y + +: +O +O +O +O +10 +extrusion direction is plotted in Fig. 2a. The {0001} pole figure shows that the Mg-Y- +Ca alloy possesses a weak texture with a strength of ~ 8.21 mrd, as displayed in Fig. +2b, compared to pure wrought Mg with a strong basal texture of >15 mrd (Yin et al., +2021). The engineering stress-strain curves of the extruded Mg-Y-Ca alloy from the +tensile and compressive tests parallel to the extrusion direction are plotted in Fig. 2c. +The scatter was very limited and the average tensile elongation was very large (≈ 32%). +Moreover, the tensile yield stress was 104 MPa, very close to the yield strength in the +compression tests (122 MPa). Thus, the Mg-Y-Ca alloy presented very low +tension/compression asymmetry in the yield strength in contrast with the marked +asymmetry in extruded Mg and Mg alloys (Sukedai and Yokoyama, 2010; Yin et al., +2021; Zhang et al., 2016b).1 It should also be noted that volume fraction of the twinned +material after tensile deformation was very low (≈ 1.8%), indicating that twining was +not a dominant deformation mechanism in the Mg alloy. + +Fig. 2. (a) IPF map of the Mg-Y-Ca along the extrusion direction. (b) {0001} Pole figure +of the Mg-Y-Ca alloy illustrating the texture characteristics before the deformation in +the TD-ED plane. (c) Engineering stress-strain curves in tension and compression +parallel to the extrusion direction of the Mg-Y-Ca alloy. + +3.2 Deformation mechanisms + +1 The comparison between both curves shows the limited tension-compression anisotropy in the yield +strength but the differences in the elastic and fully plastic regions are due to the limitations of the +compression tests. Compression tests always underestimate the elastic modulus because it is very difficult +to ensure that the specimen surface and the loading plate surface are perfectly parallel. Thus, partial +contact between both surface leads to localized plastic deformation and to an apparent elastic modulus +that is lower than the real one. Moreover, barreling of the cylindrical specimen during compression leads +to non-homogeneous plastic deformation and overestimates the strain hardening for large plastic strains. +50μm +TD +ED +Max=8.21 +ED∥ Tensile direction +8.21 +0.00 +(a) +(b) +(c) +(c) + +400 +Tension +Compression +300 +200 +0 +10 +20 +30 +40 +Engineering strain (%)Tscedan-(0001) -Magnesium +8.21 +则量计数：100008 +Subset1 +半宽：10.0* +样品对称性：三料 +使用样本对疗性：数量 +投射类型：等围积 +透射平面：XY +率球：上 +00'0 +11 +The IPF map with the crystallographic orientation of the grains in the Mg-Y-Ca +alloy is depicted in Fig. 3. The grains were larger than 150 μm, and the micropillars +were milled from the center of the grains to ensure that they were single crystals. Four +grains with appropriate orientations (Fig. 3) were selected to activate different +deformation mechanisms. The loading directions in the four grains are listed in Table 1, +as well as the maximum Schmid Factor (SF) for the corresponding slip systems ( +basal slip, prismatic slip, pyramidal Ⅰ slip, pyramidal Ⅰ slip and +pyramidal Ⅱ slip) as well as {101$2} tensile twinning. The inclination angle in Table 1 +indicates the angle between the c-axis of each grain and the compression direction, as +presented. The compression direction is nearly parallel to [112$0], [101$0], and [0001] in +grains B, C and D, respectively, and forms an angle of ~ 48.5° with respect to [0001] +axis in grain A. Herein, grain A presents the highest SF for basal slip, which is +prone to be the dominant deformation mechanism during compression. Plastic +deformation along the pyramidal I and II systems is favored in Grain D. +prismatic and pyramidal as well as pyramidal slip systems have similar SFs in +grain B, while grain C is suitably oriented to promote tensile twinning and +prismatic slip. + +Table 1. The loading direction, inclination angle, elastic modulus, and maximum +Schmid factor for each slip system and tensile twinning in the selected grains. +Grain +Loading +direction +Inclination +angle (°) +Elastic +modulus +(GPa) +Maximum Schmid factor +Basal + +Prismatic + +Pyramidal +Ⅰ +Pyramidal +Ⅰ +Pyramidal +Ⅱ +Tensile +twin +A +[112!3] +48.5 +46.04 +0.44 +0.25 +0.42 +0.36 +0.20 +0.17 +B +[112!0] +83.4 +48.14 +0.11 +0.48 +0.46 +0.47 +0.47 +0.43 +C +[101!0] +87.8 +46.48 +0.03 +0.46 +0.42 +0.43 +0.37 +0.49 +D +[0001] +4.5 +47.47 +0.06 +0.00 +0.03 +0.44 +0.47 +-* +*Tensile twinning cannot be activated during compression along the c-axis. + + +12 + +Fig. 3. Inverse pole figure (IPF) map showing the crystallographic orientation of the +grains in the Mg-Y-Ca alloy. (a) The loading direction in micropillars from grains A and +B form an angle of ~ 48.5° with the [0001] crystal orientation and are parallel to [112$0], +respectively. (b) The loading direction in micropillars from grains C is parallel to [101$0]. +(c) The loading direction in micropillars from grain D is parallel to [0001]. The +compression loading direction is perpendicular to the paper. + +3.2.1 Deformation mechanisms in micropillar of grain A +The engineering stress-strain curves obtained from the compression micropillars +carved from grain A along [112$3] orientation are plotted in Fig. 4a. For the sake of +clarity, the horizontal axis of the green curve in Fig. 4a is shifted by 0.5%. After the +initial elastic region, the curves show gradual yielding and reach a plateau in the flow +stress at an applied strain of ~ 5%, without significant work hardening afterwards. This +behavior is consistent with a plastic deformation dominated by basal slip in pure Mg +and Mg alloys (Kiener et al., 2021; Y. Liu et al., 2017; Luo et al., 2022; Wang et al., +2020, 2019a; Wu et al., 2020). Small strain bursts (noticed by sudden drops in the stress) +are present in the stress-strain curves and they are associated with the activation of +dislocation sources in particular basal slip planes. However, the magnitude of the strain +400μm +200μm +011!0 +0001 +1!21!0 +Loading direction +200μm +Grain A +Grain B +Grain C +Grain D +(a) +(b) +(c) + + +13 +bursts is much smaller than that reported in other Mg alloys. In fact, large strain bursts +are associated with the localization of deformation in a few slip planes along the +micropillar (Wang et al., 2019). However, the lateral and top views of the micropillar +after deformation (Figs. 4c and 4d, respectively) show evidence of uniform slip traces +along the length and width of the micropillar, indicating that plastic deformation was +homogeneous. A yield stress of 65 ± 11 MPa (indicated by the black stars in the inset +of Fig. 4a) was determined from the critical points in the engineering stress-strain +curves when the curves deviated from linearity, following the procedure detailed in +(Alizadeh and LLorca, 2020; Wang et al., 2019a). +Secondary electron images of lateral and top views of the deformed micropillars +were obtained in the SEM to ascertain the actual deformation mechanisms and are +shown in Figs. 4c and 4d, respectively. Many parallel slip traces appear on the top and +lateral surfaces, which were not present before deformation (Fig. 4b). The orientation +of the slip traces on the micropillar surfaces is indicated by the green dashed lines in +Figs. 4c and 4d. The slip steps were obviously observed on the top view from the top +right corner to the lower left corner, and the corresponding slip direction is determined +as marked with a white arrow in Fig. 4d. They are indicated by blue planes and red +arrows, respectively, in Figs. 4e and 4f within the crystallographic lattice. It is evident +that the slip traces in the micropillar are parallel to the basal planes and the shear +deformation takes place along the [21$1$0] direction, as shown from the top and lateral +views of the deformed micropillar. In fact, the (0001) <21$1$0> basal slip system has +highest SF (listed in Table 1) and plastic deformation along this slip system is dominant +in this micropillar. Therefore, the CRSS for basal slip (based on the yield stress and +the corresponding SF) can be estimated as 29 ± 5 MPa. + + +14 + +Fig. 4. (a) Engineering stress-strain curves obtained from micropillar compression tests +in grain A. The yield stress is marked with a black star. SEM images of the micropillar +(b) before compression and after compression from the (c) right lateral view and (d) top +view. The slip plane trace and slip direction are indicated by the green dashed lines and +the white arrow, respectively. The schematic crystallographic lattice of the +corresponding slip plane is presented (e) for right side and (f) for top side. The blue +planes indicate the theoretical basal glide planes, and the red arrows represent the +corresponding shear directions. + +3.2.2 Deformation mechanisms in micropillars of grains B and C +Representative engineering stress-strain curves obtained from micropillar + +150 +Engineering stress (MPa) +Top side +100 +:0 +50 +Right +side +0 +2um +3 +6 +9 +12 +Engineering strain (%) +Right side +Top side +Slip +Tr.Basal plane +direction +Tr.Basal plane +2μm +2μm +Basal slip plane +Basal slip plane +15 +compression tests along [112$0] in grain B and along [101$0] in grain C are plotted in +Figs. 5a and 5b, respectively. The horizontal axis of the green and blue curves was +shifted by -0.1% and +0.1%, respectively, in the inset of Fig. 5b for the sake of clarity. +The stress-strain curves are smooth, without distinct strain bursts. The initial elastic +region is followed by another linear plastic region with reduced strain hardening rate. +This behavior is radically different from that observed in micropillars with equivalent +orientation in pure Mg and several Mg alloys (Mg-Al, Mg-Zn, Mg-Y and Mg-Zn-Ca) +(Li et al., 2021b, 2021a; Y. Liu et al., 2017; Wang et al. 2021a, 2020; Wu et al., 2020), +which presented large strain bursts after the initial elastic region due to the nucleation +of tensile twins at the top of the micropillar. They are similar to those found in Mg-2Y +(wt. %) alloy at 250 ℃ (Li et al., 2021b), where prismatic slip replaced twinning +as the dominant plastic deformation mechanisms. The yield stresses (obtained as +indicated above and marked with purple stars in Fig. 5) were 219 ± 9 MPa and 228 ± +4 MPa along [112$0] and [101$0] orientations, respectively. + +Fig. 5. (a) Engineering stress-strain curves obtained from micropillar compression tests +in grain B along [112$0]. (b) Idem in grain C along [101$0]. + +The representative morphology of the micropillar deformed along [112$0] (grain B) +is depicted in the SEM images in Figs. 6a and 6b from two different sides (front and +left, respectively). Faint slip traces are visible on both lateral surfaces of the deformed +micropillars, as indicated by the blue dashed lines in Figs. 6c and 6d, which show the +rectangular zones marked by dashed lines in Figs. 6a and 6b, respectively, at higher +magnification. The slip traces are distributed homogeneously along the lateral surfaces, +Grain B:[112!0] +(a) +(b) +Grain C:[101!0] + +350 +300 +250 +200 +50 +100 +50 +0 +2 +4 +6 +8 +10 +12 +0 +Engineering strain (%)240 +220 +200 +0.8 +1.3 +1.8280 +230 +180 +1.2 +1.7 +2.2350 +Grain C:10101 +300 +D +250 +stress +200 +150 +100 +Engineering +strain +16 +indicating that plastic deformation was uniform along the micropillar. Moreover, there +are not slip steps at the surface (as opposed to the micropillar deformed along [112$3] in +Figs. 4c and 4d), in agreement with the smooth stress-strain curves. This deformation +morphology is different from that observed in other Mg alloys compressed along a-axis +(Li et al., 2021b, 2021a; Y. Liu et al., 2017; Wang et al., 2020; Wu et al., 2020), where +two regions with different contrast were always observed after the deformation due to +the nucleation of tensile twins. + +(a) +2μm +Front side +Left side +2μm +(c) +(b) +1μm +Tr. Prismatic +plane +(d) +1μm +(h) +(i) +Prismatic slip plane +Left side +Front side +2μm +4 +0 +2μm +2μm +(e) +(f) +(g) +Tr. Prismatic +plane +Before +deformation +After +deformation + +KAM +Kernel Aver. Misorient. +0 +[o] +3.96 +Y1 +5μm +光栅：498x331 +步长尺寸：0.025μm +>X12μm +光栅：243x1692 +17 +Fig. 6. SEM images of the micropillar deformed along [112$0] from grain B. (a) Lateral +front and (b) lateral left view side. The traces of the active slip planes are indicated by +the blue dashed lines in Figs. 5c and 5d, which show the rectangular zones marked by +dashed lines in Figs. 5a and 5b, respectively, at higher magnification. (e) and (f) TKD +maps of the lamella extracted from the undeformed region in grain B and along the +compression direction from the deformed micropillar, respectively. (g) KAM map of +the deformed micropillar. (h) and (i) Schematics of the crystallographic lattice showing +the corresponding slip plane for the lateral front side and left side, respectively. The red +planes indicate the theoretical prismatic glide planes, and the blue lines represent the +corresponding slip traces. + +In order to identify the deformation mechanisms, two parallel thin foils were +extracted from the undeformed region in grain B and along the loading direction from +the deformed micropillar, respectively, and their orientation was determined by TKD. +The position of the lamellae is indicated in Fig. S2 of the supplementary material. The +corresponding orientation maps in Figs. 6e and 6f show that the IPF map (∥Z) of the +undeformed and deformed thin foils share the same orientation and demonstrate that +tensile twins were not nucleated during micropillar compression up to 10% strain. +Moreover, Fig. 6g presents the kernel average misorientation (KAM) map of the whole +pillar in Fig. 6g reveals the homogeneous deformation without shear bands assuming +an angular threshold of 4°. The slip traces on the lateral surfaces of the micropillars +were associated with the prismatic planes, as indicated in Figs. 6h and 6i. Thus, +prismatic slip was triggered at the onset of the yielding and dominated plastic +deformation. The maximum SFs for prismatic slip, pyramidal I, +pyramidal II slip and tensile twinning were very similar along this orientation (Table 1) +but the presence of Y and Ca in solid solution favored the activation of prismatic slip. +It should be noted that pyramidal I slip dominated plastic deformation and +hindered the development of tensile twinning in micropillar compression tests along +[1$21$0] orientation in a Mg-4Y (wt. %) (Wu et al., 2020). The maximum SFs for +pyramidal I slip, tensile twinning and prismatic slip in this orientation were 0.41, +0.46 and 0.49 and, thus, the preference of pyramidal slip can be associated with +the higher CRSSs for tensile twin nucleation and prismatic slip in Mg-4Y alloy +(Wu et al., 2020). + + +18 +Similar deformation morphology was found in the micropillars deformed along +[101$0] in grain C. Continuous slip traces were homogeneously distributed along the +lateral surfaces, as indicated by the dashed blue lines from in Fig. 7b, which shows the +rectangular region marked by dashed lines in Fig. 7a at higher magnification. As in the +previous case, the micropillar orientation before and after deformation was assessed by +TKD carried out in a thin lamella extracted from the undeformed region (Fig. 7c) and +from the deformed micropillar along the loading direction (Fig. 7d), respectively. The +relative orientation between the two thin lamellas is shown in Fig. S3 in the +supplementary material. The corresponding orientation maps do not show any evidence +of tensile twinning and prismatic slip was again the dominant plastic deformation +mechanism. This conclusion is supported by the uniform plastic deformation without +obvious shear bands revealed by the KAM map assuming an angular threshold of 2° +(Fig. 7e) and the agreement between the slip traces on the lateral surfaces with the +orientation of the prismatic planes in the micropillar (Fig. 7f). Thus, the CRSSs for +prismatic slip (obtained from the yield stress and the SF for both micropillar +orientations) were determined to be 105 ± 4 MPa and 105 ± 2 MPa along [112$0] and +[101$0] orientations, respectively. + + +19 + +Fig. 7. SEM images of the micropillar deformed along [101$0] from grain C. (a) Lateral +left view side and (b) which shows the rectangular zone marked with a dashed line in +Fig. 7a at higher magnification. The traces of the active slip planes are indicated by the +blue dashed lines. (c) and (d) TKD maps of the lamella extracted from the undeformed +region in the grain and along the compression direction from the deformed micropillar, +respectively. (e) KAM map of the deformed micropillar in grain C. (f) Schematic of the +crystallographic lattice showing the corresponding slip plane for the lateral left view in +(a) and (b). The red plane indicates the theoretical prismatic glide plane, and the blue +2μm +(e) +Prismatic slip plane +Left side +2μm +(c) +1μm +Left side +Tr. Prismatic +plane +(a) +(b) +2μm +(d) +(f) +2 +0 +2μm +Before +deformation +After +deformation + +2KAM +Kernel Aver. Misorient. +光栅：164x268步长尺寸：0.03μm +Y1 +2μm +?X1IPF +IPF Coloring II ZO +Magnesium +0001 +-12-10 +01-10 +Y1 +2μm +光栅：220x145 +步长尺寸：0.04μm +>X1 +20 +line represents the corresponding traces. + +Further assessment of the deformation mechanisms was carried out by means of +TEM observations of the dislocation structures in a thin lamella extracted from the +micropillar deformed along [101$0] (Fig. 8). The lamella was nearly parallel to (1$21$0) +plane, as confirmed by the SADP in the inset in Fig. 8a, and there are no traces of +twinning in the micropillar. Dark field micrographs of the square region marked in Fig. +8a are depicted in Figs. 8b and 8c with g = (101$0) and g = (0002), respectively. Large +density of dislocations is observed in Fig. 8b but they disappear from this region when +g = (0002) in Fig. 8c. They are obviously dislocations with 1/3 a [112$0] or 1/3 +[21$1$0] Burgers vector, based on the dislocation extinguish condition. However, the SF +of the {011$0} [21$1$0] prismatic slip system is very low (~0.05), thus, it is reasonable to +assume that the Burgers vector of the dislocations in Fig. 8b is 1/3 [112$0]. The +screw dislocations are observed under g = (101$0) condition as marked with yellow +arrows in Fig. 8b. The Burgers vector of screw dislocation is parallel to the dislocation +line, leading to the straight dislocation lines nearly parallel to trace of the basal planes +(marked with a green line). + + +21 + +Fig. 8. TEM micrographs of the lamella extracted from the micropillar deformed along +[101$0]. The beam direction is parallel to [1$21$0] orientation. (a) Low magnification view +of the lamella. (b) and (c) High magnification dark field micrographs with g = (101$0) +and g = (0002), respectively, from the region marked with a blue square in (a). + +The activation of the prismatic slip during compression along the a-axis has +been reported recently in Mg-Zn-Ca alloy (Wang et al., 2021b) in combination with +tensile twinning. However, the activation of prismatic slip and the suppression of +tensile twinning during compression along the a-axis has not been found at ambient +temperature in pure Mg (Y. Liu et al., 2017) or any Mg alloys (Li et al., 2021b, 2021a; +Wang et al., 2021a, 2020; Wu et al., 2020). This result is very surprising because +compression of Mg and its alloys along the a-axis (or equivalent extension along the c- +axis) easily leads to the nucleation and growth of {101$2} tensile twins, because the +associated CRSS to promote tensile twin is much lower than that necessary to activate +5μm +(a) +200nm +200nm +(b) +(c) +g=(101!0) +g=(0002) + dislocations +(0001) plane +B=[1!21!0] + + +22 + pyramidal slip or prismatic slip. While the addition of Y and Ca in solid +solution leads to a large increase in the CRSS for prismatic slip with respect to pure +Mg (from 39 MPa in pure Mg (Kaya, 2013) to 105 MPa), it seems to have a much larger +effect on the CRSS for twin nucleation. In fact, considering the maximum stresses +attained in the micropillar compression tests along [112$0] and [101$0] orientations (258 +MPa in Fig. 5a and 303 MPa in Fig. 5b, respectively) and the maximum SFs for tensile +twinning in both orientations (Table 1), it can be estimated that the CRSS for twin +nucleation in the Mg-Y-Ca alloys should be higher than 148 MPa. + +3.2.3 Deformation mechanisms in micropillar of grain D +The engineering stress-strain curves obtained from the compression micropillars +carved from grain D along [0001] orientation are plotted in Fig. 9a. After the elastic +region, a strong linear hardening was observed in the plastic region. The yield stress +(marked by the purple stars in the inset) was 431 ± 15 MPa. This mechanical response +is in good agreement with the results reported in Mg-0.4Y (wt. %) and Mg-4Y (wt. %) +alloys (Wu et al., 2020) as well as in precipitation-hardened Mg-4Zn (wt. %) alloy +(Alizadeh et al., 2021) under c-axis compression. In all these cases, the presence of Y +in solid solution or of β1 +' precipitates increased the CRSS for basal slip and plastic +deformation was accommodated through pyramidal slip due to the low SF of +basal planes in this orientation. The SEM micrograph of the lateral side of deformed +micropillar in Fig. 9b, shows no slip traces but this behavior is also typical of pyramidal +slip, which does not lead to visible slip traces on the micropillar surface. The orientation +of the micropillar after deformation was assessed by TKD in a thin lamella extracted +along the compression direction. The IPF map in Fig. 9c indicates the absence of the +tensile twinning during deformation. + + +23 + +Fig. 9. (a) Engineering stress-strain curves from the micropillar deformed in +compression along [0001] in grain D. (b) SEM micropillar of the lateral side of the +deformed micropillar. (c) IPF map of the lamella extracted from the deformed +micropillar. + +To further elucidate the deformation mechanisms, the analysis of the dislocation +structures was carried out by TEM in a thin lamella extracted from the deformed +micropillar. The beam direction was parallel to [112$0] as confirmed by the SADP in the +inset in Fig. 10a. Two-beam condition imaging was performed with g = (0002) and g = +(101$0) and the corresponding dark field micrographs are depicted in Figs. 10b and Fig. +10c, respectively. A large density of dislocations (marked with blue arrows) is +observed under g = (0002) in Fig. 10b, and some components are still in contrast +at the same location when the operation vector changes to g = (101$0) in Fig. 10c. The +detail of the rectangular region marked with purple dashed lines in Fig. 10b is shown at +higher magnification in Fig. 10d. The dislocations (marked with the blue dashed +lines) are [1$1$23]/3 and [12$1$3]/3 according to the [112$0] crystal orientation in Fig. 10e. +These results are in agreement with those reported in Mg-Zn-Ca and Mg-Y alloys +(Wang et al., 2021a; Wu et al., 2020). Nevertheless, it should be noticed that it is +difficult to identify the active pyramidal plane, since both pyramidal I and pyramidal II +planes contain the same slip directions. Thus, it can be concluded that plastic +deformation along the c-axis in compression was dominated by pyramidal +dislocations. The activation of pyramidal slip was associated to homogeneous +deformation and strong strain hardening (Basu et al., 2021). This high hardening rate is +likely associated with short mean-free paths and this explains why no slip traces were +2μm +(b) +After compression +(a) +1μm +(c) + +2700 +600 +90 +500 +400 +300 +200 +100 +10 + ineeiring +Stran500 +400 +300 +2 +3.T +24 +found on the micropillar surface. It is not clear whether slip took place along pyramidal +I plane or pyramidal II plane but the SF is slightly higher for pyramidal II for this +particular orientation (Table 1), which is assumed to be the active one. Thus, the CRSS +for pyramidal Ⅱ slip can be estimated as 203 ± 7 MPa from the SF and the yield +stress. + +Fig. 10. (a) Bright field TEM image of the thin lamella extracted from the micropillar +deformed along the [0001] orientation. The beam direction is [112$0], as shown by the +SADP in the inset. (b) and (c) Dark field TEM micrographs of the square region marked +with orange dash lines in (a) under g = (0002) and g = (101$0), respectively. (d) Bright +field TEM micrograph obtained under g = (0002) from the rectangular region marked +with purple dash lines in (b). The potential pyramidal dislocations are indicated +by the crystal orientation (black box) and the pyramidal slip trace (blue line). (e) +Schematic of Mg crystal orientation and of the corresponding dislocations (blue +lines) from the [112$0] projected view. + +3.3 Effect of Y and Ca on the GSFE curves +(a) +1μm +g=(101!0) + components +(0001) plane + dislocation +100nm +100nm +(b) +(c) +(d) +(e) +50nm +g=(0002) +B=[112!0] +g=(0002) + +Basal trace +25 +The experimental evidence presented above shows that the addition of Y and Ca +affects significantly the plastic deformation mechanisms. The changes in the +deformation mechanisms are proposed to be associated with the modification of the slip +resistance of the different slip systems due to the presence of the solute atoms. The +GSFE is intimately associated with the activation barriers of the deformation modes, +hence influencing their relative contributions to the overall deformation behavior +(Sandlöbes et al., 2011). To ascertain the effect of the solute atoms (Y and/or Ca) on the +slip activities, the GSFE (γ) curves were computed for the slip systems in Mg-Y, +Mg-Ca, and Mg-Y-Ca alloy, as well as in pure Mg for comparison. +The GSFE curves for {0001}<101$0>, {11$00}<112$0> and {101$1}<1$21$0> slip +systems are presented in Figs. 11a, 11b and 11c, respectively. The curves exhibited only +one local maximum, from which the unstable stacking fault energy (γus) for each slip +system was determined (Table 2). γus is associated with the activation barrier for +dislocation slip (Ding et al., 2018; Dong et al., 2018). Evidently, the addition of Y and +Ca reduced slightly the γus for basal slip from 88 mJ/m2 in pure Mg to a minimum +of 64 mJ/m2 in Mg-Ca or of 73 mJ/m2 in Mg-Y and the γus of Mg-Y-Ca (74 mJ/m2) was +similar with that of Mg-Y. However, the reduction in γus for the prismatic slip +system was much more important, from ~235 mJ/m2 in pure Mg to γus of ~18 mJ/m2 in +the Mg-Y-Ca alloy (Table 2). This synergistic contribution of Y and Ca on γus for +prismatic slip is obvious as the sole addition of either Y or Ca only reduced γus to 120 +mJ/m2 (Table 2). The dramatic reduction of γus for prismatic slip in the Mg-Y-Ca +alloy facilitates the activation of this deformation mechanism during plastic +deformation. On the contrary, the γus for pyramidal Ⅰ only changed from 304 mJ/m2 +in pure Mg to 318 mJ/m2 in Mg-Y-Ca alloy. The sole addition of Ca (308 mJ/m2) did +not modify significantly γus for pyramidal Ⅰ while Y (359 mJ/m2) increased slightly +γus for pyramidal I. Thus, prismatic slip is favored by the addition of Y and Ca +in comparison with pyramidal I slip. + + +26 + +Fig. 11. Generalized stacking fault energy curves for (a) basal (b) prismatic, +and (c) pyramidal Ⅰ slip systems in pure Mg, Mg-Ca, Mg-Y and Mg-Y-Ca alloys. + +Table 2. The calculated γus, for basal , prismatic , and pyramidal Ⅰ , slip +systems in the Mg-Ca, Mg-Y and Mg-Y-Ca alloys compared with pure Mg. +Alloy +γus (mJ/m2) + Basal + Prismatic + Pyramidal Ⅰ +Mg48 +88 +235 +304 +Mg47Ca1 +64 +121 +308 +Mg47Y1 +73 +118 +359 +Mg46Y1Ca1 +74 +18 +318 + +4. Discussion +4.1 Effect of Y and Ca on the CRSSs +The yield stresses measured from the micropillar compression tests in different +(a) +(b) +(c) + +ikgas +Tas +Ipyranniclalsillp +300 +20 +200 +20160 +100 +60 +0.0 +0,2) +OLA +0. +1.0 +lraictonaldlspiaxeementof8<112os300 +pmshatcslp +250 +200 +160 +100 +(),, +0.6 +1.0 +bracuoneldisplakeementof 1120100 +lbersalslip +80 +6X0 +0 +0.0 +0.) +0.6 +0.8 +1.0 +Hracuonealdisplakeementof sslolcs +27 +orientations are summarized in Table 3. The CRSS for the dominant slip system in each +orientation (following slip trace analysis and TEM characterization) is also presented +in Table 3. They are basal slip in the micropillars carved from grain A, +prismatic slip in the micropillars from grains B and C, and pyramidal II slip in +the micropillars from grain D. Moreover, twin nucleation was not observed in +micropillars carved from grains B and C and this result can be used to obtain thresholds +of the CRSS for twin nucleation from the maximum stress attained during the test and +the maximum SF for tensile twinning in Table 1. These minimum values are also +included in Table 3. It should be noticed that dimensions of the micropillars selected in +this investigation follow previous results in Mg alloys (Li et al., 2021a; Wang et al., +2020; Wu et al., 2020) that indicate these values should not be very much influenced +by the “smaller is stronger” effect reported for micropillar compression tests at the +micron or sub-micron scale (Aitken et al., 2015; Chang et al., 2014). + +Table 3. Yield stress and CRSS for different slip systems from micropillar compression +tests along different orientation in the Mg-Y-Ca alloy. +Grain +Loading direction +Yield stress (MPa) +CRSS (MPa) +A +[112!3] +65 ± 11 +29 ± 5 ( basal slip) +B +[112!0] +219 ± 9 +105 ± 4 ( prismatic slip) +> 111 MPa (tensile twin*) +C +[101!0] +228 ± 4 +105 ± 2 ( prismatic slip) +> 148 MPa (tensile twin*) +D +[0001] +431 ± 15 +203 ± 7 ( pyramidal Ⅱ slip) +*: Tensile twin was not nucleated when the CRSS reached this value. + +In order to ascertain the strengthening effect of Y and Ca atoms in solid solution, +the CRSSs for the different slip systems in Mg-Y-Ca alloy are plotted in Fig. 12 along +with those reported in the literature in pure Mg (Li et al., 2021a; Wang et al., 2019a), +Mg-Al (Wang et al., 2020, 2019a), Mg-Zn (Li, 2019; Li et al., 2021a), Mg-Y (Li et al., +2021b; Wu et al., 2020), Mg-Zn-Ca (Wang et al., 2021a), Mg-Al-Ca (Luo et al., 2022) + + +28 +and Mg-Y-Zn (Chen et al., 2018) alloys. All these results were obtained from +compression tests in micropillars with a cross-section around 5 × 5 μm2 and, thus, size +effects -if any- should not affect the comparison. The results for basal slip in Fig. +12a show that the addition of Y in solid solution dramatically increases the CRSS in +comparison with pure Mg (Li et al., 2021a; Wang et al., 2019a) and with Mg-Al (Wang +et al., 2019a) or Mg-Zn (Li, 2019; Li et al., 2021a) alloys. These experimental data are +supported by the first principles simulations of the solute/dislocation interaction energy, +which showed the higher strengthening potential of Y for basal dislocations, in +comparison with Al and Zn, because of the larger atomic radius and shear modulus +misfit of Y with respect to Mg (Tehranchi et al., 2018). The addition of Ca to the Mg-Y +does not increase the CRSS for basal slip according to our results while the +strengthening effect of Ca in Mg-Al (Luo et al., 2022) or Mg-Zn (Wang et al., 2021a) +is limited and may also be attributed to the elastic interaction between Ca solute atoms +and dislocations. +Regarding pyramidal slip (Fig. 12b), Zn and Al are the alloying elements +which lead to the largest increase in the CRSS (Li et al., 2021a; Wang et al., 2020). Zn +is more efficient but the solubility of Al in Mg is larger and CRSSs in the range of 200- +250 MPa can be achieved for these binary alloys. Addition of 4 wt. % Y increases the +CRSS up to 106 MPa (Wu et al., 2020) but the combination of Y and Ca leads to a +CRSS of 203 ± 7 MPa, similar to the one found in the binary Mg-Zn alloy. Thus, Zn, +Al and Y solutes increase the CRSS for pyramidal slip due to the elastic +interaction of the solutes with the dislocations, as in the case of basal slip. It should +be noticed that the activation and glide of pyramidal dislocations is a complex +process that also depends on dislocation dissociation during gliding due to the larger +Burgers vector (Moitra et al., 2014; Tang and El-Awady, 2014). Atomistic simulations +have shown that the presence of Y and Ca favors the activation for cross-slip/double +cross-slip of pyramidal dislocations, leading to new dislocation loops which can +accommodate plastic deformation (Wu et al., 2018). + + +29 + +Fig. 12. CRSS for (a) basal slip, (b) pyramidal slip and (c) twin nucleation and prismatic +slip in Mg and Mg alloys (Chen et al., 2018; Kiener et al., 2021; Li, 2019; Li et al., +2021b, 2021a; Luo et al., 2022; Wang et al., 2021a, 2020, 2019a; Wu et al., 2020), +including the results obtained for Mg-Y-Ca alloy in this investigation. All data were +obtained from compression tests in micropillars with a cross-section around 5 × 5 μm2. +The arrow in the CRSS for twin nucleation in Mg-Y-Ca indicates that the actual CRSS +is higher than the value in the figure. + +The CRSSs for tensile twin nucleation, measured by means of micropillar +compression tests in Mg and different Mg alloys, are plotted in Fig. 12c (Kiener et al., +2021; Li et al., 2021a; Wang et al., 2020, 2021a; Wu et al., 2020). While the addition of +Y (Wu et al., 2020) and Al (Wang et al., 2020) lead to the largest enhancements in the +CRSS for twin nucleation (the latter because of the larger solid solubility), the highest +CRSS is obtained for the ternary Mg-Y-Ca alloy which -following our experimental +results- has to be higher than 148 MPa. Generally, the twin nucleation process is +dominated by the dislocation-shearing and atomic shuffle. The strong strengthening +(a) +(b) +(c) + +180 +160 +140 +$120 +100 +Prismatic slip +hg +410 +20 +8 +10200 +150 +100 +nCaamees +10410 +RSS +10 +30 +provided by Y on the CRSS for twin nucleation can be ascribed to the inhibition of +atomic shuffling due to the large atomic radius of Y (0.180 nm). Moreover, Ca has an +even larger atomic radius (0.194 nm) and it is proposed that the synergistic contribution +of both atoms in solid solution is responsible for the huge increase in the CRSS for twin +nucleation. In addition, the elastic interaction of twinning dislocations with different +solute atoms also leads to an increase in the CRSS for twin propagation (Ghazisaeidi et +al., 2014; Stanford et al., 2015), as it has been reported in previous investigations (Li et +al., 2021a, 2021b; Wang et al., 2020, 2021a). However, only the addition of Y and Ca +can inhibit twin nucleation in micropillars suitable oriented for twinning, e.g., deformed +in compression along [112$0] and [101$0] (Table 1). +The high CRSS for tensile twin nucleation in Mg-Y-Ca alloys leads to the +activation the prismatic slip, which becomes the dominant plastic deformation +mechanism under a-axis compression. There is limited information on the CRSS for +prismatic slip (because either basal slip or tensile twinning are usually activated +before prismatic slip to accommodate the plastic deformation) and the available +experimental data on Mg-Y-Zn (Chen et al., 2018) (102 MPa) and Mg-Y-Ca (105 ± 4 +MPa) are plotted in Fig. 12c. The CRSS for prismatic slip is much lower than the +CRSS for tensile twin nucleation in Mg-Y-Ca and, thus, tensile twinning is suppressed +during compression parallel to the a-axis. +The CRSSs in Fig. 12 show that the strengthening effect of the Y and Ca for +prismatic slip is much lower than the ones reported for pyramidal slip and twin +nucleation, and also, in relative terms, for basal. Moreover, evidence of +prismatic slip is unusual in Mg alloys except in the case of that they contain Ca (Zhu et +al., 2019), indicating that the presence of Ca reduces the activation barriers for +prismatic slip glide. Besides, Chen et al., (2018) found that prismatic slip was +activated during micropillar compression testing of Mg-Y-Zn alloys but it was absent +in solution-treated Mg-Zn alloys deformed along the same orientation (Li et al., 2021a; +Wang et al., 2019b), implying that the addition of Y also facilitates prismatic slip. +Although the elastic interaction of the solute atoms with prismatic dislocations is +expected to increase the CRSS, the reduction of the stacking fault energy due to the + + +31 +presence of Y and Ca reduces the activation barrier for dislocation movement on the +slip plane and facilitates the activation of this slip system. +Overall, the addition of Y and Ca leads to a marked solid solution strengthening +for basal and pyramidal slip as well as for the nucleation of tensile twins but +not for prismatic slip. + +4.2 Effect of plastic anisotropy on the ductility +In general, the tensile ductility and formability of Mg alloys during the plastic +deformation is dictated by the CRSS ratio between different slip systems, especially +between non-basal and basal slip, the latter being the dominant deformation mechanism +in most cases (G. Liu et al., 2017; Zhu et al., 2019). Therefore, the tensile ductility of +different Mg alloys is plotted as a function of the CRSS ratios between different slip +systems in Fig. 13 (Habibi et al., 2012; Huang et al., 2018; Shi et al., 2020; Wang et al., +2021b; Wu et al., 2010; Yang et al., 2022; Zhao et al., 2019a, 2019b; Zhu et al., 2020, +2019). The CRSS ratios were measured via micropillar compression tests in most of the +alloys (Agnew et al., 2003; Li et al., 2021a; Shang et al., 2021; Wang et al., 2021a, 2020, +2019a, 2018; Wu et al., 2020; Zhu et al., 2019) with a few exceptions. The CRSS ratio +between prismatic and basal slip in Mg-5Y (wt. %) (Huang et al., 2018) and +Mg-0.47Ca (wt. %) (Zhu et al., 2019) were obtained from mechanical tests in +polycrystals via slip trace analysis. Besides, those for pure Mg (Agnew et al., 2003), +Mg-0.5Ca (wt. %) (Shang et al., 2021) and Mg-3Y (wt. %) (Wang et al., 2018) alloys +were determined by the elasto-plastic self-consistent model, crystal plasticity finite +element simulations, and the elastic viscoplastic self-consistent model, respectively. +Moreover, the tensile elongation data were collected from pure Mg and wrought Mg +alloys with similar grain sizes. +In general, reduced ratios between the CRSS for non-basal slip and basal slip are +strongly associated with the improvement of the ductility of Mg alloys. This trend +agrees with the data plotted in Fig. 13b, which shows a clear link between the reduction +of CRSS pyramidal / CRSS basal and the increase in tensile elongation. However, +the limited data of the influence of CRSS prismatic / CRSS basal on the tensile ductility + + +32 +in Fig. 13a are not conclusive. Obviously, low CRSS pyramidal / CRSS basal ratios +favor isotropic deformation and limit the development of strong basal textures and both +processes help to improve ductility and formability because activation of +dislocations benefits the strain accommodation along the c-axis (Liu et al., 2019). +Besides, Wu et al., (2018) predicted that the addition of Y/Ca could significantly reduce +the cross-slip energy barriers between pyramidal I and pyramidal II planes, thus +promoting dislocation cross-slip. Enhanced non-basal slip activities and cross- +slip induce homogeneous deformation and improve the ductility. + +Fig. 13. Relation between the CRSS ratios of different slip systems (Agnew et al., 2003; +Li et al., 2021a; Shang et al., 2021; Wang et al., 2021a, 2020, 2019a, 2018; Wu et al., +2020; Zhu et al., 2019) and the tensile elongation (Habibi et al., 2012; Shi et al., 2020; +Wang et al., 2021b; Wu et al., 2010; Yang et al., 2022; Zhao et al., 2019a, 2019b; Zhu +et al., 2020, 2019) in pure Mg and Mg alloys: (a) CRSS prismatic / CRSS basal, (b) +CRSS pyramidal / CRSS basal, (c) CRSS prismatic / CRSS tensile twin. + +It should be noted that these mechanisms are particularly relevant in Mg-Y (Wang +(a) +(b) +(c) + +Pure Mg (Zhu,2020; Agnew,2003) +Pure Mg (Habibi, 2012; Agnew,2003) +<0.71 +Mg-0.5Ca(Zhu,2019;Shang,2021) +Mg-3Y(Wang,2018;Zhao,2019b) +★Mg-5Y-0.08Ca (This work) +30 +10 +0. +1.:0 +1.8 +2.0 +CRSPure Mg(Li,2021a;Wang.2019a;Habibi,2012) +PureMg(LI,2021a;Wang,2019a;Zhu,2020) +Mg-5Zn(Shi,2020;Li,2021a) +Mg-4Y(Wu,2020;Wu,2010) +Mg-4.4Al(Zhao,2019a;Wang,2020) +Mg-1.8Zn-0.2Ca (Wang,2021a;Wang,2021b) +Mg-5Y-0.08Ca (This work) +20 +1030 +10 +Mg-3Y(Wang.2018;Zhao,2019b) +Mg-0.47Ca(Zhu,2019) +Mg-5Y (Huang,2018;Yang,2020) +★Mg-5Y-0.08Ca (This work) +33 +et al., 2018) and Mg-Zn-Ca (Wang et al., 2021a, 2021b) alloys as well as in the Mg-Y- +Ca alloy analyzed in this investigation. In all these cases, the presence of Y and/or Ca +also leads to a high increase in the CRSS for twin nucleation while the CRSS for +prismatic slip is not strongly affected. As a result, the CRSS prismatic / CRSS tensile twin +is dramatically reduced and this is accompanied by a large increase in the tensile +ductility, as shown in Fig. 13c. Particularly, tensile twinning is replaced by +prismatic slip during compressive deformation along the a-axis if CRSS prismatic / +CRSS tensile twin < 1 and twinning only occurs in grains deformed in tension along the c- +axis. Moreover, as the CRSS for prismatic slip is smaller than that for +pyramidal slip, the former becomes the dominant plastic deformation mechanism in +grains suitable oriented for both. It should be noted that pyramidal slip is +associated with a large strain hardening (Fig. 9a) that it is not present for prismatic +slip (Fig. 5). Thus, pyramidal slip induced large stress concentrations at grain +boundaries that facilitate the nucleation of damage but this process is not activated if + prismatic slip is dominant. +In general, the preferential activation of basal slip and tensile twinning during +processing always introduces a strong basal texture in wrought Mg and Mg alloys, +leading to the plastic anisotropy, crack formation and limited ductility (Sabat et al., +2015; Wang et al., 2021a). The addition of Y and Ca in our alloy strongly enhanced the +activation of prismatic and pyramidal slip, which also contribute to reduce +the intensity of the texure during extrusion, as shown in Figs. 2a and 2b. This limited +texture also contributes to reduce the plastic anisotropy. +It should also be noted that the overall mechanical response of polycrystals cannot +fully ascertained by means of micromechanical tests in single crystals because other +factors (grain boundaries, grain size and texture) play a key role in the mechanical +response. However, it should be emphasized that the plastic deformation of each crystal +within the polycrystal is intrinsically related to that of a single crystal (Wang et al., +2021a) and, hence, it is important to ascertain the plastic deformation mechanisms in +single crystals to understand the complex mechanisms in bulk polycrystalline samples. +Overall, these results indicate that the presence of Y and Ca in solid solution in Mg + + +34 +alloys leads to a large increase in the CRSS for basal slip (which induces a large +reduction in CRSS pyramidal / CRSS basal) while CRSS prismatic / CRSS tensile twin +< 1. As a result, plastic deformation in polycrystals in more isotropic and localization +of the deformation in the form intense basal slips that promote fracture is suppressed +(Sandlöbes et al., 2011). Moreover, twinning and pyramidal slip are replaced by + prismatic slip in grains deformed along the a-axis. Suppression of twinning (which +induces strong anisotropy in the plastic deformation in textured alloys) and the +activation of prismatic slip (which provides an additional plastic deformation +mechanism with limited hardening) lead to an important improvement in the tensile +ductility of Mg alloys. + +5. Conclusions +The deformation mechanisms of a Mg-5Y-0.08Ca (wt. %) alloy, with a superior +tensile elongation (32%), were studied by means of micropillar compression tests, slip +trace analysis along different orientations, TEM as well as TKD. It was found that the +presence of Y and Ca in solid solution led to a huge increase in the CRSS for basal +slip (29 ± 5 MPa), pyramidal slip (203 ± 7 MPa) and tensile twin nucleation +(above 148 MPa). This behavior was attributed to the large mismatch of the atomic radii +and elastic constants of the Y and Ca atoms with respect to Mg, which leads to a strong +interaction of the dislocations with the solute atoms and hinders atomic shuffling, that +is necessary to activate twin nucleation. On the contrary, the CRSS for prismatic +slip only increases up to 105 ± 4 MPa because the hardening induced by the interaction +of the solute atoms with dislocations is partially balanced by the reduction in the +stacking fault energy associated with prismatic slip due to the presence of Y and +Ca. +The changes in the CRSS for slip and tensile twinning in Mg-Y-Ca alloys modify +the dominant deformation mechanisms. In particular, the CRSS prismatic / CRSS tensile +twin is dramatically reduced and tensile twinning is replaced by prismatic slip during +compressive deformation along the a-axis if CRSS prismatic / CRSS tensile twin < 1. +Moreover, as the CRSS for prismatic slip is smaller than that for pyramidal + + +35 +slip, the former becomes the dominant plastic deformation mechanism in grains suitable +oriented for both. As a result, reduction of twinning (which induces strong anisotropy +in the plastic deformation in textured alloys) and the activation of prismatic slip +(which provides an additional plastic deformation mechanism with limited hardening) +lead to an important improvement in the tensile ductility of Mg alloys. + +Acknowledgements +This work was supported by the National Natural Science Foundation of China +(Grant Nos. 52001199 and 51825101). Y. Cui acknowledges the support from the +Shanghai Sailing Program (Grant No. 22YF1419300). JLL acknowledges the support +from the Spanish Ministry of Science (HexaGB project, reference RTI2018-098245) +and from the MAT4.0-CM project funded by the Comunidad de Madrid under +programme S2018/NMT-4381. + +References +Agnew, S.R., Duygulu, Ö., 2005. Plastic anisotropy and the role of non-basal slip +in magnesium alloy AZ31B. Int. J. Plast. 21, 1161–1193. +Agnew, S.R., Tomé, C.N., Brown, D.W., Holden, T.M., Vogel, S.C., 2003. Study +of slip mechanisms in a magnesium alloy by neutron diffraction and modeling. Scr. +Mater. 48, 1003–1008. +Ahmad, R., Yin, B., Wu, Z., Curtin, W.A., 2019. Designing high ductility in +magnesium alloys. Acta Mater. 172, 161–184. +Aitken, Z.H., Fan, H., El-Awady, J.A., Greer, J.R., 2015. The effect of size, +orientation and alloying on the deformation of AZ31 nanopillars. J. Mech. Phys. Solids +76, 208–223. +Alizadeh, R., LLorca, J., 2020. Interactions between basal dislocations and β1′ +precipitates in Mg–4Zn alloy: Mechanisms and strengthening. 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Plast. 120, 164–179. + + diff --git a/9dFLT4oBgHgl3EQfuS8F/content/tmp_files/load_file.txt b/9dFLT4oBgHgl3EQfuS8F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cda21734a9747e92d40ced67cbbf24e11447d0f3 --- /dev/null +++ b/9dFLT4oBgHgl3EQfuS8F/content/tmp_files/load_file.txt @@ -0,0 +1,1841 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf,len=1840 +page_content='1 Critical resolved shear stresses for slip and twinning in Mg-Y-Ca alloys and their effect on the ductility Mingdi Yua, Yuchi Cuib, Jingya Wanga,*, Yiwen Chena, Zhigang Dingc, Tao Yinga, Javier Llorcad,e,*, Xiaoqin Zenga,* a National Engineering Research Center of Light Alloy Net Forming and State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, PR China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' b School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, PR China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' c Nano and Heterogeneous Materials Center, School of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' d IMDEA Materials Institute, 28906 Getafe, Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' e Department of Materials Science, Polytechnic University of Madrid, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' de Ingenieros de Caminos, 28040 Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' E-mail address: jingya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='wang@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' E-mail address: javier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='llorca@imdea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' E-mail address: xqzeng@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Abstract: The deformation mechanisms of an extruded Mg-5Y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='08Ca (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) alloy were analyzed by means of micropillar compression tests on single crystals along different orientations -selected to activate specific deformation modes- as well as slip trace analysis, transmission electron microscopy and transmission Kikuchi diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The polycrystalline alloy presented a remarkable ductility in tension (~32%) and negligible 2 differences in the yield strength between tension and compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It was found that the presence of Y and Ca in solid solution led to a huge increase in the CRSS for basal slip (29 ± 5 MPa), pyramidal slip (203 ± 7 MPa) and tensile twin nucleation (above 148 MPa), while the CRSS for prismatic slip only increases up to 105 ± 4 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The changes in the CRSS for slip and tensile twinning in Mg-Y-Ca alloys expectedly modify the dominant deformation mechanisms in polycrystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In particular, tensile twinning is replaced by prismatic slip during compressive deformation along the a-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The reduction of twinning (which generally induces strong anisotropy in the plastic deformation in textured alloys), and the activation of prismatic slip (which provides an additional plastic deformation mechanism with limited hardening) were responsible for the large tensile ductility of the alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Keywords: Mg-Y-Ca alloys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' micropillar compression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' critical resolved shear stress;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' plastic anisotropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' tension-compression asymmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' tensile ductility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Introduction Pure Mg and Mg alloys generally present poor ductility and formability, especially at room temperature (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Yaghoobi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' As a result, forming of rolled sheets and extruded bars becomes difficult and limits the application of wrought Mg alloys in different industrial sectors (Li and Fang, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, understanding the origin of the lack of ductility and formability is of paramount importance to develop new Mg alloys that overcome these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The poor ductility of Mg alloys is primarily traced to its low-symmetry hexagonal closed packed (HCP) lattice structure, which results in very large differences in the critical resolved shear stress (CRSS) between basal and non-basal slip systems as well as in the easy activation of tensile twinning (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Plastic deformation in pure Mg is initially accommodated by basal slip, which only provides two independent slip systems (Partridge, 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This process leads to the development of a strong basal texture during rolling and extrusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, plastic deformation along the c-axis (which is necessary to activate five independent slip systems to fulfil the von- 3 Mises criterion for homogeneous plastic deformation) is absorbed by tensile twinning, which is triggered at much lower CRSS than that necessary to produce pyramidal slip (Graff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Sukedai and Yokoyama, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, the plastic strain associated with tensile twinning is very limited (at most 7%), moreover, tensile twining is a polar mechanism that only occurs when the stress along the c-axis of the crystal is tensile (Mayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This leads to a large buildup of stresses to activate pyramidal slip in grains that are not suitably oriented for twinning and/or that cannot accommodate more plastic deformation by twinning (Obara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Reed-Hill and Robertson, 1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The stress concentrations in these grains facilitate the nucleation of cracks and limit the ductility (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, huge differences in the flow stress and the strain hardening rate between tension and compression appear in textured microstructures, which also lead to fracture during bending and forming operations (Agnew and Duygulu, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The strategies to improve ductility and formability of Mg alloys have been directed towards promoting the activation of multiple slip, including non-basal and non- basal slip, and to suppress deformation twinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Multiple slip leads to more homogeneous plastic deformation and limits texture development during rolling and extrusion while twinning promotes plastic anisotropy in textured microstructures (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' For instance, precipitation hardening in Mg-Zn alloys leads to large enhancements in the CRSS for basal (Alizadeh and LLorca, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Chun and Byrne, 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang and Stanford, 2015) and pyramidal slip (Alizadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021) and, thus, to an important reduction in the pyramidal-to- basal CRSS ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Nevertheless, the large increase in flow stress inherently decreases the ductility due to the strong accumulation of geometrically necessary dislocations around the precipitates (Rosalie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In addition, precipitates also increase the CRSS for twin growth but do not affect the CRSS for twin nucleation (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' As the latter is normally higher than the former, the presence of precipitates do not contribute to hinder the development of twinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The only difference induced by the precipitates is a larger number of smaller twins, as compared to the precipitate-free condition (Stanford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, precipitate is not very efficient to enhance the 4 ductility of Mg alloys (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Strategies based on solid solution hardening have been more successful to improve the ductility of Mg alloys if the alloying elements are properly chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' For instance, Sandlöbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (Sandlöbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2013, 2012, 2011) reported that the addition of 3 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' % Y led to Mg alloys with a tensile ductility > 25 %, which was associated with the presence of a large density of pyramidal dislocations in the deformed sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This behavior was mainly attributed to a reduction in the ratio between the CRSS of the < c+a > pyramidal slip and < a > basal slip, which was ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2 according to in situ high energy X-ray diffraction tests (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018) and ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='8 from micropillar compression tests (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Large ductility and formability are not achieved, however, by the addition of other elements in solid solution (such as Al or Zn) because the pyramidal-to-basal CRSS ratio in these alloys are > 10 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', (2019) found that the addition of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='47 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' % of Ca in solid solution enhanced the activity of prismatic and pyramidal I dislocations as well as the cross-slip between basal and non-basal slip planes, improving the tensile ductility to ~18 % in a Mg-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='47 Ca (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' And several authors reported a large improvement in the ductility of binary Mg-Zn and Mg-Al alloys through the addition of small amount of Ca (Hofstetter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Sandlöbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This behavior was supported by our recent micropillar compression tests that showed that the addition of Ca to Mg-Zn alloys reduced the pyramidal-to-basal CRSS ratio values, that were similar to those found in Mg-Y alloys (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Finally, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', (2018) showed that the presence of Y and Ca reduces the energy for cross-slip/double cross-slip of pyramidal dislocations, leading to new dislocation loops which accommodate plastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In contrast, the cross-slip is inhibited in pure Mg (or in Mg-Al and Mg-Zn alloys) (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018), by the favorable dissociation of edge pyramidal dislocation segments into sessile segments in the basal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Regarding the effect of solid solution on tensile twinning, several investigations reported an increase in the CRSS for twin nucleation and growth with the addition of Al (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020), Zn (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a), Y (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b) as well as Ca to Mg- 5 Zn alloys (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, the CRSS for twin nucleation and growth were lower than that for pyramidal slip in the corresponding alloy, thus, tensile twinning was still preferred over pyramidal slip to accommodate plastic deformation in grains suitable oriented for twinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In addition, the addition of 4Y (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) could significantly suppress the tensile twinning (with CRSS larger than 113 MPa) and promote the dislocations (with CRSS around 106 MPa) (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The results summarized above point to the beneficial effects of Y and Ca in solid solution to reduce the plastic anisotropy of Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, the co-addition of Ca and Y is expected to promote the homogeneous deformation and improve the plastic deformability of Mg alloys, taking advantages of the significant suppression effect of Y on the tensile twinning, the promotion effect of Ca on the non-basal slips, simultaneously the positive effect of Ca and Y on the activation of the slips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Ca enhances the activation of prismatic and pyramidal slip while Y has similar effects on slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, experimental results on the tensile behavior of an extruded Mg – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='4 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' % Y – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='3 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' % Ca (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2013) showed a very large tensile elongation (~37 %) but there is not information available in the literature -to the authors’ knowledge- on the concurrent effects of Y and Ca in solid solution on the dominant deformation mechanisms and this is the main objective of this investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, the CRSS for different slip systems and twinning was determined in a Mg-Y-Ca alloy from micropillar compression tests in single crystals with different orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The deformation mechanisms were ascertained from slip trace analysis in the scanning electron microscope (SEM), transmission electron microscopy (TEM) observations of the dislocations as well as transmission Kikuchi diffraction (TKD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This information was used to rationalize the excellent ductility of Mg-Y-Ca and to provide guidelines to design novel Mg alloys with improved ductility and formability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Materials and experimental techniques 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='1 Materials The Mg-Y-Ca alloy was prepared from pure Mg (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='99 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %), Mg-30 Ca (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) and Mg-30 Y (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) master alloys in a resistance furnace under a protective 6 atmosphere of CO2 and SF6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The actual chemical composition of the ingot, obtained by inductively coupled plasma atomic emission spectroscopy, was Mg-5Y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='08Ca (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The cast alloy was solution treated at 400 ℃ for 12 h, followed by extrusion at 300 ℃ with an extrusion ratio of ~ 18:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Afterwards, parallelepipedal samples of 10×10×5 mm3 were cut from the extruded specimens and homogenized at 550 ℃ for 20 days within quartz capsules filled with Ar to induce grain growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2 Experimental techniques Tensile and compressive tests were carried out along the extrusion direction in polycrystalline specimens at crosshead speed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5 mm/min, using a universal testing machine (Z100-TEW) at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The dimensions of the gage section of the dog-bone tensile specimens were 18×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='4×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='4 mm3 (length × width × thickness), while cylindrical specimens of 8 mm in diameter and 12 mm in length were used in the compression tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Deformation was measured with an extensometer and 3 specimens were tested in each condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The crystallographic orientation of the grains in the sample was characterized via electron back-scattered diffraction (EBSD) in a Tescan Mira-3 SEM with an Oxford Instruments Nordlys EBSD detector at an accelerating voltage of 20 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The surface of the sample was mechanically ground using abrasive SiC papers with a grit size of 1200, 2000, 3000, 5000 and 7000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Subsequently, the sample surface was electropolished in an ethanol solution with 10 (vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) perchloric acid at -30 ℃ and 30 V for 90 s to remove the surface damage induced by grinding and reveal the grain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The EBSD data were analyzed using the Channel 5 software and the Oxford Instruments AZtec Nanoanalysis software package v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0 along with AZtec Crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Several grains whose orientations were appropriate to active different deformation modes were selected to mill the micropillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Micropillars of 5 × 5 μm2 square cross and an aspect ratio 2:1 were milled from the selected grains using a FEI Helios G4 UX Focused Ion Beam (FIB)/SEM dual beam microscope operated at 30 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' These dimensions are known to minimize size effects during mechanical deformation while the time and effort to mill each micropillar are reasonable (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' An initial ion current of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='3 nA was used to remove the 7 surrounding material and it was reduced to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5 nA when the beam was getting closer to the actual dimensions of the micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' A final ion current of 80 pA was used in the final polishing step to minimize the surface damage due to Ga+ ion-implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The final taper of the micropillars was < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Micropillar compression tests were performed in ex situ using a Hysitron Triboindenter TI950 system though a diamond flat punch of 10 μm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' All the tests were conducted under displacement control up to a maximum strain of 10 % at a nominal strain rate of 10-3 s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The experimental displacement was corrected to account for the elastic deflection of the matrix material beneath the micropillars following the Sneddon correction (Sneddon, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' To this end, the elastic modulus of each grain was determined via the nanoindentation method with a Berkovich tip in the same grain where the micropillar was milled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' More details about micropillar manufacturing and the compression set-up can be found in (Sneddon, 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The engineering stress-strain curves were obtained from the load and the corrected elastic deflection of the micropillar using the initial cross-sectional area and the height of the micropillars measured in the SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The yield stress, σy , was determined from the loss of linearity in the stress-strain curve following the methodology described in (Alizadeh and LLorca, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Maaß et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' From this information, the CRSS of the active slip system was determined as CRSS = SF × σy (1) where SF is the Schmid factor of the corresponding slip system, computed from the crystallographic orientation of each crystal (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The slip traces on the top and lateral surfaces of the deformed micropillars were characterized in a Tescan Mira-3 SEM to ascertain the active slip planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The active slip plane and direction were identified from the micropillar orientation using VESTA software (Momma and Izumi, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, TEM and TKD were used to determine the dislocation activity and the orientation of the micropillar after deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' To this end, a thin lamella was lifted-out along the loading direction from the deformed pillars and thinned to < 100 nm in thickness using FIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The TKD maps were collected in a 8 Tescan Mira-3 SEM at 30 kV with a step size of 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The TEM observations were carried out using a Talos F200X G2 microscope operated at 200 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The two-beam condition was applied to obtain dislocation contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, the “g·b” visibility criterion was used to identify the types of dislocation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', the dislocation is in contrast when g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='⃗ · b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='⃗ ≠ 0, where g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='⃗ is the diffraction vector and b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='⃗ the Burgers vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='3 First-principles calculations In order to study the influence of Y and/or Ca atoms on the deformation mechanisms in Mg alloys, the generalized stacking fault energy (GSFE) curves of different slip systems were calculated via the first-principles calculations using the Vienna Ab initio Simulation Package (VASP) (Kresse and Furthmüller, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The exchange-correlation function was described using the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerholf functional (PBE), based on the projector augmented wave (PAW) (Blöchl, 1994) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' A supercell with 12-layers containing 48 atoms was defined for different slip systems, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Each supercell was separated by 15 Å vacuum to eliminate the influence of the periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The formation energy was initially calculated for different positions of the solute atoms and the configurations with lower formation energy was selected as the most stable ones (Yuasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In the binary Mg47N1 (N = Y, Ca) alloys, the most stable configuration was found when one Mg atom at the center site of the stacking fault plane was substituted by a solute atom X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In the ternary Mg46N1X1 (N = Y, and X = Ca) alloy, the most stable configuration was found when one Mg atom at the center site of the stacking fault plane was substituted by a Ca atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Then, one of the eleven nearest Mg atoms from the Ca atom was substituted by one Y atom, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' S1 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The exact position of the Y atom was determined from the formation energy (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The formation energies for every occupancy of the Y atom are listed in Table S1 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The conventional direct crystal slip methods were employed to obtain the GSFE curves of different slip systems The perfect supercell was cut into two free parts and 9 one part was displaced with respect to the other one along the slip direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The atomic positions were relaxed only along the direction perpendicular to the stacking fault plane (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' A residual force threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='01 eV/Å was performed in all geometric relaxations until the electronic energy converged to less than 10-5 eV/cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The Brillouin zone for the GSFE of the basal slip system, the prismatic slip system, and the pyramidal slip system was set as 8×8×1, 10×6×1, and 6×10×1, respectively, with an energy cutoff of 480 eV (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Schematic illustration of the models to calculate the GSFE for (a) basal slip (b) prismatic slip, and (c) pyramidal Ⅰ slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The most stable positions of Y and Ca atoms determined by the lowest formation energy are marked by blue and purple atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Stacking fault planes are noted by the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='1 Mechanical behavior of polycrystals The inverse pole figure (IPF) map of the as-extruded Mg-Y-Ca alloy along the (a) basal slip (b) prismatic slip (c) pyramidalⅠslip [11!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='00] [0001] [112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] [101!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='1] [112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] [112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='3] 105° [112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] [11!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='00] [0001] Mg Ca Y : O O O O 10 extrusion direction is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The {0001} pole figure shows that the Mg-Y- Ca alloy possesses a weak texture with a strength of ~ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='21 mrd, as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 2b, compared to pure wrought Mg with a strong basal texture of >15 mrd (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The engineering stress-strain curves of the extruded Mg-Y-Ca alloy from the tensile and compressive tests parallel to the extrusion direction are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The scatter was very limited and the average tensile elongation was very large (≈ 32%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, the tensile yield stress was 104 MPa, very close to the yield strength in the compression tests (122 MPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, the Mg-Y-Ca alloy presented very low tension/compression asymmetry in the yield strength in contrast with the marked asymmetry in extruded Mg and Mg alloys (Sukedai and Yokoyama, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='1 It should also be noted that volume fraction of the twinned material after tensile deformation was very low (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='8%), indicating that twining was not a dominant deformation mechanism in the Mg alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) IPF map of the Mg-Y-Ca along the extrusion direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (b) {0001} Pole figure of the Mg-Y-Ca alloy illustrating the texture characteristics before the deformation in the TD-ED plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (c) Engineering stress-strain curves in tension and compression parallel to the extrusion direction of the Mg-Y-Ca alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2 Deformation mechanisms 1 The comparison between both curves shows the limited tension-compression anisotropy in the yield strength but the differences in the elastic and fully plastic regions are due to the limitations of the compression tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Compression tests always underestimate the elastic modulus because it is very difficult to ensure that the specimen surface and the loading plate surface are perfectly parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, partial contact between both surface leads to localized plastic deformation and to an apparent elastic modulus that is lower than the real one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, barreling of the cylindrical specimen during compression leads to non-homogeneous plastic deformation and overestimates the strain hardening for large plastic strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 50μm TD ED Max=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='21 ED∥ Tensile direction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='00 (a) (b) (c) (c) 400 Tension Compression 300 200 0 10 20 30 40 Engineering strain (%)Tscedan-(0001) -Magnesium 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='21 则量计数：100008 Subset1 半宽：10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content="0* 样品对称性：三料 使用样本对疗性：数量 投射类型：等围积 透射平面：XY 率球：上 00'0 11 The IPF map with the crystallographic orientation of the grains in the Mg-Y-Ca alloy is depicted in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The grains were larger than 150 μm, and the micropillars were milled from the center of the grains to ensure that they were single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Four grains with appropriate orientations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 3) were selected to activate different deformation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The loading directions in the four grains are listed in Table 1, as well as the maximum Schmid Factor (SF) for the corresponding slip systems ( basal slip, prismatic slip, pyramidal Ⅰ slip, pyramidal Ⅰ slip and pyramidal Ⅱ slip) as well as {101$2} tensile twinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The inclination angle in Table 1 indicates the angle between the c-axis of each grain and the compression direction, as presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The compression direction is nearly parallel to [112$0], [101$0], and [0001] in grains B, C and D, respectively, and forms an angle of ~ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5° with respect to [0001] axis in grain A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Herein, grain A presents the highest SF for basal slip, which is prone to be the dominant deformation mechanism during compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Plastic deformation along the pyramidal I and II systems is favored in Grain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' prismatic and pyramidal as well as pyramidal slip systems have similar SFs in grain B, while grain C is suitably oriented to promote tensile twinning and prismatic slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The loading direction, inclination angle, elastic modulus, and maximum Schmid factor for each slip system and tensile twinning in the selected grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Grain Loading direction Inclination angle (°) Elastic modulus (GPa) Maximum Schmid factor Basal Prismatic Pyramidal Ⅰ Pyramidal Ⅰ Pyramidal Ⅱ Tensile twin A [112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='3] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='17 B [112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='43 C [101!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='49 D [0001] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='47 -* Tensile twinning cannot be activated during compression along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Inverse pole figure (IPF) map showing the crystallographic orientation of the grains in the Mg-Y-Ca alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) The loading direction in micropillars from grains A and B form an angle of ~ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5° with the [0001] crystal orientation and are parallel to [112$0], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (b) The loading direction in micropillars from grains C is parallel to [101$0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (c) The loading direction in micropillars from grain D is parallel to [0001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The compression loading direction is perpendicular to the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='1 Deformation mechanisms in micropillar of grain A The engineering stress-strain curves obtained from the compression micropillars carved from grain A along [112$3] orientation are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' For the sake of clarity, the horizontal axis of the green curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4a is shifted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' After the initial elastic region, the curves show gradual yielding and reach a plateau in the flow stress at an applied strain of ~ 5%, without significant work hardening afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This behavior is consistent with a plastic deformation dominated by basal slip in pure Mg and Mg alloys (Kiener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Small strain bursts (noticed by sudden drops in the stress) are present in the stress-strain curves and they are associated with the activation of dislocation sources in particular basal slip planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, the magnitude of the strain 400μm 200μm 011!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0 0001 1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='21!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0 Loading direction 200μm Grain A Grain B Grain C Grain D (a) (b) (c) 13 bursts is much smaller than that reported in other Mg alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In fact, large strain bursts are associated with the localization of deformation in a few slip planes along the micropillar (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, the lateral and top views of the micropillar after deformation (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4c and 4d, respectively) show evidence of uniform slip traces along the length and width of the micropillar, indicating that plastic deformation was homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' A yield stress of 65 ± 11 MPa (indicated by the black stars in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4a) was determined from the critical points in the engineering stress-strain curves when the curves deviated from linearity, following the procedure detailed in (Alizadeh and LLorca, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Secondary electron images of lateral and top views of the deformed micropillars were obtained in the SEM to ascertain the actual deformation mechanisms and are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4c and 4d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Many parallel slip traces appear on the top and lateral surfaces, which were not present before deformation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The orientation of the slip traces on the micropillar surfaces is indicated by the green dashed lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4c and 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The slip steps were obviously observed on the top view from the top right corner to the lower left corner, and the corresponding slip direction is determined as marked with a white arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' They are indicated by blue planes and red arrows, respectively, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4e and 4f within the crystallographic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It is evident that the slip traces in the micropillar are parallel to the basal planes and the shear deformation takes place along the [21$1$0] direction, as shown from the top and lateral views of the deformed micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In fact, the (0001) <21$1$0> basal slip system has highest SF (listed in Table 1) and plastic deformation along this slip system is dominant in this micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Therefore, the CRSS for basal slip (based on the yield stress and the corresponding SF) can be estimated as 29 ± 5 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) Engineering stress-strain curves obtained from micropillar compression tests in grain A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The yield stress is marked with a black star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' SEM images of the micropillar (b) before compression and after compression from the (c) right lateral view and (d) top view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The slip plane trace and slip direction are indicated by the green dashed lines and the white arrow, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The schematic crystallographic lattice of the corresponding slip plane is presented (e) for right side and (f) for top side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The blue planes indicate the theoretical basal glide planes, and the red arrows represent the corresponding shear directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2 Deformation mechanisms in micropillars of grains B and C Representative engineering stress-strain curves obtained from micropillar 150 Engineering stress (MPa) Top side 100 :0 50 Right side 0 2um 3 6 9 12 Engineering strain (%) Right side Top side Slip Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Basal plane direction Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Basal plane 2μm 2μm Basal slip plane Basal slip plane 15 compression tests along [112$0] in grain B and along [101$0] in grain C are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5a and 5b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The horizontal axis of the green and blue curves was shifted by -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='1% and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='1%, respectively, in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5b for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The stress-strain curves are smooth, without distinct strain bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The initial elastic region is followed by another linear plastic region with reduced strain hardening rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This behavior is radically different from that observed in micropillars with equivalent orientation in pure Mg and several Mg alloys (Mg-Al, Mg-Zn, Mg-Y and Mg-Zn-Ca) (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 2021a, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020), which presented large strain bursts after the initial elastic region due to the nucleation of tensile twins at the top of the micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' They are similar to those found in Mg-2Y (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) alloy at 250 ℃ (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b), where prismatic slip replaced twinning as the dominant plastic deformation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The yield stresses (obtained as indicated above and marked with purple stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5) were 219 ± 9 MPa and 228 ± 4 MPa along [112$0] and [101$0] orientations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) Engineering stress-strain curves obtained from micropillar compression tests in grain B along [112$0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (b) Idem in grain C along [101$0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The representative morphology of the micropillar deformed along [112$0] (grain B) is depicted in the SEM images in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 6a and 6b from two different sides (front and left, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Faint slip traces are visible on both lateral surfaces of the deformed micropillars, as indicated by the blue dashed lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 6c and 6d, which show the rectangular zones marked by dashed lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 6a and 6b, respectively, at higher magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The slip traces are distributed homogeneously along the lateral surfaces, Grain B:[112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] (a) (b) Grain C:[101!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] 350 300 250 200 50 100 50 0 2 4 6 8 10 12 0 Engineering strain (%)240 220 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='8280 230 180 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2350 Grain C:10101 300 D 250 stress 200 150 100 Engineering strain 16 indicating that plastic deformation was uniform along the micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, there are not slip steps at the surface (as opposed to the micropillar deformed along [112$3] in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4c and 4d), in agreement with the smooth stress-strain curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This deformation morphology is different from that observed in other Mg alloys compressed along a-axis (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020), where two regions with different contrast were always observed after the deformation due to the nucleation of tensile twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) 2μm Front side Left side 2μm (c) (b) 1μm Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Prismatic plane (d) 1μm (h) (i) Prismatic slip plane Left side Front side 2μm 4 0 2μm 2μm (e) (f) (g) Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Prismatic plane Before deformation After deformation KAM Kernel Aver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Misorient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 0 [o] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='96 Y1 5μm 光栅：498x331 步长尺寸：0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='025μm >X12μm 光栅：243x1692 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' SEM images of the micropillar deformed along [112$0] from grain B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) Lateral front and (b) lateral left view side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The traces of the active slip planes are indicated by the blue dashed lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5c and 5d, which show the rectangular zones marked by dashed lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5a and 5b, respectively, at higher magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (e) and (f) TKD maps of the lamella extracted from the undeformed region in grain B and along the compression direction from the deformed micropillar, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (g) KAM map of the deformed micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (h) and (i) Schematics of the crystallographic lattice showing the corresponding slip plane for the lateral front side and left side, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The red planes indicate the theoretical prismatic glide planes, and the blue lines represent the corresponding slip traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In order to identify the deformation mechanisms, two parallel thin foils were extracted from the undeformed region in grain B and along the loading direction from the deformed micropillar, respectively, and their orientation was determined by TKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The position of the lamellae is indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' S2 of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The corresponding orientation maps in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 6e and 6f show that the IPF map (∥Z) of the undeformed and deformed thin foils share the same orientation and demonstrate that tensile twins were not nucleated during micropillar compression up to 10% strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 6g presents the kernel average misorientation (KAM) map of the whole pillar in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 6g reveals the homogeneous deformation without shear bands assuming an angular threshold of 4°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The slip traces on the lateral surfaces of the micropillars were associated with the prismatic planes, as indicated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 6h and 6i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, prismatic slip was triggered at the onset of the yielding and dominated plastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The maximum SFs for prismatic slip, pyramidal I, pyramidal II slip and tensile twinning were very similar along this orientation (Table 1) but the presence of Y and Ca in solid solution favored the activation of prismatic slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It should be noted that pyramidal I slip dominated plastic deformation and hindered the development of tensile twinning in micropillar compression tests along [1$21$0] orientation in a Mg-4Y (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The maximum SFs for pyramidal I slip, tensile twinning and prismatic slip in this orientation were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='41, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='46 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='49 and, thus, the preference of pyramidal slip can be associated with the higher CRSSs for tensile twin nucleation and prismatic slip in Mg-4Y alloy (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 18 Similar deformation morphology was found in the micropillars deformed along [101$0] in grain C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Continuous slip traces were homogeneously distributed along the lateral surfaces, as indicated by the dashed blue lines from in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 7b, which shows the rectangular region marked by dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 7a at higher magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' As in the previous case, the micropillar orientation before and after deformation was assessed by TKD carried out in a thin lamella extracted from the undeformed region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 7c) and from the deformed micropillar along the loading direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 7d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The relative orientation between the two thin lamellas is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' S3 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The corresponding orientation maps do not show any evidence of tensile twinning and prismatic slip was again the dominant plastic deformation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This conclusion is supported by the uniform plastic deformation without obvious shear bands revealed by the KAM map assuming an angular threshold of 2° (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 7e) and the agreement between the slip traces on the lateral surfaces with the orientation of the prismatic planes in the micropillar (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 7f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, the CRSSs for prismatic slip (obtained from the yield stress and the SF for both micropillar orientations) were determined to be 105 ± 4 MPa and 105 ± 2 MPa along [112$0] and [101$0] orientations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 19 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' SEM images of the micropillar deformed along [101$0] from grain C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) Lateral left view side and (b) which shows the rectangular zone marked with a dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 7a at higher magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The traces of the active slip planes are indicated by the blue dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (c) and (d) TKD maps of the lamella extracted from the undeformed region in the grain and along the compression direction from the deformed micropillar, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (e) KAM map of the deformed micropillar in grain C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (f) Schematic of the crystallographic lattice showing the corresponding slip plane for the lateral left view in (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The red plane indicates the theoretical prismatic glide plane, and the blue 2μm (e) Prismatic slip plane Left side 2μm (c) 1μm Left side Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Prismatic plane (a) (b) 2μm (d) (f) 2 0 2μm Before deformation After deformation 2KAM Kernel Aver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Misorient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 光栅：164x268步长尺寸：0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='03μm Y1 2μm ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='X1IPF IPF Coloring II ZO Magnesium 0001 12-10 01-10 Y1 2μm 光栅：220x145 步长尺寸：0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='04μm >X1 20 line represents the corresponding traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Further assessment of the deformation mechanisms was carried out by means of TEM observations of the dislocation structures in a thin lamella extracted from the micropillar deformed along [101$0] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The lamella was nearly parallel to (1$21$0) plane, as confirmed by the SADP in the inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 8a, and there are no traces of twinning in the micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Dark field micrographs of the square region marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 8a are depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 8b and 8c with g = (101$0) and g = (0002), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Large density of dislocations is observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 8b but they disappear from this region when g = (0002) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' They are obviously dislocations with 1/3 a [112$0] or 1/3 [21$1$0] Burgers vector, based on the dislocation extinguish condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, the SF of the {011$0} [21$1$0] prismatic slip system is very low (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='05), thus, it is reasonable to assume that the Burgers vector of the dislocations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 8b is 1/3 [112$0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The screw dislocations are observed under g = (101$0) condition as marked with yellow arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The Burgers vector of screw dislocation is parallel to the dislocation line, leading to the straight dislocation lines nearly parallel to trace of the basal planes (marked with a green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' TEM micrographs of the lamella extracted from the micropillar deformed along [101$0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The beam direction is parallel to [1$21$0] orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) Low magnification view of the lamella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (b) and (c) High magnification dark field micrographs with g = (101$0) and g = (0002), respectively, from the region marked with a blue square in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The activation of the prismatic slip during compression along the a-axis has been reported recently in Mg-Zn-Ca alloy (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b) in combination with tensile twinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, the activation of prismatic slip and the suppression of tensile twinning during compression along the a-axis has not been found at ambient temperature in pure Mg (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2017) or any Mg alloys (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This result is very surprising because compression of Mg and its alloys along the a-axis (or equivalent extension along the c- axis) easily leads to the nucleation and growth of {101$2} tensile twins, because the associated CRSS to promote tensile twin is much lower than that necessary to activate 5μm (a) 200nm 200nm (b) (c) g=(101!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0) g=(0002) dislocations (0001) plane B=[1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='21!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] 22 pyramidal slip or prismatic slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' While the addition of Y and Ca in solid solution leads to a large increase in the CRSS for prismatic slip with respect to pure Mg (from 39 MPa in pure Mg (Kaya, 2013) to 105 MPa), it seems to have a much larger effect on the CRSS for twin nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In fact, considering the maximum stresses attained in the micropillar compression tests along [112$0] and [101$0] orientations (258 MPa in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5a and 303 MPa in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5b, respectively) and the maximum SFs for tensile twinning in both orientations (Table 1), it can be estimated that the CRSS for twin nucleation in the Mg-Y-Ca alloys should be higher than 148 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='3 Deformation mechanisms in micropillar of grain D The engineering stress-strain curves obtained from the compression micropillars carved from grain D along [0001] orientation are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' After the elastic region, a strong linear hardening was observed in the plastic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The yield stress (marked by the purple stars in the inset) was 431 ± 15 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This mechanical response is in good agreement with the results reported in Mg-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='4Y (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) and Mg-4Y (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) alloys (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020) as well as in precipitation-hardened Mg-4Zn (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) alloy (Alizadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021) under c-axis compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=" In all these cases, the presence of Y in solid solution or of β1 ' precipitates increased the CRSS for basal slip and plastic deformation was accommodated through pyramidal slip due to the low SF of basal planes in this orientation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The SEM micrograph of the lateral side of deformed micropillar in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 9b, shows no slip traces but this behavior is also typical of pyramidal slip, which does not lead to visible slip traces on the micropillar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The orientation of the micropillar after deformation was assessed by TKD in a thin lamella extracted along the compression direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The IPF map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 9c indicates the absence of the tensile twinning during deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) Engineering stress-strain curves from the micropillar deformed in compression along [0001] in grain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (b) SEM micropillar of the lateral side of the deformed micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (c) IPF map of the lamella extracted from the deformed micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' To further elucidate the deformation mechanisms, the analysis of the dislocation structures was carried out by TEM in a thin lamella extracted from the deformed micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The beam direction was parallel to [112$0] as confirmed by the SADP in the inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Two-beam condition imaging was performed with g = (0002) and g = (101$0) and the corresponding dark field micrographs are depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 10b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 10c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' A large density of dislocations (marked with blue arrows) is observed under g = (0002) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 10b, and some components are still in contrast at the same location when the operation vector changes to g = (101$0) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 10c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The detail of the rectangular region marked with purple dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 10b is shown at higher magnification in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 10d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The dislocations (marked with the blue dashed lines) are [1$1$23]/3 and [12$1$3]/3 according to the [112$0] crystal orientation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 10e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' These results are in agreement with those reported in Mg-Zn-Ca and Mg-Y alloys (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Nevertheless, it should be noticed that it is difficult to identify the active pyramidal plane, since both pyramidal I and pyramidal II planes contain the same slip directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, it can be concluded that plastic deformation along the c-axis in compression was dominated by pyramidal dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The activation of pyramidal slip was associated to homogeneous deformation and strong strain hardening (Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This high hardening rate is likely associated with short mean-free paths and this explains why no slip traces were 2μm (b) After compression (a) 1μm (c) 2700 600 90 500 400 300 200 100 10 ineeiring Stran500 400 300 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='T 24 found on the micropillar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It is not clear whether slip took place along pyramidal I plane or pyramidal II plane but the SF is slightly higher for pyramidal II for this particular orientation (Table 1), which is assumed to be the active one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, the CRSS for pyramidal Ⅱ slip can be estimated as 203 ± 7 MPa from the SF and the yield stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (a) Bright field TEM image of the thin lamella extracted from the micropillar deformed along the [0001] orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The beam direction is [112$0], as shown by the SADP in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (b) and (c) Dark field TEM micrographs of the square region marked with orange dash lines in (a) under g = (0002) and g = (101$0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (d) Bright field TEM micrograph obtained under g = (0002) from the rectangular region marked with purple dash lines in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The potential pyramidal dislocations are indicated by the crystal orientation (black box) and the pyramidal slip trace (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' (e) Schematic of Mg crystal orientation and of the corresponding dislocations (blue lines) from the [112$0] projected view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='3 Effect of Y and Ca on the GSFE curves (a) 1μm g=(101!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0) components (0001) plane dislocation 100nm 100nm (b) (c) (d) (e) 50nm g=(0002) B=[112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] g=(0002) Basal trace 25 The experimental evidence presented above shows that the addition of Y and Ca affects significantly the plastic deformation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The changes in the deformation mechanisms are proposed to be associated with the modification of the slip resistance of the different slip systems due to the presence of the solute atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The GSFE is intimately associated with the activation barriers of the deformation modes, hence influencing their relative contributions to the overall deformation behavior (Sandlöbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' To ascertain the effect of the solute atoms (Y and/or Ca) on the slip activities, the GSFE (γ) curves were computed for the slip systems in Mg-Y, Mg-Ca, and Mg-Y-Ca alloy, as well as in pure Mg for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The GSFE curves for {0001}<101$0>, {11$00}<112$0> and {101$1}<1$21$0> slip systems are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 11a, 11b and 11c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The curves exhibited only one local maximum, from which the unstable stacking fault energy (γus) for each slip system was determined (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' γus is associated with the activation barrier for dislocation slip (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Evidently, the addition of Y and Ca reduced slightly the γus for basal slip from 88 mJ/m2 in pure Mg to a minimum of 64 mJ/m2 in Mg-Ca or of 73 mJ/m2 in Mg-Y and the γus of Mg-Y-Ca (74 mJ/m2) was similar with that of Mg-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, the reduction in γus for the prismatic slip system was much more important, from ~235 mJ/m2 in pure Mg to γus of ~18 mJ/m2 in the Mg-Y-Ca alloy (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This synergistic contribution of Y and Ca on γus for prismatic slip is obvious as the sole addition of either Y or Ca only reduced γus to 120 mJ/m2 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The dramatic reduction of γus for prismatic slip in the Mg-Y-Ca alloy facilitates the activation of this deformation mechanism during plastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' On the contrary, the γus for pyramidal Ⅰ only changed from 304 mJ/m2 in pure Mg to 318 mJ/m2 in Mg-Y-Ca alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The sole addition of Ca (308 mJ/m2) did not modify significantly γus for pyramidal Ⅰ while Y (359 mJ/m2) increased slightly γus for pyramidal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, prismatic slip is favored by the addition of Y and Ca in comparison with pyramidal I slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 26 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Generalized stacking fault energy curves for (a) basal (b) prismatic, and (c) pyramidal Ⅰ slip systems in pure Mg, Mg-Ca, Mg-Y and Mg-Y-Ca alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The calculated γus, for basal , prismatic , and pyramidal Ⅰ , slip systems in the Mg-Ca, Mg-Y and Mg-Y-Ca alloys compared with pure Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Alloy γus (mJ/m2) Basal Prismatic Pyramidal Ⅰ Mg48 88 235 304 Mg47Ca1 64 121 308 Mg47Y1 73 118 359 Mg46Y1Ca1 74 18 318 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='1 Effect of Y and Ca on the CRSSs The yield stresses measured from the micropillar compression tests in different (a) (b) (c) ikgas Tas Ipyranniclalsillp 300 20 200 20160 100 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0 0,2) OLA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0 lraictonaldlspiaxeementof8<112os300 pmshatcslp 250 200 160 100 (),, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0 bracuoneldisplakeementof 1120100 lbersalslip 80 6X0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0 Hracuonealdisplakeementof sslolcs 27 orientations are summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The CRSS for the dominant slip system in each orientation (following slip trace analysis and TEM characterization) is also presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' They are basal slip in the micropillars carved from grain A, prismatic slip in the micropillars from grains B and C, and pyramidal II slip in the micropillars from grain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, twin nucleation was not observed in micropillars carved from grains B and C and this result can be used to obtain thresholds of the CRSS for twin nucleation from the maximum stress attained during the test and the maximum SF for tensile twinning in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' These minimum values are also included in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It should be noticed that dimensions of the micropillars selected in this investigation follow previous results in Mg alloys (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020) that indicate these values should not be very much influenced by the “smaller is stronger” effect reported for micropillar compression tests at the micron or sub-micron scale (Aitken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Yield stress and CRSS for different slip systems from micropillar compression tests along different orientation in the Mg-Y-Ca alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Grain Loading direction Yield stress (MPa) CRSS (MPa) A [112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='3] 65 ± 11 29 ± 5 ( basal slip) B [112!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] 219 ± 9 105 ± 4 ( prismatic slip) > 111 MPa (tensile twin*) C [101!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0] 228 ± 4 105 ± 2 ( prismatic slip) > 148 MPa (tensile twin*) D [0001] 431 ± 15 203 ± 7 ( pyramidal Ⅱ slip) : Tensile twin was not nucleated when the CRSS reached this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In order to ascertain the strengthening effect of Y and Ca atoms in solid solution, the CRSSs for the different slip systems in Mg-Y-Ca alloy are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 12 along with those reported in the literature in pure Mg (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019a), Mg-Al (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020, 2019a), Mg-Zn (Li, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a), Mg-Y (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020), Mg-Zn-Ca (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a), Mg-Al-Ca (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022) 28 and Mg-Y-Zn (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018) alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' All these results were obtained from compression tests in micropillars with a cross-section around 5 × 5 μm2 and, thus, size effects -if any- should not affect the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The results for basal slip in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 12a show that the addition of Y in solid solution dramatically increases the CRSS in comparison with pure Mg (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019a) and with Mg-Al (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019a) or Mg-Zn (Li, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a) alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' These experimental data are supported by the first principles simulations of the solute/dislocation interaction energy, which showed the higher strengthening potential of Y for basal dislocations, in comparison with Al and Zn, because of the larger atomic radius and shear modulus misfit of Y with respect to Mg (Tehranchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The addition of Ca to the Mg-Y does not increase the CRSS for basal slip according to our results while the strengthening effect of Ca in Mg-Al (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022) or Mg-Zn (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a) is limited and may also be attributed to the elastic interaction between Ca solute atoms and dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Regarding pyramidal slip (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 12b), Zn and Al are the alloying elements which lead to the largest increase in the CRSS (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zn is more efficient but the solubility of Al in Mg is larger and CRSSs in the range of 200- 250 MPa can be achieved for these binary alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Addition of 4 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' % Y increases the CRSS up to 106 MPa (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020) but the combination of Y and Ca leads to a CRSS of 203 ± 7 MPa, similar to the one found in the binary Mg-Zn alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, Zn, Al and Y solutes increase the CRSS for pyramidal slip due to the elastic interaction of the solutes with the dislocations, as in the case of basal slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It should be noticed that the activation and glide of pyramidal dislocations is a complex process that also depends on dislocation dissociation during gliding due to the larger Burgers vector (Moitra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Tang and El-Awady, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Atomistic simulations have shown that the presence of Y and Ca favors the activation for cross-slip/double cross-slip of pyramidal dislocations, leading to new dislocation loops which can accommodate plastic deformation (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 29 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' CRSS for (a) basal slip, (b) pyramidal slip and (c) twin nucleation and prismatic slip in Mg and Mg alloys (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Kiener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Li, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a, 2020, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020), including the results obtained for Mg-Y-Ca alloy in this investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' All data were obtained from compression tests in micropillars with a cross-section around 5 × 5 μm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The arrow in the CRSS for twin nucleation in Mg-Y-Ca indicates that the actual CRSS is higher than the value in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The CRSSs for tensile twin nucleation, measured by means of micropillar compression tests in Mg and different Mg alloys, are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 12c (Kiener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' While the addition of Y (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020) and Al (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020) lead to the largest enhancements in the CRSS for twin nucleation (the latter because of the larger solid solubility), the highest CRSS is obtained for the ternary Mg-Y-Ca alloy which -following our experimental results- has to be higher than 148 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Generally, the twin nucleation process is dominated by the dislocation-shearing and atomic shuffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The strong strengthening (a) (b) (c) 180 160 140 $120 100 Prismatic slip hg 410 20 8 10200 150 100 nCaamees 10410 RSS 10 30 provided by Y on the CRSS for twin nucleation can be ascribed to the inhibition of atomic shuffling due to the large atomic radius of Y (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='180 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, Ca has an even larger atomic radius (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='194 nm) and it is proposed that the synergistic contribution of both atoms in solid solution is responsible for the huge increase in the CRSS for twin nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In addition, the elastic interaction of twinning dislocations with different solute atoms also leads to an increase in the CRSS for twin propagation (Ghazisaeidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Stanford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2015), as it has been reported in previous investigations (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a, 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, only the addition of Y and Ca can inhibit twin nucleation in micropillars suitable oriented for twinning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', deformed in compression along [112$0] and [101$0] (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The high CRSS for tensile twin nucleation in Mg-Y-Ca alloys leads to the activation the prismatic slip, which becomes the dominant plastic deformation mechanism under a-axis compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' There is limited information on the CRSS for prismatic slip (because either basal slip or tensile twinning are usually activated before prismatic slip to accommodate the plastic deformation) and the available experimental data on Mg-Y-Zn (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018) (102 MPa) and Mg-Y-Ca (105 ± 4 MPa) are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 12c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The CRSS for prismatic slip is much lower than the CRSS for tensile twin nucleation in Mg-Y-Ca and, thus, tensile twinning is suppressed during compression parallel to the a-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The CRSSs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 12 show that the strengthening effect of the Y and Ca for prismatic slip is much lower than the ones reported for pyramidal slip and twin nucleation, and also, in relative terms, for basal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, evidence of prismatic slip is unusual in Mg alloys except in the case of that they contain Ca (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019), indicating that the presence of Ca reduces the activation barriers for prismatic slip glide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Besides, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', (2018) found that prismatic slip was activated during micropillar compression testing of Mg-Y-Zn alloys but it was absent in solution-treated Mg-Zn alloys deformed along the same orientation (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019b), implying that the addition of Y also facilitates prismatic slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Although the elastic interaction of the solute atoms with prismatic dislocations is expected to increase the CRSS, the reduction of the stacking fault energy due to the 31 presence of Y and Ca reduces the activation barrier for dislocation movement on the slip plane and facilitates the activation of this slip system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Overall, the addition of Y and Ca leads to a marked solid solution strengthening for basal and pyramidal slip as well as for the nucleation of tensile twins but not for prismatic slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2 Effect of plastic anisotropy on the ductility In general, the tensile ductility and formability of Mg alloys during the plastic deformation is dictated by the CRSS ratio between different slip systems, especially between non-basal and basal slip, the latter being the dominant deformation mechanism in most cases (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Therefore, the tensile ductility of different Mg alloys is plotted as a function of the CRSS ratios between different slip systems in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 13 (Habibi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019a, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The CRSS ratios were measured via micropillar compression tests in most of the alloys (Agnew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a, 2020, 2019a, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019) with a few exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The CRSS ratio between prismatic and basal slip in Mg-5Y (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018) and Mg-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='47Ca (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019) were obtained from mechanical tests in polycrystals via slip trace analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Besides, those for pure Mg (Agnew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2003), Mg-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5Ca (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) (Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021) and Mg-3Y (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018) alloys were determined by the elasto-plastic self-consistent model, crystal plasticity finite element simulations, and the elastic viscoplastic self-consistent model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, the tensile elongation data were collected from pure Mg and wrought Mg alloys with similar grain sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In general, reduced ratios between the CRSS for non-basal slip and basal slip are strongly associated with the improvement of the ductility of Mg alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This trend agrees with the data plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 13b, which shows a clear link between the reduction of CRSS pyramidal / CRSS basal and the increase in tensile elongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, the limited data of the influence of CRSS prismatic / CRSS basal on the tensile ductility 32 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 13a are not conclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Obviously, low CRSS pyramidal / CRSS basal ratios favor isotropic deformation and limit the development of strong basal textures and both processes help to improve ductility and formability because activation of dislocations benefits the strain accommodation along the c-axis (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Besides, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', (2018) predicted that the addition of Y/Ca could significantly reduce the cross-slip energy barriers between pyramidal I and pyramidal II planes, thus promoting dislocation cross-slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Enhanced non-basal slip activities and cross- slip induce homogeneous deformation and improve the ductility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Relation between the CRSS ratios of different slip systems (Agnew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a, 2020, 2019a, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019) and the tensile elongation (Habibi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019a, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020, 2019) in pure Mg and Mg alloys: (a) CRSS prismatic / CRSS basal, (b) CRSS pyramidal / CRSS basal, (c) CRSS prismatic / CRSS tensile twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It should be noted that these mechanisms are particularly relevant in Mg-Y (Wang (a) (b) (c) Pure Mg (Zhu,2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Agnew,2003) Pure Mg (Habibi, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Agnew,2003) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='71 Mg-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='5Ca(Zhu,2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Shang,2021) Mg-3Y(Wang,2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Zhao,2019b) ★Mg-5Y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='08Ca (This work) 30 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=':0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='0 CRSPure Mg(Li,2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Habibi,2012) PureMg(LI,2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Wang,2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Zhu,2020) Mg-5Zn(Shi,2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Li,2021a) Mg-4Y(Wu,2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Wu,2010) Mg-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='4Al(Zhao,2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Wang,2020) Mg-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='8Zn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2Ca (Wang,2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Wang,2021b) Mg-5Y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='08Ca (This work) 20 1030 10 Mg-3Y(Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Zhao,2019b) Mg-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='47Ca(Zhu,2019) Mg-5Y (Huang,2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='Yang,2020) ★Mg-5Y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='08Ca (This work) 33 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2018) and Mg-Zn-Ca (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a, 2021b) alloys as well as in the Mg-Y- Ca alloy analyzed in this investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In all these cases, the presence of Y and/or Ca also leads to a high increase in the CRSS for twin nucleation while the CRSS for prismatic slip is not strongly affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' As a result, the CRSS prismatic / CRSS tensile twin is dramatically reduced and this is accompanied by a large increase in the tensile ductility, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 13c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Particularly, tensile twinning is replaced by prismatic slip during compressive deformation along the a-axis if CRSS prismatic / CRSS tensile twin < 1 and twinning only occurs in grains deformed in tension along the c- axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, as the CRSS for prismatic slip is smaller than that for pyramidal slip, the former becomes the dominant plastic deformation mechanism in grains suitable oriented for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It should be noted that pyramidal slip is associated with a large strain hardening (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 9a) that it is not present for prismatic slip (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Thus, pyramidal slip induced large stress concentrations at grain boundaries that facilitate the nucleation of damage but this process is not activated if prismatic slip is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In general, the preferential activation of basal slip and tensile twinning during processing always introduces a strong basal texture in wrought Mg and Mg alloys, leading to the plastic anisotropy, crack formation and limited ductility (Sabat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The addition of Y and Ca in our alloy strongly enhanced the activation of prismatic and pyramidal slip, which also contribute to reduce the intensity of the texure during extrusion, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 2a and 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This limited texture also contributes to reduce the plastic anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It should also be noted that the overall mechanical response of polycrystals cannot fully ascertained by means of micromechanical tests in single crystals because other factors (grain boundaries, grain size and texture) play a key role in the mechanical response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' However, it should be emphasized that the plastic deformation of each crystal within the polycrystal is intrinsically related to that of a single crystal (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a) and, hence, it is important to ascertain the plastic deformation mechanisms in single crystals to understand the complex mechanisms in bulk polycrystalline samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Overall, these results indicate that the presence of Y and Ca in solid solution in Mg 34 alloys leads to a large increase in the CRSS for basal slip (which induces a large reduction in CRSS pyramidal / CRSS basal) while CRSS prismatic / CRSS tensile twin < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' As a result, plastic deformation in polycrystals in more isotropic and localization of the deformation in the form intense basal slips that promote fracture is suppressed (Sandlöbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, twinning and pyramidal slip are replaced by prismatic slip in grains deformed along the a-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Suppression of twinning (which induces strong anisotropy in the plastic deformation in textured alloys) and the activation of prismatic slip (which provides an additional plastic deformation mechanism with limited hardening) lead to an important improvement in the tensile ductility of Mg alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Conclusions The deformation mechanisms of a Mg-5Y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='08Ca (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' %) alloy, with a superior tensile elongation (32%), were studied by means of micropillar compression tests, slip trace analysis along different orientations, TEM as well as TKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' It was found that the presence of Y and Ca in solid solution led to a huge increase in the CRSS for basal slip (29 ± 5 MPa), pyramidal slip (203 ± 7 MPa) and tensile twin nucleation (above 148 MPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' This behavior was attributed to the large mismatch of the atomic radii and elastic constants of the Y and Ca atoms with respect to Mg, which leads to a strong interaction of the dislocations with the solute atoms and hinders atomic shuffling, that is necessary to activate twin nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' On the contrary, the CRSS for prismatic slip only increases up to 105 ± 4 MPa because the hardening induced by the interaction of the solute atoms with dislocations is partially balanced by the reduction in the stacking fault energy associated with prismatic slip due to the presence of Y and Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' The changes in the CRSS for slip and tensile twinning in Mg-Y-Ca alloys modify the dominant deformation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' In particular, the CRSS prismatic / CRSS tensile twin is dramatically reduced and tensile twinning is replaced by prismatic slip during compressive deformation along the a-axis if CRSS prismatic / CRSS tensile twin < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Moreover, as the CRSS for prismatic slip is smaller than that for pyramidal 35 slip, the former becomes the dominant plastic deformation mechanism in grains suitable oriented for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' As a result, reduction of twinning (which induces strong anisotropy in the plastic deformation in textured alloys) and the activation of prismatic slip (which provides an additional plastic deformation mechanism with limited hardening) lead to an important improvement in the tensile ductility of Mg alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 52001199 and 51825101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Cui acknowledges the support from the Shanghai Sailing Program (Grant No.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' {112$2}<112$3> Slip system in magnesium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Acta Metall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 21, 845–853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Partridge, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 1967.' metadata={'source': 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On the role of non-basal deformation mechanisms for the ductility of Mg and Mg-Y alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Acta Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 59, 429–439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Shang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Zhu, G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Plast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 144, 103040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Shi, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Deng, K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Towards high ductility in magnesium alloys - The role of intergranular deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Plast.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' On the inhomogeneous deformation behavior of magnesium alloy beam subjected to bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Plast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 150, 103180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Tang, Y.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Deformation mechanisms 39 of Mg-Ca-Zn alloys studied by means of micropillar compression tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Acta Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Li, N.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Zeng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Predicting grain boundary damage by machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Plast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 150, 103186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Tu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Dai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Shin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Pan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Strain Hardening Behavior in Mg–Al Alloys at Room Temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 21, 1801062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Tu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Luo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Cheng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Atrens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Pan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Strain hardening behavior of Mg–Y alloys after extrusion process.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Highly deformable Mg–Al–Ca alloy with Al2Ca precipitates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Acta Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 200, 236–245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Zhu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Shen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Tu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Zhu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Jin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', Zeng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Improving ductility of a Mg alloy via non-basal slip induced by Ca addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' Plast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} +page_content=' 120, 164–179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf'} diff --git a/9dFPT4oBgHgl3EQfYjT_/content/tmp_files/2301.13074v1.pdf.txt b/9dFPT4oBgHgl3EQfYjT_/content/tmp_files/2301.13074v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4256aab145bfcc841a45b54124dfc7b7348cc94 --- /dev/null +++ b/9dFPT4oBgHgl3EQfYjT_/content/tmp_files/2301.13074v1.pdf.txt @@ -0,0 +1,2263 @@ +Prepared for submission to JHEP +Dynamic Radius Jet Clustering Algorithm +Biswarup Mukhopadhyayaa, Tousik Samuia, and Ritesh K. Singha +aDepartment of Physical Sciences, Indian Institute of Science Education and Research Kolkata, +Mohanpur, 741246, India. +E-mail: biswarup@iiserkol.ac.in, tousiksamui@gmail.com, +ritesh.singh@iiserkol.ac.in +Abstract: +The study of standard QCD jets produced along with fat jets, which may +appear as a result of the decay of a heavy particle, has become an essential part of collider +studies. +Current jet clustering algorithms, which use a fixed radius parameter for the +formation of jets from the hadrons of an event, may be inadequate to capture the differing +radius features. +In this work, we develop an alternative jet clustering algorithm that +allows the radius to vary dynamically based on local kinematics and distribution in the η-φ +plane inside each evolving jet. We present the usefulness of this dynamic radius clustering +algorithm through two Standard Model processes, and thereafter illustrate it for a scenario +beyond the Standard Model at the 13 TeV LHC. +arXiv:2301.13074v1 [hep-ph] 30 Jan 2023 + +Contents +1 +Introduction +1 +2 +Methodology +3 +2.1 +Standard Sequential Recombination Algorithms +3 +2.2 +Our Proposal: Dynamic Radius Jet Clustering Algorithm +4 +3 +Application to Standard Model Processes +7 +3.1 +Illustration I: pp → tj process +8 +3.2 +Illustration II: pp → V j Subprocess +15 +4 +Usefulness in BSM signals +19 +5 +Summary and Outlook +25 +1 +Introduction +The physics extraction capacity of any high-energy collider depends crucially on the han- +dling of coloured particles in various final states. These are produced as partons via ei- +ther short-distance interactions of quantum chromodynamics (QCD) or electroweak pro- +cesses [1, 2]. The partons, however, hadronize through long-distance QCD effects which +are not calculable ab initio. One rather uses semi-empirical methods to predict the prob- +ability that energetic partons will fragment into more low-energy partons and ultimately +form colour-neutral hadrons which are observable in the detector. Groups of closely spaced +hadrons with varied degrees of collimation form ‘jets’ whose identification, isolation, and +merger are predicted once more with the help of semi-empirical (and by no means uniquely +decided) algorithms called jet clustering algorithms [3–7]. The aim always remains to define +jets with such algorithms which most accurately elicit the short-distance physics underly- +ing the events that are studied. They thus constitute some of our most important tools in +the analysis of phenomena at colliders. +In the context of the Large Hadron Collider (LHC), a widely used class of jet criteria is +based on so-called kt-type sequential recombination jet algorithms [7–13]. These algorithms +(briefly discussed in the next section) typically try to merge ‘neighbouring’ hadrons to +identify the group as a jet. The neighbourhood of a hadron is defined by a single radius +parameter R0 in the η-φ plane of the detector, which is used to quantify the radius (or size) +of a jet. This is because the hadrons within R0 are merged to form a jet while the hadrons +outside R0 are not included in that jet. The choices for the value of R0 in these algorithms +depend on the physics searches one is carrying out. At the 13 TeV LHC, the typical choices +for R0 are 0.4 or 0.8 for a ‘narrow’ or a ‘fat’ jet, respectively. There are, in addition, jet +isolation criteria depending on whether one is trying to separate a jet from a hard lepton +or another hadronic jet. However, the sequential recombination algorithms generally do +not accommodate varying choices of radii on a jet-by-jet basis in a single event since they +– 1 – + +have a single constant parameter that determines the radius of a jet. Separate classifiers +for a ‘narrow’ jet and a ‘fat’ jet in a single event in the current kt-type algorithms are thus +difficult to set. An important improvement over the current fixed radius algorithms would +be to make them adapt the jet radii dynamically jet-by-jet in each event. We make an +attempt in this direction in this work. +Our central idea of choosing the radius dynamically of a jet, especially for a boosted fat +jet, is based on the kinematics of the decay products of the initiating heavy particle. From +the theoretical side, the formation of boosted fat jet occurs due to the high collimation of +the on-shell decay products – and their showering and subsequent hadronization – of the +energetic and therefore boosted heavy particles. This is very different from the formation of +light quark- or gluon-initiated jets, whose collimation is primarily due to parton showering +and subsequent hadronization. On the other hand, at the operational level, as per the +standard kt-type algorithms, the fat jets are formed in the same way as the regular ‘narrow’ +jets, which are initiated by light quarks or gluons. However, the kinematics of on-shell decay +products and their radiation pattern of a heavy particle is different from the showering of +energetic light quarks or gluons. Therefore, the internal structure of a fat jet is very different +from a narrow one. These internal structure has been used to tag different heavy and +light jets in the LHC context. For example, jet substructure (JSS) observable generalized +angularities λκ +β [14, 15] is used to distinguish between quark- and gluon-initiated jets [16– +26]. The same variable was used in the classification among the narrow jet, fat W jet, +or boosted top jet [27–30]. Another important set of JSS observables, namely the energy +correlation functions (ECFs) [31, 32], was shown to be useful in classifying different types +of jets [29, 33–37]. The observable N-subjettiness (τN) [38, 39] has been used to find the +multi-pronged nature of light or heavy jets [40–65]. These variables have also been used +extensively by the experimental collaborations at the 13 TeV LHC [66–68]. These examples +try to exploit the energy distribution pattern inside a jet to distinguish a heavy object from +a QCD jet. The common theme of these jet substructure variables is the utilization of the +‘multi-pronged’ nature of the fat jets. Due to this multi-pronged nature, one expects the +variance of inter-constituent distance ∆R of a fat jet to be significantly different compared +to the narrow QCD jets. +This variance of a jet can be used to grow the radius of a +jet starting from an initial radius. Earlier attempts to make the jet radius variable, albeit +with somewhat different motivations and formalisms, can be found in references [69, 70]. In +Ref. [69], the effective radius of a pseudojet during their evolution was taken to be inversely +proportional to the pT with a maximum cut-off on the radius. Essentially, this algorithm +starts from a big effective radius and the size shrinks as a process of evolution. On the +other hand, in Ref. [70], an expectation-maximization approach was taken for clustering +the hadrons into a pre-determined number of clusters (jets). Our approach, in this work, is +to modify the standard fixed radius kt-type algorithms to make the radius grow depending +on the local kinematics and distribution (in the η-φ plane) of the hadrons. +The rest of the article is organized as follows. In section 2, we briefly outline the kt-type +sequential recombination algorithms followed by our improvement to the same. We test +the efficacy of our algorithms on two SM processes and discuss them in section 3. Section 4 +deals with one application in the BSM scenario. We summarize and conclude in section 5. +– 2 – + +2 +Methodology +2.1 +Standard Sequential Recombination Algorithms +At the operational level, a jet is constituted by a bunch of four-momenta obtained using +some clustering algorithm. Among various possible ways of grouping up the four-momenta +of an event, we need to choose those relevant to physics at the collider. It is important +that the clustering algorithm should ensure infrared and collinear (IRC) safety, which, in +our context, can be defined in terms of the following conditions [7]: +Infrared (IR) safety: The output of the algorithm should not be affected by the intro- +duction of a four-momentum with p → 0. +Collinear (C) safety: The output of the algorithm should not be affected by a collinear +splitting of any four-momentum. +The algorithm that best takes care of the issue of IRC safety is known as kt-type sequential +recombination jet clustering algorithms [7]. We briefly outline these algorithms below1. +If an event consists of N final state particles, whose four-momenta are taken in a list as +an input of the kt-type algorithms. The distance dij between the ith and jth four-momenta +and the distance diB between the ith and the beam are then defined as +dij = min +� +p2p +Ti, p2p +Tj +� +∆R2 +ij, +(2.1) +diB = p2p +TiR2 +0, +(2.2) +where R0 is the radius parameter of the algorithm, ∆Rij is the Euclidean distance between +the ith and jth four-momenta in the η-φ plane, and pTi is pT of ith four-momenta. The +exponent p sets the weight factor to the Euclidean distance in the η-φ plane. The three +choices of p = 1, 0 and −1 correspond to the kt (KT) [8–10], Cambridge-Aachen (CA) [11, +12], and anti-kt (AK) [13] algorithms, respectively. The algorithm for combining nearby +four-momenta with respect to the above distance measures to form jets has the following +steps. +Step 1. The distances dij for all the possible pairs and beam distances diB for all the +four-momenta are calculated first. +Step 2. The minimum among all the dij and diB’s is determined. +Step 3a. If the minimum occurs at one of the i, j pairs, the corresponding ith and jth +four-momenta are merged to form a new four-momentum. The older ones, ith and +jth four-momenta are removed from the list and the newly merged one is added to +the list and goes back to Step 1. +Step 3b. On the other hand, if the minimum distance is one of the diB, the ith four- +momenta is declared as a final jet, and it is removed from the list and goes back to +Step 1. +1Here, we only discuss the inclusive algorithms in the LHC context. For other jet clustering algorithms, +please see Ref. [7]. +– 3 – + +Step 4. The process is stopped once the list gets empty. +This class of algorithms is seedless because the clustering of four-momenta to form a +jet does not start from a particular seed. Rather, the algorithms try to merge the closest +pair first. A group of hadrons is then declared as a jet when an appropriate size is reached. +The essential difference among the three different algorithms, viz. AK, CA, and KT is +that they give different weights to the Euclidean distance in the η-φ plane. This typically +sets some sort of seed to the clustering algorithms in the sense that it gives a preference +to a hadron around which four-momenta merge to give rise to a final jet. In the case of +the KT algorithm, it is the softer (in terms of pT ) constituent which merges first and then +the harder ones get attached to it. As a result, the shape of the final jet may not be +circular in the η-φ plane. On the other hand, in the AK algorithm, the hardest particle in +a neighbourhood becomes some sort of seed for the jet and the softer ones merge at a later +stage. Hence the final jet looks circular in the η-φ plane. In the CA algorithm, the merging +is purely angular. Among the three algorithms, the AK algorithm is the most popular one +owing to its circular shape. Importantly, in the kt-type algorithms, there is a fixed radius +parameter R0, whose value dictates the typical size of all the jets in a particular event. +We note that these algorithms are unable to capture the essential features of the events +where narrow and fat jets may simultaneously arise. In our proposed algorithm, we have +modified these algorithms to bring out the features of varying sizes of the jets. +2.2 +Our Proposal: Dynamic Radius Jet Clustering Algorithm +The usual kt-type algorithms take a fixed radius as an input parameter, and hence the +algorithms return all the jets to be of the same size (or narrower) in a single event. This +lack of dynamicity in choosing a radius can be overcome by setting the radius parameter +dynamically during the construction of each jet. +In any kt-type algorithm, the starting point is a list of N four-momenta of particles. +We will refer to these as fundamental particles or, sometimes, fundamental four-momenta. +The algorithm follows Steps 1 to 3b, as defined in section 2.1, iteratively until the list +gets empty. At every iteration, the number of contents of the list gets reduced by one. +The reduction happens in two ways: (1) via the merger of two four-momenta, (2) via the +declaration of four-momentum as a final jet. Thus at an intermediate iteration, the list +contains two different types of objects. These two types of objects are (1) fundamental +four-momenta, and (2) composite four-momenta, generated through the merger of two or +more fundamental four-momenta. These composite objects evolve through iterations to +give rise to the final jets. For our convenience, let us label these composite evolving objects +as pseudojets. We borrowed the name pseudojet from the PseudoJet class in the FastJet3 +package [71], where all the types of four-momenta are called pseudojet. However, we will +call them by different names: fundamental, pseudojet (composite or evolving), and jet (or +final jet). +Our proposal is to change the constant nature of the radius parameter R0 in Eq. (2.2) +to a dynamic quantity depending on the distribution, in the η-φ plane, of the fundamental +objects inside each evolving pseudojet. Therefore, the modified distance measure for the +– 4 – + +dynamic radius algorithm takes the form +dij = min +� +p2p +Ti, p2p +Tj +� +∆R2 +ij, +(2.3) +diB = p2p +Ti R2 +di, +(2.4) +where Rdi is the dynamical radius parameter, defined as +Rdi = R0 + σi. +(2.5) +The constant R0 is an input parameter similar to the standard kt-type algorithm and it +is the starting point of the dynamical growth of the radius of an evolving jet. For the ith +pseudojet, σi is calculated as +σ2 +i = +� +a 5 GeV) +jets. We then tag the energetic jets, event by event, as reconstructed ‘top’ or reconstructed +‘jet’ with the help of MG5 parton-level information. The events are classified into two +categories, as described below. +A1. Category A1 consists of events satisfying the following conditions. +• A jet should have mass in the range (150, 200) GeV and have ∆R(toptruth, jet) < +0.5. This jet is identified as a reconstructed top jet. We label these reconstructed +objects as ‘top (A1)’ in the subsequent discussions. +• After the tagging of the top jet, another jet should have pT > 300 GeV and +should be within 0.5 distance from the original jet as generated by MG5. These +jets are labelled as ‘jet (A1)’ in further discussions. +A2. Category A2 are the events which satisfy the following conditions. +• Two separate jets within 1.0 distance of the original top quark and having +an invariant mass between 150 and 200 GeV. These two jets are tagged as +constituent jets of the reconstructed top jet, which is a combination of these +two constituents. These combinations are labelled as ‘top (A2)’. +• Another jet having pT > 300 GeV and within 0.5 radius from the original jet. +This is labelled as ‘jet (A2)’. +In general, any inclusive kt-type clustering algorithm yields as output many soft jets +along with the hard ones. The origin of these soft jets is primarily the soft radiation due +to underlying events and wide angle parton shower. These jets are expected in both the +category A1 and A2 events. Any jet having pT > 5 GeV and labelled neither as top nor as +jet is labelled as soft jet. +The two categories have been chosen to demonstrate the usefulness of the dynamic +radius jet algorithm. Category A1 captures the whole top jet by the jet clustering algorithm +while the events in category A2 need post-processing after the jet clustering. Therefore, a +desirable criterion of a better-performing jet clustering algorithm would be to have more +events in category A1. In order to illustrate that, for a given category, we define acceptance +efficiency +A = number of events accepted in a particular category +total number of events +. +(3.1) +After the classification of the events into the above two categories, the distribution +of distances between the MG5 parton-level objects and reconstructed ones are plotted +in Fig. 2. In both the panels of the figure, the blue and brown histograms are for top +jets, and the green and red ones are for energetic jets. The corresponding categories of +the histograms are mentioned alongside the legends. The distributions are shown for jets +clustered using the AK algorithm with (a) R0 = 0.5, and (b) R0 = 0.8. Since this distance +between the MG5 parton-level and reconstructed ones are features of parton showering +and hadronization, the normalized distributions are kind of identical for different radius +– 10 – + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +∆R(parton, reconstructed) +0 +2 +4 +6 +8 +10 +12 +14 +frequency (normalized) +(a) +AK, R0 = 0.5 +top (A1) +top (A2) +jet (A1) +jet (A2) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +∆R(parton, reconstructed) +0 +2 +4 +6 +8 +10 +12 +14 +frequency (normalized) +(b) +AK, R0 = 0.8 +top (A1) +top (A2) +jet (A1) +jet (A2) +Figure 2: Normalized distribution of ∆R between the MG5 parton-level object and cor- +responding reconstructed jet. The jets were clustered using the AK algorithm with radius +parameters (a) 0.5 and (b) 0.8. +choices. These ∆R distributions are very similar even with different choices of standard or +dynamic radius sequential recombination algorithms and, therefore, are not shown to avoid +repetition. This distribution also justified the choice of 0.5 radius to find reconstructed +objects from the MG5 partons. +500 +1000 +1500 +2000 +2500 +jet energy [GeV] +10−4 +10−3 +frequency (normalized) +(a) +AK, R0 = 0.5 +Category A1 +top +jet +500 +1000 +1500 +2000 +2500 +jet energy [GeV] +10−4 +10−3 +frequency (normalized) +(b) +AK, R0 = 0.5 +Category A2 +top +jet +Figure 3: Normalized distribution of jet energy for categories (a) A1 and (b) A2. The +blue and green histograms are respectively for the reconstructed top and the high-pT jet. +We show in Fig. 3 the jet energy distributions for the objects of our study. The left +and right panels show the distributions for categories A1 and A2, respectively. The blue +and green histograms are for the top and the high-pT jet produced in association with it. +– 11 – + +0 +50 +100 +150 +200 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(a) +R0 = 0.5 +Category A1 +DR-AK A = 48.79% +AK A = 14.71% +DR-AK top +DR-AK jet +DR-AK soft +AK top +AK jet +AK soft +0 +50 +100 +150 +200 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(b) +R0 = 0.5 +Category A2 +DR-AK A = 20.21% +AK A = 52.23% +DR-AK top +DR-AK jet +DR-AK soft +AK top +AK jet +AK soft +0 +50 +100 +150 +200 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(c) +R0 = 0.5 +Category A1 +DR-CA A = 32.52% +CA A = 14.67% +DR-CA top +DR-CA jet +DR-CA soft +CA top +CA jet +CA soft +0 +50 +100 +150 +200 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(d) +R0 = 0.5 +Category A2 +DR-CA A = 36.48% +CA A = 50.43% +DR-CA top +DR-CA jet +DR-CA soft +CA top +CA jet +CA soft +0 +50 +100 +150 +200 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(e) +R0 = 0.5 +Category A1 +DR-KT A = 38.87% +KT A = 17.71% +DR-KT top +DR-KT jet +DR-KT soft +KT top +KT jet +KT soft +0 +50 +100 +150 +200 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(f) +R0 = 0.5 +Category A2 +DR-KT A = 24.70% +KT A = 47.15% +DR-KT top +DR-KT jet +DR-KT soft +KT top +KT jet +KT soft +Figure 4: Normalized distribution of jet mass for the process pp → tj. The left panel +shows the distribution for category A1 events while the right panel is the distribution for +category A2 events. The blue, green, and red histograms are for reconstructed top, hard +jet, and soft jets (defined in the text), respectively. The histograms, from top to bottom, +are for AK, CA, and KT algorithms. The filled histograms correspond to fixed radius +algorithms and the unfilled ones correspond to their dynamic radius (DR) counterparts. +– 12 – + +One of the primary obligations of choosing the appropriate size for jets according to +requirements is to avoid the rise of jet mass even with soft but widely separated constituents +inside a jet. We, therefore, choose to show the distribution of masses of reconstructed top +jets, reconstructed energetic jets in Fig. 4. The jet energy ranges corresponding to the +mass distributions shown can be approximately 500-2000 GeV, as seen in Fig. 3. The left +panel of the figure represents the distribution for category A1 events while the right panel +represents the distribution for category A2 events. The blue, green, and red histograms +are for reconstructed top, hard jet, and soft jets, respectively. The histograms, from top to +bottom, are for anti-kt, C/A, and kt algorithms. The filled histograms are for standard jet +clustering algorithms and the unfilled ones are their dynamic radius counterparts. In the +legends, the prefix ‘DR’ to AK, CA, or KT stands for dynamic radius. In all the panels, the +starting radius parameter has been taken to be R0 = 0.5. For standard kt-type algorithms, +the starting radius is the fixed constant radius parameter, i.e., Rd = R0. The values for +A for different algorithms and different categories are quoted inside each panel of Fig. 4. +In all the panels, it is seen that the acceptance efficiencies for A1 category events in the +cases with dynamic radius algorithms are higher than their fixed radius counterparts. +An interesting feature to notice is that the mass distribution for the energetic jet re- +mains almost the same for both the standard and dynamic radius jet clustering algorithms. +The similarity between these two are more prominent for AK and CA algorithms and less +so for the KT algorithm. This is expected as the KT algorithm starts to merge softer +momenta first and then capture the harder ones almost at the end. As a result, this al- +gorithm lets the size of the dynamic radius grow in the beginning and hence allows the +softer hadron, even if they are a little wider, to merge with the evolving jet. The top jet +mass distribution is also a little off with respect to their fixed radius analogue. These are +not very problematic since jet grooming [77–83], trimming [84], or pruning [85, 86] methods +help in cleaning soft and wide-angle radiation. A similar strategy of grooming is useful in +the removal of soft jets as well. +The change in mass distribution for top jet but not for the energetic jet can easily +be understood from the behaviour of the final radius Rd = (R0 + σ) [Eq. (2.5)] a jet has +acquired. We, therefore, show the distribution of the final radii of the three different types +of jets in Fig. 5. The three plots in the top panel are for category A1 events while those in +the bottom panel are for category A2 events. For category A2 events, ‘top c1’ and ‘top c2’ +labels represent the two constituent jets of reconstructed top. The distributions are shown +for DR-AK, DR-CA, and DR-KT algorithms in Figs. 5(a,d), 5(b,e), and 5(c,f), respectively, +with R0 = 0.5 in each. +From all the histograms in Fig. 5, some clear features emerge. For the case of category +A1 top jets, the final radius Rd grows to more than 0.6 with a peak at Rd ≃ 0.75, (ap- +proximately 50% increase with respect to the starting radius). On the other hand, for the +energetic jets, Rd does not grow by much. This indicates that the radius grows dynamically +according to the distribution of constituents inside the jet. The growth of the soft jets is +higher compared to the hard jets candidates. In general, this is will not be a problem in the +heavy object finding since they can easily be eliminated by choosing an appropriate pT or +mass cuts. The story for the category A2 events is similar for jets and soft jets. The only +– 13 – + +0.5 +0.6 +0.7 +0.8 +0.9 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(a) +Category A1 +DR-AK +R0 = 0.5 +top +jet +soft +0.5 +0.6 +0.7 +0.8 +0.9 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(c) +Category A1 +DR-KT +R0 = 0.5 +top +jet +soft +0.5 +0.6 +0.7 +0.8 +0.9 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(b) +Category A1 +DR-CA +R0 = 0.5 +top +jet +soft +0.5 +0.6 +0.7 +0.8 +0.9 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(d) +Category A2 +DR-AK +R0 = 0.5 +top c1 +top c2 +jet +soft +0.5 +0.6 +0.7 +0.8 +0.9 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(f) +Category A2 +DR-KT +R0 = 0.5 +top c1 +top c2 +jet +soft +0.5 +0.6 +0.7 +0.8 +0.9 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(e) +Category A2 +DR-CA +R0 = 0.5 +top c1 +top c2 +jet +soft +Figure 5: Normalized distribution of the final radius Rd of three different types of jets. +The three plots in the top panel are for category A1 events while those in the bottom panel +are for category A2 events. The conventions for the colours and labels ‘top’, ‘jet’, and ‘soft’ +are the same as in Fig. 4. For category A2, ‘topc1’ and ‘topc2’ labels represent the two +constituent jets of reconstructed top. The distributions are shown for DR-AK, DR-CA, +and DR-KT algorithms in the panels (a,d), (b,e), and (e,f), respectively, with R0 = 0.5. +difference is that the whole top could not be reconstructed as a single jet in these events. +The normalized distributions of the final radii of these two constituent jets of reconstructed +tops are plotted. These constituents tend to grow more than the energetic jets. +The values of acceptance efficiencies A [Eq. (3.1)] for different category events vary +with the choice of the value for the starting radius R0. If the starting radius is small, the +algorithms fail to capture the fat jet. On the other hand, the large starting radius R0 will +capture the unwanted contamination coming from underlying events or radiations from +other nearby showers. As a result, the jets will be unnecessarily fat and massive. There +is a suitable range for R0 within which the algorithms work better. We, therefore, show +the variation of acceptance efficiencies A as a function of starting radius R0 in Fig. 6 for +both categories A1 (blue) and A2 (red). The variations are shown for (DR-) AK, CA, and +KT algorithms in panels (a), (b), and (c), respectively. As expected, for small R0 values, +the efficiencies for category A1 (blue lines) are negligible in both dynamic radius and fixed +– 14 – + +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +10 +20 +30 +40 +50 +60 +A [%] +(a) +A1, DR-AK +A2, DR-AK +A1, AK +A2, AK +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +10 +20 +30 +40 +50 +60 +A [%] +(c) +A1, DR-KT +A2, DR-KT +A1, KT +A2, KT +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +10 +20 +30 +40 +50 +60 +A [%] +(b) +A1, DR-CA +A2, DR-CA +A1, CA +A2, CA +Figure 6: The variation of acceptance efficiency A [Eq. (3.1)] as a function of starting +radius R0 for pp → tj SM process. The blue and red lines represent the variations of A for +categories A1 and A2 events, respectively. The dashed lines are for (a) AK, (b) CA, and +(c) KT algorithm and the solid lines are for their dynamic radius versions. +radius analyses since the constituents of the entire top jet could not be captured with these +small values of R0. Rather, the category A2 (red lines) which form the top with the help +of two jets yields more A . This picture changes once we tend towards higher values for +R0 ≃ 0.5 as more and more top jets are being reconstructed in the A1 category. As a result, +the values of A for the A2 category get reduced. In all the panels of Fig. 6, it is interesting +to note that the dynamic radius algorithms (solid) yield higher values for A than their +fixed radius counterparts (dashed). This is indicative of the usefulness of the dynamic +radius algorithm over the fixed radius ones. +The dip in the blue solid lines after near +R0 = 0.7 is not essentially the failure of the algorithm. Rather, it is because of the capture +of unwanted contaminations along with the radiation coming from the top. Therefore, the +jet mass goes beyond 200 GeV, at which point we stop labelling them as a reconstructed +top jet. Furthermore, a rough comparison among the curves in the three panels of Fig. 6 +indicates that DR-AK is better suited than DR-CA and DR-KT algorithms. +3.2 +Illustration II: pp → V j Subprocess +A similar study has been performed in SM pp → V j, (V = W or Z) processes. In order +to ensure the formation of fat jets, a lower cut of 500 GeV on the pT of the jet has been +imposed at the time of generation of parton-level events via MG5. +These events were +then passed on to Pythia8 with Monash 2013 Tune [76] tune for parton showering and +hadronization. The final state hadrons of these events were then sent to FastJet3 for jet +clustering with starting radius R0 = 0.4. +As before, we label the energetic jets coming from a jet clustering algorithm, as recon- +structed ‘V’ or reconstructed ‘jet’ with the help of MG5 parton-level information. The rest +– 15 – + +0 +50 +100 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(a) +R0 = 0.4 +Category B1 +DR-AK A = 72.18% +AK A = 61.96% +DR-AK V +DR-AK jet +DR-AK soft +AK V +AK jet +AK soft +0 +50 +100 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(b) +R0 = 0.4 +Category B2 +DR-AK A = 13.81% +AK A = 26.23% +DR-AK V +DR-AK jet +DR-AK soft +AK V +AK jet +AK soft +0 +50 +100 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(c) +R0 = 0.4 +Category B1 +DR-CA A = 68.46% +CA A = 62.30% +DR-CA V +DR-CA jet +DR-CA soft +CA V +CA jet +CA soft +0 +50 +100 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(d) +R0 = 0.4 +Category B2 +DR-CA A = 18.10% +CA A = 25.25% +DR-CA V +DR-CA jet +DR-CA soft +CA V +CA jet +CA soft +0 +50 +100 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(e) +R0 = 0.4 +Category B1 +DR-KT A = 70.41% +KT A = 62.58% +DR-KT V +DR-KT jet +DR-KT soft +KT V +KT jet +KT soft +0 +50 +100 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(f) +R0 = 0.4 +Category B2 +DR-KT A = 11.99% +KT A = 24.02% +DR-KT V +DR-KT jet +DR-KT soft +KT V +KT jet +KT soft +Figure 7: Normalized distributions of jet mass for the process pp → V j. The left panel +shows jet mass distributions of category B1 events and the right panel is the distribution for +category B2 events. The blue, green, and red histograms are for reconstructed V, energetic +jet and soft jets (defined in the text), respectively. The histograms, from top to bottom, +are for AK, CA, and KT algorithms, respectively. The filled histograms correspond to +fixed radius algorithms and the unfilled ones correspond to their dynamic radius (DR) +analogues. +– 16 – + +of the jets having pT > 5 GeV are tagged as ‘soft jets’. As in the previous illustration, we +classify the events into two separate categories based on the following criteria. +B1. An event was labelled as a category B1 event if it satisfies the following two conditions. +• A jet should have mass in the range (65, 105) GeV and ∆R(VMG5, jet) < 0.5. +This jet was identified as a reconstructed V jet and we label them as ‘V (B1)’ +in further discussions. +• After the tagging of the V jet, another jet should have pT > 300 GeV and +∆R(jMG5, jet) < 0.5. These jets are labelled as ‘jet (B1)’ in further discussions. +B2. An event, after failing to satisfy the criteria for the category B1, could be classified +as a category B2 event subject to satisfying the below conditions. +• Two separate jets within 1.0 distance from the original vector boson (W or Z) +and should have an invariant mass between 65 and 105 GeV. These two jets are +tagged as constituent jets of the reconstructed ‘V’ jet. The final reconstructed +‘V’ jet should be within 0.5 distance from the original boson. This combination +is labelled as ‘V (B2)’. +• Another jet having pT > 300 GeV and within 0.5 radii of the original jet and +this is labelled as ‘jet (B2)’. +We show the jet mass distribution in Fig. 7 for SM pp → V j process. All the distribu- +tions in the left panel of the figure represent the category B1 events and the distributions +in the right panel are for category B2. The blue, green, and red histograms are for recon- +structed ‘V’, jet and soft jets, respectively. The histograms, from top to bottom, are for +AK, CA, and KT algorithms, respectively. The filled histograms are for standard jet clus- +tering algorithms and the unfilled ones are their dynamic radius analogues. Quite clearly, +the two peaks in the blue histograms, in all the distributions, correspond to the mass peaks +of W and Z bosons. The jet mass distribution of the energetic jets using dynamic radius +algorithms remains similar to their fixed radius counterparts. The increment in the per- +centage of the acceptance efficiencies A [Eq. (3.1)] of category B1 events is representative +of the appropriateness of using the dynamic radius algorithms over the standard ones in +these types of scenarios. +We next show in Fig. 8 the normalized distributions of the final radius for three different +types of jets, viz. ‘V’ jets, energetic jets, and soft jets. As in the pp → tj process, the fat +V jets acquires a larger radius than the energetic jets after the dynamical expansion of the +jet size. Here, again, the soft jets acquire a higher radius compared to the energetic jets. +These soft jetsare not of much concern since they are rather soft and hence they can be +removed easily from the analysis. +In Fig. 9, we show the variation of A [Eq. (3.1)] as a function of starting radius R0. In +all the panels of the figure, the blue and red lines correspond to the variations for categories +B1 and B2 events, respectively. The dashed lines are for fixed radius algorithms and the +solid lines are for dynamic radius jet algorithms. The variations are shown for (a) AK, (b) +– 17 – + +0.4 +0.5 +0.6 +0.7 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(a) +Category B1 +DR-AK +R0 = 0.4 +V +jet +soft +0.4 +0.5 +0.6 +0.7 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(c) +Category B1 +DR-KT +R0 = 0.4 +V +jet +soft +0.4 +0.5 +0.6 +0.7 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(b) +Category B1 +DR-CA +R0 = 0.4 +V +jet +soft +0.4 +0.5 +0.6 +0.7 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(d) +Category B2 +DR-AK +R0 = 0.4 +V c1 +V c2 +jet +soft +0.4 +0.5 +0.6 +0.7 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(f) +Category B2 +DR-KT +R0 = 0.4 +V c1 +V c2 +jet +soft +0.4 +0.5 +0.6 +0.7 +Rd +0 +5 +10 +15 +20 +frequency (normalized) +(e) +Category B2 +DR-CA +R0 = 0.4 +V c1 +V c2 +jet +soft +Figure 8: Normalized distribution of final radius Rd for the three different types of jets. +The top panel represents the distributions of Rd in the category B1 events and the whole +bottom panel is for category B2 events. The conventions for the colours and labels V, jet, +and soft are the same as Fig. 7. For category B2 events, ‘V c1’ and ‘V c2’ labels represent +the two constituent jets of the reconstructed vector bosons. The distributions are shown for +DR-AK, DR-CA, and DR-KT algorithms in the panels (a,d), (b,e), and (c,f), respectively, +with R0 = 0.4. +CA, and (c) KT algorithms. A quick observation of the curves tells us that the behaviour +of these curves is similar to that of the curves in Fig. 6 except the monotonic decreasing +nature of the category B2 curves. The reason is as follows: in the case of V jets, the jets +are ‘two-pronged’ in nature. Therefore, the small radius jets can capture one of the two +prongs of V jets, and thereby these two jets are able to reconstruct V jets in B2 category. +However, as the starting radius R0 is increasing, more and more events are migrating to +category B1. The declining nature of the curves for large radii after 0.5 is because of the +fact that the jets capture more hadrons than are required for their optimal size. As a +result, the mass of the V jets tends to go beyond the mass window set to label them as V +jets. Again, more variables than just the jet mass can help one to improve the tagger and +hence the acceptance efficiency. +We conclude this section with the note that the dynamic radius jet algorithms are +– 18 – + +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Acceptance [%] +(a) +B1, DR-AK +B2, DR-AK +B1, AK +B2, AK +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Acceptance [%] +(c) +B1, DR-KT +B2, DR-KT +B1, KT +B2, KT +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Acceptance [%] +(b) +B1, DR-CA +B2, DR-CA +B1, CA +B2, CA +Figure 9: The variation of A [Eq. (3.1)] as a function of the starting radius R0 for pp → V j +SM process. The blue and red lines represent the values of A for categories B1 and B2 +events, respectively. The dashed lines are for (a) AK, (b) CA, and (c) KT algorithms. The +solid lines are for their dynamic radius versions. +useful in finding fat as well as narrow jet in a single event in the colliders. +We have +successfully illustrated this in two SM processes, viz. pp → tj and pp → V j, at the 13 TeV +LHC. A comparison among the three dynamic radius analogues of the standard kt-type +algorithm reveals that the DR-AK algorithm performs better compared to the DR-CA or +the DR-KT algorithms. +4 +Usefulness in BSM signals +We now illustrate the usefulness of the dynamic radius jet algorithm in the context of +a scenario beyond the standard model (BSM). This is a scenario where an additional +vectorlike singlet quark b′ of charge −1/3 exists along with (d, s, b). Such quarks occur, for +example, in E(6) grand unified theories, as also in some seesaw models of quark masses [87– +92]. The b′ can mix with the three SM down-type quarks when electroweak symmetry +breaking takes place2. This causes the mass eigenstate dominated by b′ to decay into a top +quark and a W boson. In addition, the mixing between a T3 = −1/2 quark and one with +T3 = 0 induces flavour-changing Z- and Higgs-couplings in the b-b′ sector. Thus the b′, +produced via strong interactions at the LHC, has the decays b′ → tW, b′ → bZ, b′ → bh. +The detailed theoretical framework and the resulting phenomenology have been discussed +widely in the literature [59, 93–101]. +The currently available data from the LHC restrict mb′ to be no less than 1.3–1.5 TeV +[102–105]. When such a massive quark decays thereafter, its decay products are consider- +2In the following discussion, we shall (a) denote this mass eigenstate itself by b′, (b) assume that +ordinary-exotic quark mixing takes place involving only the third family sequential quark, namely, b, and +(c) parametrize the b-b′ mixing by the angle θ. +– 19 – + +ably lighter compared to it. Therefore the b′ decay products are considerably boosted, so +as to produce fat jets. Furthermore, the difference in mass between two product particles +leads to jets of varying degrees of fatness. +Since our purpose here is to show the efficacy of the dynamic radius jet algorithm, we +illustrate our main points in the context of pp → b′¯b′ followed by each b′ decaying into a +top quark and a W boson. The t’s and the W’s thus give rise to energetic jets of different +radii. We demonstrate below how our newly developed algorithm can capture the identity +of the ensuing final state. While the present work is aimed at capturing the essence of our +proposed jet algorithm, a more detailed discussion, including combinations of all the three +aforementioned decay channels of the b′, is going to be presented in a separate work [106]. +mb′ +sin θL +sin θR +1.3 TeV +0.12 +8.02 × 10−3 +Table 1: Values of some important parameters of the vectorlike singlet b′ model considered +for the illustration. +The model has been implemented in a Mathematica-based package SARAH [107–109]. +The Universal FeynRules Output (UFO) [110] generated by SARAH is then used in MG5 +for the generation of parton-level events. The parameter card for MG5 has been generated +using spectrum generator SPheno [111, 112]. The values for the important parameters of +the model are tabulated in Table 1. The angles θL and θR in the table represent the mixing +angle between SM b quark and exotic b′ quark of chirality left and right, respectively. After +the generation of the MG5 parton-level events, the rest of the analysis pipeline is the same +as the previous illustrations of SM processes. +In this illustration, we choose DR-AK, based on the discussion in the previous section. +We show the resultant jets having pT > 30 GeV formed out of the hadrons generated by +Pythia8 in Fig. 10. The left panel shows the positions of the generated hadrons and jets +constructed using the AK algorithm with R0 = 0.5. The right panel shows the same for +the DR-AK algorithm. In both panels, the red dots represent the position of final state +hadrons in the η-φ plane and the size of each dot is proportional to the √pT of the hadron. +The unfilled circles represent the final radius (Rd) of a jet. The teal dots represent the +constituents of boosted fat ‘W’ jets. The green, blue, and purple (wherever applicable) dots +represent the constituents of the fat ‘top’ jet. The yellow dots containing texts represent +the position of the MG5 parton-level pT -hard quarks after the decay of top or W. The +mothers of the q or b are mentioned in the subscripts of q or b. +An interesting point to observe in Fig. 10(b) is that the DR-AK yields only 4 jets, +which are representative of 2 fat W and 2 fat t jets. However, in Fig. 10, the fixed radius +algorithm could form the fat W jets but fails to capture the entirety of the two fat t jets. +One, of course, can use a bigger radius in the AK algorithm to capture the whole of the +top jet. However, this will make the W jet unnecessarily fat. This demonstrates the utility +of the dynamic radius jet algorithm. +– 20 – + +−4 +−2 +0 +2 +4 +η +0 +1 +2 +3 +4 +5 +6 +φ +pp → b′¯b′ → tW −¯tW + +AK, R0 = 0.5 +(a) +Hadrons +−4 +−2 +0 +2 +4 +η +0 +1 +2 +3 +4 +5 +6 +φ +pp → b′¯b′ → tW −¯tW + +DR-AK, R0 = 0.5 +(b) +Hadrons +Figure 10: The distribution of final state hadrons and jets in η-φ plane for an example +event. The colours and sizes of the dots and circles follow the same convention as Fig. 1. +The teal coloured dots represent the constituents of hard fat ‘W’ jets. The green and blue +(wherever applicable) dots represent the constituents of the fat ‘top’ jet. The yellow dots +containing texts represent the position of the hard quarks after the decay of top or W +which are mentioned as the subscripts of q or b. The plots are shown for (a) AK and (b) +DR-AK algorithms. +To study the goodness of DR-AK quantitatively, we define the following criteria for +tagging of top and W jets. +• A jet having mass in the range (150, 200) GeV and having ∆R(toptruth, jet) < 0.5 is +identified as a reconstructed top jet. +• A jet will be called W jet if it has a mass in the range (65, 105) GeV and is within +0.5 distance from the original MG5 parton-level W boson. +Similar to the illustrations with SM processes, we classify the events into different +categories. Due to the complex nature of the final states, we have classified the events into +more than two categories in the present scenario. The realization is based on the following +understanding. +• Out of the two W’s coming directly from b′ in an event, the number of reconstructed +W as fat jet from the algorithm could be 0, 1, or 2. We call these reconstructed fat +W jets as primary W jets. +• Similarly, out of the two t quarks, the number of reconstructed t as fat jets can be 0, +1, or 2. +• In some particular cases, the whole top may not be reconstructed, but the W boson +coming from the top quarks may be reconstructed. These are referred to as secondary +W jets in the subsequent discussions. +– 21 – + +Based on the above observations, we classify the events into different categories, whose +generic name is given as Cij, where i and j are two integers encoding the number of +reconstructed top and reconstructed W’s, respectively. For the present scenario, the allowed +value for i does not exceed two. For a given i, the values for j should not exceed 4 − i. +That is, to say, i ≤ 2 and j ≤ 4−i. An exhaustive list of all possible categories is tabulated +in Table 2. For example, the event shown in Fig. 10 would be categorized as C22 for the +DR-AK algorithm while the same event would be classified as C03 for the AK algorithm. +One may again subdivide some of the categories into subcategories based on how many W +jets are coming directly from b′ (primary W) and how many of them are coming from the +decay of the top quark (secondary W). Therefore, the generic name for the subcategories +can be given as Cijk with i, j, and k being the numbers of reconstructed top, primary W, +and secondary W jets. The possible ranges for i, j, and k are 0 ≤ i, j ≤ 2 and 0 ≤ k ≤ 2−i. +Category +Subcategory +No. of top jet +No. of primary +No. of secondary +W jet +W jet +C22 +C220 +2 +2 +0 +C21 +C210 +2 +1 +0 +C20 +C200 +2 +0 +0 +C13 +C121 +1 +2 +1 +C12 +C120 +1 +2 +0 +C111 +1 +1 +1 +C11 +C110 +1 +1 +0 +C101 +1 +0 +1 +C10 +C100 +1 +0 +0 +C04 +C022 +0 +2 +2 +C03 +C021 +0 +2 +1 +C012 +0 +1 +2 +C02 +C020 +0 +2 +0 +C011 +0 +1 +1 +C002 +0 +0 +2 +C01 +C010 +0 +1 +0 +C001 +0 +0 +1 +C00 +C000 +0 +0 +0 +Table 2: The definitions of the list of categories and subcategories as according to how +many fat jets can be reconstructed from the jet algorithm. +– 22 – + +0 +50 +100 +150 +200 +jet mass [GeV] +0.00 +0.05 +0.10 +0.15 +0.20 +frequency (normalized) +(a) +R0 = 0.5 +Category C22 +DR-AK A = 5.47% +AK A = 1.62% +DR-AK top +DR-AK W +DR-AK soft +AK top +AK W +AK soft +0.5 +0.6 +0.7 +0.8 +0.9 +Rd +0 +5 +10 +15 +frequency (normalized) +(b) +Category C22 +DR-AK +top +W +soft +Figure 11: (a) The normalized distribution of jet mass of the category C22 events for the +pp → b′¯b′ → tW −¯tW + process. The blue, green, and red histograms are reconstructed top, +W, and soft jets, respectively. The unfilled histograms are for the jets clustered using the +DR-AK algorithm while the filled ones are for the jets using the AK clustering algorithm. +(b) The normalized distribution of the final radii of top, W, and soft jets with blue, green, +and red colours, respectively. For both panels, R0 = 0.5 was used and any additional jets +having pT >5 GeV were considered as a soft jet. +We plot the normalized distribution of jet mass of the category C22 events in Fig. 11(a). +Jets were clustered using R0 = 0.5. In the plot, the blue, green, and red histograms are +reconstructed top, W, and soft jets, respectively. The unfilled histograms are for the jets +clustered using the DR-AK algorithm, and the filled ones are for the jets using the AK +clustering algorithm. Any untagged jet with pT > 5 GeV was considered to be a soft jet. +Fig. 11(b) shows the normalized distribution for the finally acquired radii of different jets +for category C22 events. The desirable feature of the reconstructed W jets being narrower +than the reconstructed top jets is clearly apparent in the figure. Here, again, the soft jets +are growing to larger radius are expected. However, as discussed in the previous section, +they can be removed from an analysis by pT or jet mass cuts. +The values of A [Eq. (3.1)] for the two different algorithms, viz. DR-AK and AK are +also quoted in Fig. 11(a). These values (1.62% for AK and 5.47% for DR-AK), clearly, +indicate that the dynamic radius jet algorithm is working better while probing the correct +mass windows for the particles. The shift of the mass distribution towards larger values is +indicative of capturing little extra than required. As discussed previously, this can be recti- +fied by the techniques of grooming [77–81], trimming [84], or pruning [85, 86]. Furthermore, +going beyond just the jet mass to tag the topor W jets would further help in extracting +signals. +The variation of A as a function of initial radius R0 is shown in Fig. 12 for six categories, +namely C22, C21, C20, C13, C12, and C11. These categories have at least one top jet +identified within 0.5 distance from the MG5 parton-level top quark. The solid blue lines +– 23 – + +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +5 +10 +15 +20 +Acceptance [%] +C22 +DR-AK +AK +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +5 +10 +15 +20 +Acceptance [%] +C21 +DR-AK +AK +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +5 +10 +15 +20 +Acceptance [%] +C20 +DR-AK +AK +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +5 +10 +15 +20 +Acceptance [%] +C13 +DR-AK +AK +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +5 +10 +15 +20 +Acceptance [%] +C12 +DR-AK +AK +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +R0 +0 +5 +10 +15 +20 +Acceptance [%] +C11 +DR-AK +AK +Figure 12: The variation of A [Eq. (3.1)] as a function of initial radius R0 for six categories, +namely C22, C21, C20, C13, C12, and C11. +The solid blue lines are for the DR-AK +algorithm, and the dashed lines are for the AK algorithm. The jets are clustered with +R0 = 0.5. +represent the efficiencies for the DR-AK algorithm, and the dashed lines are representative +of the AK algorithm. For the case of dynamic radius, the quintessential feature is the +initial increment in the acceptance efficiencies A up to R0 = 0.5, and, beyond this value, +the efficiencies decrease. The reason for this is an unnecessary accumulation of hadrons +and making the jets bigger than their optimal size. However, for the AK algorithm, the +efficiencies keep on increasing until R0 = 0.7, which is kind of the optimal radius for this +scenario. +The most important point to note is that up to R0=0.5, the efficiencies for +the DR-AK algorithm are higher than those for the AK algorithm. This feature, again, +establishes the utility of using dynamic radius algorithms over fixed radius ones. +In the end, we look at the bar plot of the acceptance efficiencies A for all the categories +in Fig. 13. The blue and green bars are for DR-AK and AK algorithms, respectively. The +initial radius R0 is taken to be 0.5. The numbers under the curly braces below the x-axis +represent the values of A for the categories which capture 2 tops, 1 top, 0 top, and none of +the top or W jets. The important observation in this regard is that the categories containing +2 top and 1 top jets have better efficiencies for the dynamic radius algorithm than the fixed +radius one. This means that the events, where the AK algorithm could not capture the +whole of the top constituents, the DR-AK algorithm could capture the full tops. Thus the +credence of our proposed algorithm is established in a BSM context as well. +– 24 – + +C22 C21 C20 C13 C12 C11 C10 C04 C03 C02 C01 C00 +Categories +0 +5 +10 +15 +20 +25 +30 +Acceptance [%] +R0 = 0.5 +DR-AK: +AK: +� +�� +� +9.86% +4.38% +� +�� +� +35.33% +27.31% +� +�� +� +34.02% +53.75% +���� +20.79% +14.56% +DR-AK +AK +Figure 13: Bar plot of A for different categories for jet algorithms with R0 = 0.5. The +blue, and green bars are for DR-AK and AK algorithm respectively. From left to right, +The numbers under the braces represent the values of A for the categories which capture +2 top, 1 top, 0 top, and none of the top or W jets. +5 +Summary and Outlook +We go beyond the most popular jet clustering algorithms, where the formation of jets +is performed using a fixed radius parameter. +These algorithms return fixed-sized jets +corresponding to the input radius parameter. In this work, an attempt is made to make +the radius of each jet variable depending on the kinematics and hadronic activity in the +neighbourhood of an evolving jet. The proposed method is based on the standard kt-type +sequential recombination jet clustering algorithms with the incorporation of the dynamic +nature of the radius parameter. +Starting from a reasonable radius parameter, during the process of formation of a jet, +the radius of each evolving jet is allowed to grow based on fuzziness inside it. For this +work, the measure of the fuzziness of each evolving jet is chosen to be the ‘pT -weighted’ +standard deviation of the inter-particle distances (in the η-φ plane) of the particles inside +the evolving jet. +After describing the proposed method, we have presented two different SM processes, +viz. pp → tj and pp → Wj +Zj, to demonstrate some applicabilities of the dynamic radius +jet clustering algorithm. In these two processes, differently-sized jets are expected in a sin- +gle event. In the two SM process examples, we observe that the jets are being formed with +radii varying in size on a jet-by-jet basis. In terms of the acceptance efficiency [Eq. (3.1)], +we show that the performance of the dynamic radius algorithm is better compared to their +– 25 – + +fixed radius counterparts. We take up a scenario beyond the Standard Model for further +illustration, where a vectorlike SU(2)L singlet charge −1/3 quark b′ is added. We study +jet clustering in pp → b′¯b′ followed by each b′ decaying into tW. Once more, our proposed +method turns out to be effective in the reconstruction of the final state particles. +In the examples given above, the dynamicity has been incorporated in the radius +parameter of the standard kt-type sequential recombination algorithm. The central idea +is the usage of fuzziness of an evolving jet to appropriately increase its radius starting +from a starting radius R0. Although examples with only one measure of fuzziness have +been shown in this work, one may consider other appropriate measures. depending upon +the underlying physics process or the final goal of the analysis. Therefore, the idea of the +dynamic radius jet algorithm should not be restricted only to this particular measure. The +applicability of these possibilities will be presented in a separate work. 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Commun. 183 (2012) 2458 [1104.1573]. +– 32 – + diff --git a/9dFPT4oBgHgl3EQfYjT_/content/tmp_files/load_file.txt b/9dFPT4oBgHgl3EQfYjT_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98f340e530b1b2234b78a954c800c8d6086fc1fe --- /dev/null +++ b/9dFPT4oBgHgl3EQfYjT_/content/tmp_files/load_file.txt @@ -0,0 +1,1812 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf,len=1811 +page_content='Prepared for submission to JHEP Dynamic Radius Jet Clustering Algorithm Biswarup Mukhopadhyayaa, Tousik Samuia, and Ritesh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Singha aDepartment of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur, 741246, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' E-mail: biswarup@iiserkol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='in, tousiksamui@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='com, ritesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='singh@iiserkol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='in Abstract: The study of standard QCD jets produced along with fat jets, which may appear as a result of the decay of a heavy particle, has become an essential part of collider studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Current jet clustering algorithms, which use a fixed radius parameter for the formation of jets from the hadrons of an event, may be inadequate to capture the differing radius features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' In this work, we develop an alternative jet clustering algorithm that allows the radius to vary dynamically based on local kinematics and distribution in the η-φ plane inside each evolving jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' We present the usefulness of this dynamic radius clustering algorithm through two Standard Model processes, and thereafter illustrate it for a scenario beyond the Standard Model at the 13 TeV LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='13074v1 [hep-ph] 30 Jan 2023 Contents 1 Introduction 1 2 Methodology 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='1 Standard Sequential Recombination Algorithms 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='2 Our Proposal: Dynamic Radius Jet Clustering Algorithm 4 3 Application to Standard Model Processes 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='1 Illustration I: pp → tj process 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='2 Illustration II: pp → V j Subprocess 15 4 Usefulness in BSM signals 19 5 Summary and Outlook 25 1 Introduction The physics extraction capacity of any high-energy collider depends crucially on the han- dling of coloured particles in various final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' These are produced as partons via ei- ther short-distance interactions of quantum chromodynamics (QCD) or electroweak pro- cesses [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The partons, however, hadronize through long-distance QCD effects which are not calculable ab initio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' One rather uses semi-empirical methods to predict the prob- ability that energetic partons will fragment into more low-energy partons and ultimately form colour-neutral hadrons which are observable in the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Groups of closely spaced hadrons with varied degrees of collimation form ‘jets’ whose identification, isolation, and merger are predicted once more with the help of semi-empirical (and by no means uniquely decided) algorithms called jet clustering algorithms [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The aim always remains to define jets with such algorithms which most accurately elicit the short-distance physics underly- ing the events that are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' They thus constitute some of our most important tools in the analysis of phenomena at colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' In the context of the Large Hadron Collider (LHC), a widely used class of jet criteria is based on so-called kt-type sequential recombination jet algorithms [7–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' These algorithms (briefly discussed in the next section) typically try to merge ‘neighbouring’ hadrons to identify the group as a jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The neighbourhood of a hadron is defined by a single radius parameter R0 in the η-φ plane of the detector, which is used to quantify the radius (or size) of a jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' This is because the hadrons within R0 are merged to form a jet while the hadrons outside R0 are not included in that jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The choices for the value of R0 in these algorithms depend on the physics searches one is carrying out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' At the 13 TeV LHC, the typical choices for R0 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='4 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='8 for a ‘narrow’ or a ‘fat’ jet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' There are, in addition, jet isolation criteria depending on whether one is trying to separate a jet from a hard lepton or another hadronic jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' However, the sequential recombination algorithms generally do not accommodate varying choices of radii on a jet-by-jet basis in a single event since they – 1 – have a single constant parameter that determines the radius of a jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Separate classifiers for a ‘narrow’ jet and a ‘fat’ jet in a single event in the current kt-type algorithms are thus difficult to set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' An important improvement over the current fixed radius algorithms would be to make them adapt the jet radii dynamically jet-by-jet in each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' We make an attempt in this direction in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Our central idea of choosing the radius dynamically of a jet, especially for a boosted fat jet, is based on the kinematics of the decay products of the initiating heavy particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' From the theoretical side, the formation of boosted fat jet occurs due to the high collimation of the on-shell decay products – and their showering and subsequent hadronization – of the energetic and therefore boosted heavy particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' This is very different from the formation of light quark- or gluon-initiated jets, whose collimation is primarily due to parton showering and subsequent hadronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' On the other hand, at the operational level, as per the standard kt-type algorithms, the fat jets are formed in the same way as the regular ‘narrow’ jets, which are initiated by light quarks or gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' However, the kinematics of on-shell decay products and their radiation pattern of a heavy particle is different from the showering of energetic light quarks or gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Therefore, the internal structure of a fat jet is very different from a narrow one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' These internal structure has been used to tag different heavy and light jets in the LHC context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' For example, jet substructure (JSS) observable generalized angularities λκ β [14, 15] is used to distinguish between quark- and gluon-initiated jets [16– 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The same variable was used in the classification among the narrow jet, fat W jet, or boosted top jet [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Another important set of JSS observables, namely the energy correlation functions (ECFs) [31, 32], was shown to be useful in classifying different types of jets [29, 33–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The observable N-subjettiness (τN) [38, 39] has been used to find the multi-pronged nature of light or heavy jets [40–65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' These variables have also been used extensively by the experimental collaborations at the 13 TeV LHC [66–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' These examples try to exploit the energy distribution pattern inside a jet to distinguish a heavy object from a QCD jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The common theme of these jet substructure variables is the utilization of the ‘multi-pronged’ nature of the fat jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Due to this multi-pronged nature, one expects the variance of inter-constituent distance ∆R of a fat jet to be significantly different compared to the narrow QCD jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' This variance of a jet can be used to grow the radius of a jet starting from an initial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Earlier attempts to make the jet radius variable, albeit with somewhat different motivations and formalisms, can be found in references [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' [69], the effective radius of a pseudojet during their evolution was taken to be inversely proportional to the pT with a maximum cut-off on the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Essentially, this algorithm starts from a big effective radius and the size shrinks as a process of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' On the other hand, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' [70], an expectation-maximization approach was taken for clustering the hadrons into a pre-determined number of clusters (jets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Our approach, in this work, is to modify the standard fixed radius kt-type algorithms to make the radius grow depending on the local kinematics and distribution (in the η-φ plane) of the hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' In section 2, we briefly outline the kt-type sequential recombination algorithms followed by our improvement to the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' We test the efficacy of our algorithms on two SM processes and discuss them in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Section 4 deals with one application in the BSM scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' We summarize and conclude in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' – 2 – 2 Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='1 Standard Sequential Recombination Algorithms At the operational level, a jet is constituted by a bunch of four-momenta obtained using some clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Among various possible ways of grouping up the four-momenta of an event, we need to choose those relevant to physics at the collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' It is important that the clustering algorithm should ensure infrared and collinear (IRC) safety, which, in our context, can be defined in terms of the following conditions [7]: Infrared (IR) safety: The output of the algorithm should not be affected by the intro- duction of a four-momentum with p → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Collinear (C) safety: The output of the algorithm should not be affected by a collinear splitting of any four-momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The algorithm that best takes care of the issue of IRC safety is known as kt-type sequential recombination jet clustering algorithms [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' We briefly outline these algorithms below1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' If an event consists of N final state particles, whose four-momenta are taken in a list as an input of the kt-type algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The distance dij between the ith and jth four-momenta and the distance diB between the ith and the beam are then defined as dij = min � p2p Ti, p2p Tj � ∆R2 ij, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='1) diB = p2p TiR2 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='2) where R0 is the radius parameter of the algorithm, ∆Rij is the Euclidean distance between the ith and jth four-momenta in the η-φ plane, and pTi is pT of ith four-momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The exponent p sets the weight factor to the Euclidean distance in the η-φ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The three choices of p = 1, 0 and −1 correspond to the kt (KT) [8–10], Cambridge-Aachen (CA) [11, 12], and anti-kt (AK) [13] algorithms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The algorithm for combining nearby four-momenta with respect to the above distance measures to form jets has the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The distances dij for all the possible pairs and beam distances diB for all the four-momenta are calculated first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The minimum among all the dij and diB’s is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Step 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' If the minimum occurs at one of the i, j pairs, the corresponding ith and jth four-momenta are merged to form a new four-momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The older ones, ith and jth four-momenta are removed from the list and the newly merged one is added to the list and goes back to Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Step 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' On the other hand, if the minimum distance is one of the diB, the ith four- momenta is declared as a final jet, and it is removed from the list and goes back to Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' 1Here, we only discuss the inclusive algorithms in the LHC context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' For other jet clustering algorithms, please see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' – 3 – Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The process is stopped once the list gets empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' This class of algorithms is seedless because the clustering of four-momenta to form a jet does not start from a particular seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Rather, the algorithms try to merge the closest pair first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' A group of hadrons is then declared as a jet when an appropriate size is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The essential difference among the three different algorithms, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' AK, CA, and KT is that they give different weights to the Euclidean distance in the η-φ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' This typically sets some sort of seed to the clustering algorithms in the sense that it gives a preference to a hadron around which four-momenta merge to give rise to a final jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' In the case of the KT algorithm, it is the softer (in terms of pT ) constituent which merges first and then the harder ones get attached to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' As a result, the shape of the final jet may not be circular in the η-φ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' On the other hand, in the AK algorithm, the hardest particle in a neighbourhood becomes some sort of seed for the jet and the softer ones merge at a later stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Hence the final jet looks circular in the η-φ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' In the CA algorithm, the merging is purely angular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Among the three algorithms, the AK algorithm is the most popular one owing to its circular shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Importantly, in the kt-type algorithms, there is a fixed radius parameter R0, whose value dictates the typical size of all the jets in a particular event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' We note that these algorithms are unable to capture the essential features of the events where narrow and fat jets may simultaneously arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' In our proposed algorithm, we have modified these algorithms to bring out the features of varying sizes of the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='2 Our Proposal: Dynamic Radius Jet Clustering Algorithm The usual kt-type algorithms take a fixed radius as an input parameter, and hence the algorithms return all the jets to be of the same size (or narrower) in a single event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' This lack of dynamicity in choosing a radius can be overcome by setting the radius parameter dynamically during the construction of each jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' In any kt-type algorithm, the starting point is a list of N four-momenta of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' We will refer to these as fundamental particles or, sometimes, fundamental four-momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The algorithm follows Steps 1 to 3b, as defined in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='1, iteratively until the list gets empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' At every iteration, the number of contents of the list gets reduced by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' The reduction happens in two ways: (1) via the merger of two four-momenta, (2) via the declaration of four-momentum as a final jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Thus at an intermediate iteration, the list contains two different types of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' These two types of objects are (1) fundamental four-momenta, and (2) composite four-momenta, generated through the merger of two or more fundamental four-momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' These composite objects evolve through iterations to give rise to the final jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' For our convenience, let us label these composite evolving objects as pseudojets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' We borrowed the name pseudojet from the PseudoJet class in the FastJet3 package [71], where all the types of four-momenta are called pseudojet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' However, we will call them by different names: fundamental, pseudojet (composite or evolving), and jet (or final jet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Our proposal is to change the constant nature of the radius parameter R0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='2) to a dynamic quantity depending on the distribution, in the η-φ plane, of the fundamental objects inside each evolving pseudojet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' Therefore, the modified distance measure for the – 4 – dynamic radius algorithm takes the form dij = min � p2p Ti, p2p Tj � ∆R2 ij, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='3) diB = p2p Ti R2 di, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='4) where Rdi is the dynamical radius parameter, defined as Rdi = R0 + σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content='5) The constant R0 is an input parameter similar to the standard kt-type algorithm and it is the starting point of the dynamical growth of the radius of an evolving jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf'} +page_content=' For the ith pseudojet, σi is calculated as σ2 i = � a 𝜙(𝑟t). The function 𝐸𝛾(𝑔, 𝑥) +represents 𝑒𝑥 for 𝑔 = 0 and 𝑒𝑥𝛾(𝑔, 𝑥)/Γ(𝑔) for 𝑔 > 0, where 𝛾(𝑔, 𝑥) +is the lower incomplete gamma function and Γ(𝑔) stands for the +gamma function. This distribution function depends on the specific +energy 𝐸 and the specific angular momentum 𝐽. The function 𝜙 is the +gravitational potential and 𝑟t is the truncation radius. The parameter +𝑔 is called the truncation parameter, and it regulates the energy +truncation of the model. The parameter 𝑟a is the anisotropic radius, +and it determines how anisotropic a system is. When 𝑟a grows, the +model is less anisotropic, and 𝑟a → ∞ corresponds to an isotropic +model. The constants 𝐴 and 𝑠 are used to set the physical scale of the +model. The density can be obtained by integrating the distribution +function 𝑓 (𝐸, 𝐽) over the velocity space: +𝜌 = +∫ +𝑓 (𝐸, 𝐽) d3𝑣. +(2) +Since 𝐸 = 𝑣2/2 + 𝜙(𝑟) and the distribution function is zero for 𝐸 > +𝜙(𝑟t), it can be just integrated from 0 to 𝑣max = [2𝜙(𝑟t) − 2𝜙(𝑟)]1/2 +at each 𝑟. This 𝑣max becomes zero when 𝑟 = 𝑟t and the density +vanishes for 𝑟 ≥ 𝑟t. Hence, the truncation radius 𝑟t represents the +distance where the density comes to zero. +The gravitational potential 𝜙 is subjected to the Poisson equation. +For spherical systems such as globular clusters, the equation results +in the following form: +d2𝜙 +d𝑟2 + 2 +𝑟 +d𝜙 +d𝑟 = 4𝜋𝐺𝜌, +(3) +where 𝑟 is the radial coordinate and 𝐺 is the gravitational constant. +The relevant quantities were first turned into dimensionless ones for +solving the Poisson equation. The dimensionless potential is defined +as ˆ𝜙 = [𝜙(𝑟t) − 𝜙]/𝑠2. The dimensionless density and radius are +ˆ𝜌 = 𝜌/𝜌0 and ˆ𝑟 = 𝑟/𝑟0, where 𝜌0 and 𝑟0 satisfy 4𝜋𝐺𝑟2 +0𝜌0/𝑠2 = 9. +Then, the Poisson equation becomes +d2 ˆ𝜙 +dˆ𝑟2 + 2 +ˆ𝑟 +d ˆ𝜙 +dˆ𝑟 = −9 ˆ𝜌. +(4) +The equation is solved with the boundary conditions that, at ˆ𝑟 = 0, +d ˆ𝜙/dˆ𝑟 = 0 and ˆ𝜙 = 𝑊0, where 𝑊0 is a constant that specifies a +particular solution. Hence,𝑊0 is also a parameter of the LIMEPY model, +called the concentration parameter. It characterizes the concentration +of the model. +As previously mentioned, LIMEPY models provide an extended fam- +ily of isothermal models. Those famous models are included as sub- +families. For example, the Woolley model (Woolley 1954) can be +produced by setting 𝑔 = 0, 𝑟a → ∞. When 𝑔 = 1 and 𝑟a → ∞, +the King model (King 1966) is obtained. The Wilson model (Wilson +1975), which is more extended, corresponds to 𝑔 = 2 and 𝑟a → ∞. +Models with 𝑊0 → ∞ or 𝑔 → ∞ become the isothermal spheres. In +addition, the polytrope can be represented as 𝑊0 → 0. It includes the +Plummer model (Plummer 1911) which corresponds to the model +with 𝑔 = 3.5. It has a finite mass but infinite extents. In general, the +model with appropriate 𝑊0 and 𝑟a can be finite in extent if 𝑔 < 3.5 +and conversely infinite in extent with 𝑔 ≥ 3.5. In addition, Gieles & +Zocchi (2015) also showed that one kind of finite model is unsuitable +for star clusters. These systems have an upturn in the density far from +the center, so there is a large amount of mass in the halo. The ratio +of the virial radius and half-mass radius 𝑟v/𝑟h is a crucial parameter +for these models. They suggested that the models with 𝑟v/𝑟h ≥ 0.64 +can adequately describe star clusters. +The LIMEPY models describe spherical systems with different con- +centrations, truncation, and radial anisotropy. In general, the model +is isotropic near the center but could be anisotropic in the middle +part of the system. The energy truncation limits the contribution of +anisotropy to radial orbits with 𝐸 ≈ 𝜙(𝑟t) and thus suppresses the +degree of radial anisotropy near the edge. The corresponding physi- +cal picture is that a cluster under the interaction of an external tidal +field has a preferential mass loss on stars with radial orbits. This re- +duces the amount of anisotropy in the outer region (Oh & Lin 1992; +Takahashi et al. 1997). Simulations of star clusters in the tidal field +confirmed this isotropic behavior near the edge (Tiongco et al. 2016). +Thus, the energy truncation acts as a role of the tidal field. In fact, +MNRAS 000, 1–15 (20XX) + +Dynamical Properties of Globular Clusters +3 +the tidal field can also make the outer region profiles tangentially +anisotropic (Baumgardt & Makino 2003). +In addition to the anisotropic radius 𝑟a, there is a convenient +anisotropic parameter 𝜅 ≡ 2𝐾r/𝐾t, where 𝐾r is the total radial ki- +netic energy and 𝐾t is the total tangential kinetic energy. If 𝜅 > 1, +the system is radially anisotropic, and if 𝜅 < 1, the system is tangen- +tially anisotropic. When 𝜅 = 1, it is an isotropic system. Therefore, 𝜅 +represents a simple and global measure of the anisotropy. We mainly +used 𝜅 to determine the amount of the anisotropy of clusters. +In Zocchi et al. (2016), the comparisons with N-body simulations +illustrated the variation of model parameters of a cluster during the +evolution. The cluster started with the Plummer model and the sim- +ulation snapshots at different time were fitted with LIMEPY models. +The concentration parameter tended to increase with time, which +was also suggested previously by King (1966). The truncation pa- +rameter 𝑔 decreased roughly from 2.5 to 0.5 during the evolution. It +corresponded to an increased truncation by the tidal field as a cluster +gradually filled the Roche volume. Thus, a cluster tends to become +more concentrated and truncated with time. In addition, the degree +of radial anisotropy increased due to radial diffusion but decreased +later during the core collapse. +3 THE OBSERVATIONAL DATA +One of our primary goals is to provide updated results with a complete +inclusion of all available observational data for globular clusters. The +observational data of 𝑉-band surface brightness 𝜇 were taken from +Trager et al. (1995), which provided a catalog of surface brightness +profiles for over a hundred Galactic globular clusters. Some proce- +dures were needed before the data were ready for the fitting. There +was a correction related to extinction. The method is based on the +global mean curve discussed in Fitzpatrick (1999), which uses the +mean value for the ratio of the extinction 𝐴𝑉 and the reddening +𝐸(𝐵 − 𝑉) so that 𝐴𝑉 = 3.1𝐸(𝐵 − 𝑉). We took the reddening in +the catalog of Harris (1996) (2010 version) and then computed the +corrected surface brightness by 𝜇𝑖 = 𝜇𝑖,0 − 𝐴𝑉 , where 𝜇𝑖,0 denotes +the data before the correction. The data with 𝑤𝑖 < 0.15 were not +adopted according to McLaughlin & van der Marel (2005), where +𝑤𝑖 is the weight of each data given in Trager et al. (1995). +Because the data number was large, which might make the surface +brightness dominate the fitting, we sliced the radial range with equal +logarithmic width and averaged the surface brightness and the weight +in each bin. The bin number was 55 which equaled the largest data +number of the velocity dispersion. To compute the uncertainty for +each data, we followed the method in McLaughlin & van der Marel +(2005). The uncertainty of the data was obtained by 𝜖𝜇,𝑖 = 𝜖𝜇,b/𝑤𝑖, +where 𝜖𝜇,b is the base error bar for each cluster. +For line-of-sight velocity dispersion, we used the profiles derived +from the collected literature (Baumgardt 2017), the data from un- +published spectra of stars in the ESO and Keck Science archives +(Baumgardt & Hilker 2018), and the dispersion from the integral- +field-unit data from the WAGGS project (Dalgleish et al. 2020). The +above data are expressed by open circles in Fig. 3. The data from the +MUSE survey (Kamann et al. 2018) were also used and denoted by +solid triangles. Some additional data were supplemented and marked +as crosses, such as those from McLaughlin et al. (2006) for NGC 104 +and Larson & Seth (2015, private communication) for NGC 1851 +and NGC 2808. (The data of McLaughlin et al. (2006) and Larson & +Seth (2015, private communication) were collected from the compi- +lation in Watkins et al. (2015b) and others were collected from the +compilation in the updated web catalog (third version) of Baumgardt +& Hilker (2018).) +For proper-motion velocity dispersion, we mainly took the data +from the Hubble Space Telescope from Watkins et al. (2015a) and the +Gaia data from Vasiliev & Baumgardt (2021). Open circles expressed +the former, and solid triangles expressed the latter in Fig. 4. Some +additional data were supplemented and denoted by crosses, which +include Häberle et al. (2021) for NGC 6441, McLaughlin et al. (2006) +for NGC 104, McNamara et al. (2003) for NGC 7078, McNamara +et al. (2012) for NGC 6266, and Zloczewski et al. (2012) for NGC +6656 and NGC 6752. (The data of Vasiliev & Baumgardt (2021) and +Häberle et al. (2021) were collected from the updated web catalog of +Baumgardt & Hilker (2018), and the data of McLaughlin et al. (2006), +McNamara et al. (2003), McNamara et al. (2012), and Zloczewski +et al. (2012) were collected from Watkins et al. (2015b).) +Some proper motion data were downloaded in units of km/s, +which depends on the cluster distance written in the literature. +These data were transformed into mas/yr as the observational val- +ues for our work here. The transformation is 𝑣 = 𝑣0/𝐷𝐶, where 𝑣 +and 𝑣0 are the velocity in mas/yr and km/s, 𝐷 is the distance and +𝐶 = 4.74047 km yr kpc−1 mas−1 s−1 which is a factor for the unit +conversion (van Leeuwen 2009; Watkins et al. 2015b). The values +of cluster distances were taken from the corresponding literature. By +taking the root mean square of the upper and lower error bars from +the literature, we obtained a symmetric uncertainty for each data for +our work. Finally, to focus on the systems with enough observational +information, we studied 18 clusters with more than five data points +in each type of the above observational profiles. +4 THE DETERMINATION OF PHYSICAL PARAMETERS +It was shown in Zocchi et al. (2017) that models with different +amounts of anisotropy could give the same surface brightness but +different kinematic profiles. Thus, using the surface brightness data +alone can lead to some degeneracy. Therefore, here we included +the surface brightness, the light-of-sight velocity dispersion, and the +proper-motion velocity dispersion data to obtain complete pictures of +the physical structures and kinematic properties of globular clusters +by determining related physical parameters through the data-model +fitting. +Following the method in Zocchi et al. (2017), we employed the +one-step fitting procedure with the single-mass LIMEPY models in this +paper. With all three considered types of observational data, a single +step of the fitting was performed to determine all cluster parameters. +The fitting was done through the minimization of the 𝜒2 function: +𝜒2 = 𝜒2 +sb + 𝜒2 +los + 𝜒2 +pm, +(5) +where 𝜒2 +sb, 𝜒2 +los, 𝜒2pm are the contributions from surface brightness, +line-of-sight velocity dispersion, and proper-motion velocity disper- +sion, respectively. They are defined by +𝜒2 +sb = +𝑛sb +∑︁ +𝑖=1 +[𝜇𝑖 − ¯𝜇(𝑟𝑖)]2 +𝜖2 +𝜇,𝑖 +, +(6) +𝜒2 +los = +𝑛los +∑︁ +𝑖=1 +[𝜎los,𝑖 − ¯𝜎los(𝑟𝑖)]2 +𝜖2 +los,𝑖 +, +(7) +and +𝜒2 +pm = +𝑛pm +∑︁ +𝑖=1 +[𝜎pm,𝑖 − ¯𝜎pm(𝑟𝑖)]2 +𝜖2 +pm,𝑖 +, +(8) +MNRAS 000, 1–15 (20XX) + +4 +Cheng and Jiang +where 𝜇𝑖 is the 𝑖-th observational data of a surface brightness profile, +¯𝜇(𝑟𝑖) is the theoretical surface brightness at that radial coordinate +𝑟𝑖, and 𝜖𝜇,𝑖 is the error bar of the data 𝜇𝑖. Similarly, 𝜎los,𝑖, ¯𝜎los(𝑟𝑖), +𝜖los,𝑖 are the corresponding quantities for line-of-sight velocity dis- +persion, and 𝜎pm,𝑖, ¯𝜎pm(𝑟𝑖), 𝜖pm,𝑖 are the observational data, the- +oretical value, and error bar for proper-motion velocity dispersion, +respectively. The numbers of observational data are 𝑛sb, 𝑛los, 𝑛pm, +respectively, for the surface brightness, line-of-sight velocity disper- +sion, and proper-motion velocity dispersion, individually. +The LIMEPY code was employed to obtain the above theoretical pro- +files. This code needed five input parameters, including the concen- +tration parameter 𝑊0, the truncation parameter 𝑔, the dimensionless +anisotropy radius ˆ𝑟a, the cluster mass 𝑀, and the half-mass radius +𝑟h. The LIMEPY code generated several profiles, such as the surface +mass density Σ(𝑟𝑖), line-of-sight mean-square velocity 𝑢2 +L(𝑟𝑖), radial +and tangential mean-square velocity on the projected plane 𝑢2 +R(𝑟𝑖) +and 𝑢2 +T(𝑟𝑖). Thus, the value of ¯𝜎los(𝑟𝑖) is simply the square root of +𝑢2 +L(𝑟𝑖), and ¯𝜎pm(𝑟𝑖) is the square root of [𝑢2 +R(𝑟𝑖) + 𝑢2 +T(𝑟𝑖)]/2. +To complete the data-model fitting, two more parameters were +needed. The cluster distance 𝐷 is a parameter that converts the radial +coordinate of the theoretical profile from pc to arcsec and the ob- +servational proper-motion velocity dispersion from mas/yr to km/s. +The V-band mass-to-light ratio Υ is a parameter for producing the +luminosity density Σ(𝑟𝑖)/Υ, and the surface brightness ¯𝜇(𝑟𝑖) can be +obtained by +¯𝜇(𝑟𝑖) = 𝑀V,⊙ − 5(1 + log 𝑐) − 2.5 log(Σ(𝑟𝑖)/Υ), +(9) +where 𝑀V,⊙ = 4.83 mag is the V-band absolute magnitude of the +Sun and 𝑐 = 𝜋/648000 rad/arcsec is a factor for the unit conversion +(Watkins et al. 2015b). +Through the minimization of the 𝜒2 function, the best-fit values +of seven parameters 𝑊0, 𝑔, ˆ𝑟a, 𝑀, 𝑟h, 𝐷, Υ can be obtained. We +used the code EMCEE (Foreman-Mackey et al. 2013) to perform the 𝜒2 +minimization. It is an affine-invariant ensemble sampler that employs +the Markov chain Monte Carlo (MCMC) process (Goodman & Weare +2010). One has to decide the initial distribution and the parameters +range for the EMCEE samples. For the concentration parameter 𝑊0, the +range was set to 1 < 𝑊0 < 15. It covers a similar range in Table II +of King (1966) and represents various degrees of concentration of +star clusters. Figure 4 in Gieles & Zocchi (2015) showed the relevant +models for star clusters and the corresponding parameters; hence +we set 0 < 𝑔 < 3 for the truncation parameter accordingly. The +dimensionless anisotropy radius ˆ𝑟a needs a wide range to include the +isotropic models. Therefore, we set a large range for log ˆ𝑟a as −1 < +log ˆ𝑟a < 20. For the remained parameters, we checked the literature +values and considered wider ranges to include more possibilities. The +ranges of these parameters were set to be 0.1 < 𝑀 < 50 (105 M⊙), +0.1 < 𝑟h < 15 (pc), 0.1 < 𝐷 < 35 (kpc), and 0.1 < Υ < 5 (Υ⊙). +Finally, the initial distributions of all parameters are set to be uniform. +5 RESULTS AND DISCUSSION +The best-fit results are displayed in Table 1. The first column shows +the names of the clusters. Seven fitting parameters are listed from the +second to eighth columns. The second column presents the concen- +tration parameter 𝑊0 and the values range roughly from 3 to 9 for +these clusters. The third and the fourth columns show the truncation +parameter 𝑔 and the logarithm of the dimensionless anisotropy radius +log ˆ𝑟a. The fifth and sixth columns list the cluster mass 𝑀 and the +half-mass radius 𝑟h. These clusters have 𝑟h ≲ 10 pc. Among them, +NGC 5139 has the largest mass and radius. The heliocentric distance +𝐷 is shown in the seventh column. Most clusters have 𝐷 ≲ 12 kpc +except for NGC 6715, which is roughly two times distant. The eighth +column reveals the V-band mass-to-light ratio Υ. To understand the +anisotropy conveniently, the quantity 𝜅 is shown in the ninth column. +NGC 5139 and NGC 7078 have 𝜅 > 1, indicating the anisotropic +behavior. The quantity in the last column is the reduced chi-square +𝜒2r defined by +𝜒2 +r = +𝜒2 +𝑛 − 𝑛p +, +(10) +where 𝑛 is the total number of data and 𝑛p is the number of parame- +ters. +5.1 Comparison with Previous Work +To compare our results with the previous work, we used the mea- +surable physical properties estimated in the published literature, as +listed in Table 2. We first considered the comparison of the cluster’s +total mass. In general, the masses estimated by Baumgardt & Hilker +(2018) are larger than those estimated by Watkins et al. (2015b), and +our results are usually between their values. Almost all of our results +are very close to the masses estimated in Watkins et al. (2015b). +We also compared our half-mass radius with the one in the catalog +of Baumgardt & Hilker (2018). Generally, our results are smaller, +consistent with the results of total mass, since our masses are lower +than those in Baumgardt & Hilker (2018). Therefore, the radii of the +clusters tend to be smaller to fit the line-of-sight velocity dispersion. +Some differences between the radius might come from the mass +spectrum. The radial distributions of different species may introduce +additional variation between the half-mass radii. Nevertheless, the +mass-to-light ratios obtained in our work are consistent with the +values in Baumgardt et al. (2020) and Watkins et al. (2015b). +For distance comparison, we compared with the values in Watkins +et al. (2015b), Baumgardt & Vasiliev (2021), and Harris (1996). +Watkins et al. (2015b) derived the distance by comparing their proper +motion velocity dispersion with the line-of-sight velocity dispersion +from the literature. Baumgardt & Vasiliev (2021) calculated the mean +distance from several methods, such as the Gaia EDR3 parallaxes, +the method by fitting nearby subdwarfs to globular cluster main +sequences, the color-magnitude diagram fitting, and the distances +from the period-luminosity relation of RR Lyrae stars. The distances +in Harris (1996) are a compilation of the distance measurements +from the literature. +Fig. 1 shows the ratio of our distance 𝐷 and the one published +in literature 𝐷lit, i.e., 𝐷/𝐷lit, for each considered cluster. For each +panel, the compared literature is labeled at the top-right corner. Each +point represents a particular cluster studied in the compared literature +and this work. The dashed line represents the unity, and the solid line +is the average value of the ratio. Two numbers are shown in the +bottom-right of the panels, the left number is the averaged 𝐷/𝐷lit, +and the right one is the averaged |𝐷/𝐷lit−1|. These numbers indicate +that our results are closer to Harris (1996) and Watkins et al. (2015b), +and slightly lower than Baumgardt & Vasiliev (2021). In general, our +results agree with the values from these studies. +5.2 The Profiles +Fig. 2 to 4 show the profiles of surface brightness, line-of-sight veloc- +ity dispersion, and proper-motion velocity dispersion. The horizontal +axis is the distance from the cluster’s center in arcsec. The vertical +axis gives the surface brightness in mag/arcsec2 in Fig. 2, and ve- +locity dispersion in km/s from Fig. 3 to 4. It can be seen that LIMEPY +MNRAS 000, 1–15 (20XX) + +Dynamical Properties of Globular Clusters +5 +Table 1. The properties of the clusters. The first column lists the names of the clusters. Columns two to eight show the fitting parameters, which are concentration +parameter 𝑊0, truncation parameter 𝑔, the logarithm of the dimensionless anisotropy radius log ˆ𝑟a, cluster mass 𝑀, half-mass radius 𝑟h, distance 𝐷, and V-band +mass-to-light ratio Υ. Column nine presents the quantity 𝜅 which measures the amount of anisotropy, and the final column gives 𝜒2r . +cluster +𝑊0 +𝑔 +log ˆ𝑟a +𝑀 +𝑟h +𝐷 +Υ +𝜅 +𝜒2r +(105 M⊙) +(pc) +(kpc) +(Υ⊙) +NGC 104 +8.36 ± 0.06 +1.31 ± 0.03 +11.13+6.02 +−6.10 +6.87 ± 0.15 +5.21 ± 0.12 +4.33 ± 0.03 +1.53 ± 0.03 +1.00 +2.10 +NGC 288 +4.46+0.47 +−0.82 +1.55+0.52 +−0.38 +10.40+6.44 +−6.41 +1.02+0.11 +−0.10 +8.26+0.33 +−0.32 +9.80+0.37 +−0.36 +2.32 ± 0.12 +1.00 +1.18 +NGC 362 +7.20 ± 0.10 +1.67 ± 0.06 +11.24+5.87 +−6.48 +2.09+0.11 +−0.10 +2.36+0.08 +−0.07 +8.71+0.16 +−0.15 +1.22 ± 0.03 +1.00 +4.74 +NGC 1851 +7.33+0.19 +−0.20 +2.04 ± 0.09 +10.77+6.26 +−6.12 +2.28+0.10 +−0.09 +2.15+0.15 +−0.13 +10.82+0.15 +−0.14 +1.73+0.09 +−0.08 +1.00 +1.61 +NGC 2808 +6.27+0.17 +−0.16 +2.02+0.10 +−0.07 +11.81+5.01 +−6.78 +6.57+0.25 +−0.19 +2.69+0.08 +−0.06 +9.63+0.12 +−0.10 +1.56+0.05 +−0.04 +1.00 +1.63 +NGC 3201 +5.89+0.31 +−0.34 +2.45 ± 0.09 +11.00+6.19 +−6.29 +1.21+0.08 +−0.07 +5.21+0.42 +−0.33 +4.38+0.10 +−0.09 +2.33+0.12 +−0.11 +1.00 +2.74 +NGC 5139 +4.02+0.48 +−0.65 +1.94+0.27 +−0.26 +0.41+0.08 +−0.10 +32.82+0.65 +−0.67 +8.82+0.19 +−0.17 +5.32 ± 0.03 +2.38 ± 0.09 +1.15 +3.86 +NGC 5904 +7.03+0.09 +−0.10 +1.56+0.05 +−0.04 +10.39+6.55 +−6.07 +3.03+0.16 +−0.15 +4.54+0.12 +−0.11 +7.24 ± 0.13 +1.39+0.04 +−0.03 +1.00 +1.85 +NGC 6121 +7.52+0.16 +−0.13 +0.46+0.31 +−0.21 +9.80+6.94 +−5.59 +0.81+0.05 +−0.04 +3.20+0.17 +−0.13 +1.85+0.04 +−0.03 +2.11+0.10 +−0.08 +1.00 +1.12 +NGC 6218 +5.77+0.29 +−0.35 +1.51+0.25 +−0.22 +10.69+6.36 +−6.53 +0.75 ± 0.06 +3.02+0.14 +−0.13 +4.59+0.15 +−0.14 +1.78+0.09 +−0.08 +1.00 +1.19 +NGC 6266 +7.84+0.08 +−0.09 +0.62+0.11 +−0.10 +10.90 ± 6.23 +5.98+0.25 +−0.24 +2.55 ± 0.08 +6.33+0.09 +−0.08 +1.85 ± 0.05 +1.00 +1.57 +NGC 6388 +7.09+0.10 +−0.11 +1.68+0.09 +−0.08 +10.86+6.16 +−6.13 +7.79 ± 0.20 +2.07 ± 0.05 +10.35 ± 0.10 +1.68 ± 0.03 +1.00 +2.94 +NGC 6397 +9.17 ± 0.17 +0.87 ± 0.08 +10.96+6.13 +−6.11 +0.79+0.04 +−0.03 +3.73 ± 0.19 +2.40 ± 0.04 +2.47 ± 0.12 +1.00 +1.97 +NGC 6441 +7.75 ± 0.06 +1.24+0.10 +−0.09 +10.74+6.31 +−6.27 +10.54+0.28 +−0.27 +2.90+0.07 +−0.06 +11.91 ± 0.11 +1.82 ± 0.03 +1.00 +3.35 +NGC 6656 +6.48+0.23 +−0.26 +1.87+0.34 +−0.39 +10.73+6.33 +−6.50 +3.57+0.22 +−0.20 +4.40+0.29 +−0.20 +3.10 ± 0.05 +1.85 ± 0.07 +1.00 +1.04 +NGC 6715 +6.99 ± 0.07 +2.21+0.02 +−0.03 +11.26+6.00 +−6.28 +17.79+1.12 +−1.06 +5.28+0.29 +−0.25 +25.08 ± 0.53 +2.07 ± 0.06 +1.00 +2.83 +NGC 6752 +8.35+0.12 +−0.13 +1.38 ± 0.06 +10.95+6.16 +−6.21 +1.92 ± 0.09 +3.45 ± 0.16 +4.13 ± 0.06 +2.24 ± 0.08 +1.00 +1.20 +NGC 7078 +8.30+0.12 +−0.13 +0.86+0.15 +−0.13 +1.16+0.07 +−0.06 +5.08 ± 0.17 +4.05+0.18 +−0.17 +10.40 ± 0.12 +1.53 ± 0.05 +1.16 +1.46 +Table 2. The literature parameters of the clusters. The number in the parentheses represents the literature, (1) stands for Baumgardt & Hilker (2018), (2) refers +to Watkins et al. (2015b), (3) corresponds to Baumgardt & Vasiliev (2021), (4) represents Harris (1996), and (5) is Baumgardt et al. (2020). The updated values +for (1) and (5) are picked from the web catalog of Baumgardt & Hilker (2018). +cluster +𝑀 +𝑀 +𝑟h +𝐷 +𝐷 +𝐷 +Υ +Υ +(105 M⊙) +(105 M⊙) +(pc) +(kpc) +(kpc) +(kpc) +(Υ⊙) +(Υ⊙) +(1) +(2) +(1) +(2) +(3) +(4) +(5) +(2) +NGC 104 +8.95 ± 0.06 +5.57+0.33 +−0.28 +6.30 +4.15 ± 0.08 +4.521 ± 0.031 +4.5 +1.96 ± 0.09 +1.40 ± 0.03 +NGC 288 +0.934 ± 0.026 +0.79+0.13 +−0.11 +8.37 +9.03+0.48 +−0.56 +8.988+0.089 +−0.088 +8.9 +2.16 ± 0.10 +2.20+0.13 +−0.10 +NGC 362 +2.84 ± 0.04 +... +3.79 +... +8.829 ± 0.096 +8.6 +1.44 ± 0.05 +... +NGC 1851 +3.18 ± 0.04 +1.78+0.10 +−0.11 +2.90 +10.32+0.20 +−0.24 +11.951+0.134 +−0.133 +12.1 +1.66 ± 0.06 +1.51 ± 0.03 +NGC 2808 +8.64 ± 0.06 +5.91+0.22 +−0.25 +3.89 +9.45+0.13 +−0.15 +10.060+0.112 +−0.111 +9.6 +1.51 ± 0.06 +1.56 ± 0.02 +NGC 3201 +1.60 ± 0.03 +... +6.78 +... +4.737+0.043 +−0.042 +4.9 +2.16 ± 0.09 +... +NGC 5139 +36.4 ± 0.4 +34.52+1.45 +−1.43 +10.36 +5.19+0.07 +−0.08 +5.426 ± 0.047 +5.2 +2.58 ± 0.10 +2.66 ± 0.04 +NGC 5904 +3.94 ± 0.06 +3.65 ± 0.75 +5.68 +7.79+0.47 +−0.61 +7.479 ± 0.060 +7.5 +1.81 ± 0.06 +1.43+0.09 +−0.10 +NGC 6121 +0.871 ± 0.011 +... +3.69 +... +1.851+0.015 +−0.016 +2.2 +1.59 ± 0.06 +... +NGC 6218 +1.07 ± 0.03 +... +4.05 +... +5.109+0.049 +−0.048 +4.8 +1.92 ± 0.09 +... +NGC 6266 +6.10 ± 0.04 +6.09+0.39 +−0.33 +2.43 +6.42 ± 0.14 +6.412+0.105 +−0.104 +6.8 +1.99 ± 0.11 +2.22 ± 0.04 +NGC 6388 +12.5 ± 0.1 +8.27+0.89 +−0.95 +4.34 +10.90+0.40 +−0.45 +11.171+0.162 +−0.161 +9.9 +2.19 ± 0.06 +1.68+0.06 +−0.07 +NGC 6397 +0.966 ± 0.013 +0.70+0.09 +−0.08 +3.90 +2.39+0.13 +−0.11 +2.482 ± 0.019 +2.3 +1.66 ± 0.07 +2.23+0.10 +−0.09 +NGC 6441 +13.2 ± 0.1 +... +3.47 +... +12.728+0.163 +−0.162 +11.6 +1.77 ± 0.13 +... +NGC 6656 +4.76 ± 0.05 +2.49+0.44 +−0.37 +5.29 +2.84 ± 0.16 +3.303 ± 0.037 +3.2 +2.05 ± 0.08 +1.88+0.12 +−0.10 +NGC 6715 +17.8 ± 0.3 +11.83+0.62 +−0.53 +5.20 +22.57+0.44 +−0.39 +26.283+0.328 +−0.325 +26.5 +2.10 ± 0.12 +1.94 ± 0.03 +NGC 6752 +2.76 ± 0.04 +1.82 ± 0.12 +5.27 +4.02+0.10 +−0.08 +4.125 ± 0.041 +4.0 +2.34 ± 0.11 +2.14+0.05 +−0.06 +NGC 7078 +6.33 ± 0.07 +4.95 ± 0.19 +4.30 +10.36+0.15 +−0.16 +10.709+0.096 +−0.095 +10.4 +1.58 ± 0.10 +1.49 ± 0.02 +MNRAS 000, 1–15 (20XX) + +6 +Cheng and Jiang +5 +10 +15 +20 +25 +D [kpc] +0.8 +0.9 +1.0 +1.1 +1.2 +D / Dlit +Watkins et al. (2015b) +1.02, 0.05 +5 +10 +15 +20 +25 +D [kpc] +0.8 +0.9 +1.0 +1.1 +1.2 +Baumgardt & Vasiliev (2021) +0.96, 0.05 +5 +10 +15 +20 +25 +D [kpc] +0.8 +0.9 +1.0 +1.1 +1.2 +Harris (1996) +0.98, 0.05 +Figure 1. The comparison of the cluster distance with the values mentioned in earlier studies. The horizontal axis is the cluster distance obtained in this work, +and the vertical axis shows the ratio of our value to the distance given in the literature. The dashed line and the solid line represent the unity and the average. Each +panel is for comparison with the particular publication, as labeled at the top-right corner. At the bottom-right corner, the left number is the averaged 𝐷/𝐷lit, +and the right number is the averaged |𝐷/𝐷lit − 1|. +models can produce similar profiles as observational ones. To ex- +amine these clusters more quantitatively, we classified the results by +𝜒2r . Many clusters were found to have 𝜒2r < 2. These clusters have +suitable fittings for all three profiles, as shown in the figures. +NGC 362 has the largest 𝜒2r , and the model profiles agree with the +observations in surface brightness and line-of-sight velocity disper- +sion. However, the central part of the modeled proper-motion velocity +dispersion is slightly larger than the observations. Data with a small +error bar in the outer part located much higher than the profile, mak- +ing the fitting worse. NGC 6441 also has a larger 𝜒2r . The model +agrees well with the surface brightness and the outer part of the +proper motion velocity dispersion but predicts larger values for the +inner part. The model can also fit the rough trend of the line-of-sight +velocity dispersion, but some points lie below the model. +For NGC 3201, the model has smaller line-of-sight velocity dis- +persion for radius above 100 arcsec. There are also some under +estimations for the proper motions in the outermost region, where +the observational profile tends to level off rather than continue to +decrease. Some scenarios were proposed to explain the higher ve- +locity dispersion in the outer part, such as the orbital history with +accretion and the embedding by a dark matter halo (Bianchini et al. +2019). It was also found that binary stars could contribute to part of +the effect (Wan et al. 2021). For NGC 6715, the model agrees with +the observations, except for the outermost region of the line-of-sight +velocity dispersion, where the observational profile grows. This rise +is probably caused by the stars in the nucleus of the Sagittarius dwarf +galaxy, where NGC 6715 inhabits (Bellazzini et al. 2008). +NGC 5139 has large central velocity dispersions, which the model +cannot explain well. For NGC 6388, the model has a steeper proper- +motion velocity dispersion profile than the observational one. Further +discussions of these two clusters will be made in the following sub- +section. +5.3 Possible Intermediate-Mass Black Hole ? +Stellar black holes exist in astrophysical systems such as X-ray bina- +ries (Mikolajewska et al. 2022). In addition, supermassive black holes +are also confirmed to exist at the centers of our Milky Way (GRAV- +ITY Collaboration et al. 2019) and other galaxies (Blandford et al. +2019). Whether there are any intermediate-mass black holes in the +universe is one of the most important questions in astronomy. Globu- +lar clusters are considered good candidates to host intermediate-mass +black holes and thus attract much attention. Among 18 globular clus- +ters in the present work, NGC 5139 was discussed previously as a +likely candidate. +For our work here, the data-model fitting of NGC 5139 led to +two groups of model parameters, as shown in Fig. 5. These groups +have very different concentration parameters 𝑊0 and logarithm of +the dimensionless anisotropy radius log ˆ𝑟a. One has smaller 𝑊0 and +log ˆ𝑟a, and the other has larger values. Hence, we do further fittings +with narrower ranges as 1 < 𝑊0 < 8, −1 < log ˆ𝑟a < 2, and 8 < 𝑊0 < +15, 2 < log ˆ𝑟a < 20, separately. The results are shown in Table 3. +We denote the one with lower 𝜒2r as Model A, the result previously +listed in Table 1 and presented in Fig. 2 to 4. Model A has a low +concentration. It also has a small dimensionless anisotropy radius +with 𝜅 = 1.15, making it more anisotropic. In contrast, Model B is +isotropic with a high concentration. +The best-fit profiles are shown in Fig. 6. Model A fits the surface +brightness well but predicts lower central velocity dispersion, espe- +cially for the proper motion. On the other hand, Model B has good +fittings on both velocity dispersion but a poor fitting on the surface +brightness. The deviation in surface brightness leads to a larger 𝜒2r . +Although the data and radial range of the observational kinematic +profiles differs, the parameters from Model A agree with those in the +best-fit model in Zocchi et al. (2017). +These results obtained with two models show that it is difficult +to perfectly and simultaneously fit all profiles of NGC 5139 with +the current considered model. This could indicate the existence of +central dark objects which can cause an increase in central velocities. +These objects could be an intermediate-mass black hole (Noyola et al. +2010; Baumgardt 2017) or a group of stellar-mass black holes at the +cluster center (Baumgardt et al. 2019b). Both can also suppress the +mass segregation of the stars (Gill et al. 2008; Peuten et al. 2016) +and render the cluster to have a larger core (Baumgardt et al. 2005; +Peuten et al. 2017). The main difference is that the intermediate- +mass black hole could produce some stars faster than 60 km/s in the +central 20 arcsec of NGC 5139, which was not confirmed in current +observations (Baumgardt et al. 2019b). +NGC 6388 is another candidate cluster that may host a central +intermediate-mass black hole. The study of the integrated light spec- +tra revealed a high central LOS velocity dispersion ∼25 km/s within 2 +arcsec (Lützgendorf et al. 2011). However, there was also a result that +suggests a dispersion ∼15 km/s in the same region derived from stars’ +radial velocities (Lanzoni et al. 2013). Hence, the actual kinematic +behavior of the cluster center is not clear. The data we used have +the extension to nearly 5 arcsec with a velocity dispersion ∼20 km/s. +Our results show that the surface brightness and line-of-sight velocity +MNRAS 000, 1–15 (20XX) + +Dynamical Properties of Globular Clusters +7 +100 +101 +102 +103 +15 +20 +25 +μ [mag/arcsec2] +NGC 104 +100 +101 +102 +103 +20 +25 +30 +NGC 288 +100 +101 +102 +103 +15 +20 +25 +NGC 362 +100 +101 +102 +103 +15 +20 +25 +μ [mag/arcsec2] +NGC 1851 +100 +101 +102 +103 +15 +20 +25 +NGC 2808 +100 +101 +102 +103 +20 +25 +NGC 3201 +101 +102 +103 +15 +20 +25 +μ [mag/arcsec2] +NGC 5139 +101 +102 +103 +15 +20 +25 +30 +NGC 5904 +100 +101 +102 +103 +15 +20 +25 +NGC 6121 +100 +101 +102 +103 +15 +20 +25 +μ [mag/arcsec2] +NGC 6218 +100 +101 +102 +103 +15 +20 +NGC 6266 +100 +101 +102 +15 +20 +25 +NGC 6388 +100 +101 +102 +103 +15 +20 +25 +μ [mag/arcsec2] +NGC 6397 +100 +101 +102 +12.5 +15.0 +17.5 +20.0 +22.5 +NGC 6441 +100 +101 +102 +103 +15.0 +17.5 +20.0 +22.5 +25.0 +NGC 6656 +100 +101 +102 +103 +r [arcsec] +15 +20 +25 +μ [mag/arcsec2] +NGC 6715 +100 +101 +102 +103 +r [arcsec] +15 +20 +25 +NGC 6752 +100 +101 +102 +103 +r [arcsec] +15 +20 +25 +NGC 7078 +Figure 2. The surface brightness profiles of the clusters. The observations are shown as crosses, and the models are expressed by grey lines. For each panel, the +name of the cluster is mentioned at the top-right corner. +MNRAS 000, 1–15 (20XX) + +8 +Cheng and Jiang +101 +102 +103 +0 +5 +10 +15 +σlos [km/s] +NGC 104 +102 +103 +0 +1 +2 +3 +4 +NGC 288 +101 +102 +103 +0.0 +2.5 +5.0 +7.5 +10.0 +NGC 362 +101 +102 +103 +0 +5 +10 +σlos [km/s] +NGC 1851 +101 +102 +0 +5 +10 +15 +NGC 2808 +101 +102 +103 +0 +2 +4 +NGC 3201 +101 +102 +103 +0 +10 +20 +30 +σlos [km/s] +NGC 5139 +101 +102 +103 +0.0 +2.5 +5.0 +7.5 +10.0 +NGC 5904 +102 +103 +0 +2 +4 +6 +NGC 6121 +101 +102 +0 +2 +4 +σlos [km/s] +NGC 6218 +101 +102 +0 +5 +10 +15 +20 +NGC 6266 +101 +102 +0 +10 +20 +NGC 6388 +101 +102 +103 +0 +2 +4 +6 +σlos [km/s] +NGC 6397 +101 +102 +0 +10 +20 +NGC 6441 +101 +102 +103 +0 +5 +10 +NGC 6656 +101 +102 +103 +r [arcsec] +0 +5 +10 +15 +20 +σlos [km/s] +NGC 6715 +101 +102 +103 +r [arcsec] +0.0 +2.5 +5.0 +7.5 +10.0 +NGC 6752 +101 +102 +103 +r [arcsec] +0 +5 +10 +15 +NGC 7078 +Figure 3. The line-of-sight velocity dispersion profiles of the clusters. The open circles represent the data of Baumgardt (2017), Baumgardt & Hilker (2018), +and Dalgleish et al. (2020). The data of Kamann et al. (2018) are shown in solid triangles. The crosses are used for additional data of some clusters mentioned +in Section 3. The models are expressed by grey lines. For each panel, the name of the cluster is mentioned at the top-right corner. +MNRAS 000, 1–15 (20XX) + +Dynamical Properties of Globular Clusters +9 +101 +102 +103 +0 +5 +10 +15 +σpm [km/s] +NGC 104 +101 +102 +0 +2 +4 +NGC 288 +101 +102 +0.0 +2.5 +5.0 +7.5 +10.0 +NGC 362 +101 +102 +103 +0.0 +2.5 +5.0 +7.5 +10.0 +σpm [km/s] +NGC 1851 +101 +102 +103 +0 +5 +10 +15 +NGC 2808 +102 +103 +0 +1 +2 +3 +4 +NGC 3201 +101 +102 +103 +0 +10 +20 +30 +σpm [km/s] +NGC 5139 +100 +101 +102 +0.0 +2.5 +5.0 +7.5 +10.0 +NGC 5904 +102 +103 +0 +2 +4 +NGC 6121 +102 +103 +0 +1 +2 +3 +4 +σpm [km/s] +NGC 6218 +100 +101 +102 +0 +10 +20 +NGC 6266 +101 +102 +0 +5 +10 +15 +20 +NGC 6388 +101 +102 +103 +0 +2 +4 +6 +σpm [km/s] +NGC 6397 +100 +101 +102 +0 +10 +20 +NGC 6441 +101 +102 +103 +0.0 +2.5 +5.0 +7.5 +10.0 +NGC 6656 +101 +102 +r [arcsec] +0 +5 +10 +15 +20 +σpm [km/s] +NGC 6715 +101 +102 +103 +r [arcsec] +0.0 +2.5 +5.0 +7.5 +10.0 +NGC 6752 +100 +101 +102 +r [arcsec] +0 +5 +10 +15 +20 +NGC 7078 +Figure 4. The proper-motion velocity dispersion profiles of the clusters. The open circles represent the data of Watkins et al. (2015a). The data of Vasiliev & +Baumgardt (2021) is shown in solid triangles. The crosses are used for additional data of some clusters mentioned in Section 3. The models are expressed by +grey lines. For each panel, the name of the cluster is mentioned at the top-right corner. +MNRAS 000, 1–15 (20XX) + +10 +Cheng and Jiang +dispersion can be fitted well without the central black hole. However, +the model predicts a steeper proper-motion velocity-dispersion pro- +file than the observations, being higher inside but lower outside. This +behavior can also be seen in Figure 9 of Watkins et al. (2015b). +NGC 7078 is also a candidate cluster that could host an +intermediate-mass black hole. The increase in central velocity dis- +persion found in Hubble Space Telescope was explained by an +intermediate-mass black hole (Gerssen et al. 2002). However, the +cluster can also be fitted with a group of dark stellar remnants (den +Brok et al. 2014) or N-body simulations without intermediate-mass +black holes (Baumgardt 2017). In our results, the cluster could be fit- +ted well without central black holes, and some degree of anisotropy +was observed, which can raise the central velocities. In addition, +although there are raised velocity dispersions in observation, the un- +certainties of the data are also large. Therefore, we obtain a better +fitting than NGC 5139. +5.4 The Anisotropy +Two clusters, NGC 5139 and NGC 7078, possess small dimensionless +anisotropy radius and reveal some degree of anisotropy. The former +has 𝜅 = 1.15 and the latter has 𝜅 = 1.16. Other clusters have isotropic +behavior with 𝜅 = 1.00 and a large anisotropy radius. One effect of +radial anisotropy is that it can increase the central velocity dispersion. +The rise in central velocity dispersions can be seen in Fig. 3 and 4. On +the other hand, the amount of anisotropy estimated from our fittings +could be underestimated, since the difference between tangential and +radial proper motions will be averaged out in the combined proper +motion velocity dispersion. +The results are reasonable compared with some previous studies. +For example, the parameters of NGC 5139 are similar to those es- +timated in Zocchi et al. (2017) which the fittings were carried out +with both radial and tangential proper motion velocity dispersions. +The weak anisotropy in many clusters were also reported by Watkins +et al. (2015a) and Watkins et al. (2015b), in which most of our +samples were also studied. Watkins et al. (2015b) showed that their +distance estimation had good agreement with Harris (1996) and con- +cluded that the assumption of isotropy for their samples is reasonable. +Watkins et al. (2015a) examined the ratio 𝜎T/𝜎R, which compared +the tangential and radial components of the proper motion velocity +dispersion at different radii. They found that the cluster centers are +relatively isotropic, and the behavior of the increasing anisotropy +with the radius was very moderate. From their figures, it can be seen +that the decreasing of 𝜎T/𝜎R with a growing radius is more evident +for NGC 5139 and NGC 7078. +In recent years, Gaia has provided the proper motion data in the +outer parts of globular clusters, and the behavior of 𝜎T/𝜎R reveals +more evidence of anisotropy (Jindal et al. 2019; Vasiliev & Baum- +gardt 2021). In both studies, NGC 5904 appears to be isotropic, and +NGC 104, NGC 5139, and NGC 7078 show radial anisotropy. Some +clusters are anisotropic in one study but are isotropic or uncertain +in another; these include NGC 2808, NGC 6121, NGC 6397, NGC +6656, and NGC 6752. +In addition, the anisotropy profiles 𝜎T/𝜎R−1 from the observations +and our models are plotted in Fig. 7. The observational data was +mainly from a recent report on the globular-cluster survey through +Hubble Space Telescope (Libralato et al. 2022). It includes 16 clusters +of our samples. The remaining two clusters were supplemented with +the data from Watkins et al. (2015a). The data from Gaia (Jindal +et al. 2019) which contains half of our samples were also used. In +Fig. 7, the data of the above-discussed literature are expressed by +open circles, crosses, and solid triangles; the profiles are roughly +isotropic or slightly radial anisotropic within 𝑟 ≲ 100 arcsec. The +larger anisotropy appears mainly in the outer regions. The radial +anisotropy of NGC 5139 tends to increase from near 100 arcsec and +later decrease to isotropy in 𝑟 ≳ 1000 arcsec. Our model predicts the +decrease in radial anisotropy at a larger radius. For NGC 7078, the +model shows a similar and milder profile to the observational one. +NGC 6121 shows isotropy inside but grows to tangential anisotropy at +a larger radius. The cluster was also found to be tangential anisotropy +in Vasiliev & Baumgardt (2021). It could imply a more substantial +influence from the tidal field, which is consistent with our results that +this cluster has a smaller truncation parameter than others. +5.5 The Imprint of Galactic Tidal Field +As mentioned earlier, the truncation parameter has the effect of mak- +ing the extent of the system finite, and also drives the profile to be +isotropic near the edge. These make the truncation parameter play +a similar role as the external tidal field for the cluster. The exter- +nal field generally becomes weaker for a larger distance from the +Galactic center. Thus, clusters at larger distances from the Galactic +center might be more extended and have larger values of truncation +parameter 𝑔. +In addition, Chernoff et al. (1986) found that the tidal field can +increase the evolution rate of the cluster through relaxation and shock +heating. Therefore, clusters closer to the Galactic center tend to evolve +faster. They also suggested that inner regions of the Galaxy could be +good places to look for the core-collapsed clusters. This agreed with +Djorgovski & King (1986) who found that the mean and median +distances of core-collapsed clusters from the Galactic center are +smaller than 5 kpc. +Moreover, the simulation in Zocchi et al. (2016) showed some +related properties during the evolution of a globular cluster in an +external tidal field. For example, the truncation parameter 𝑔 and the +cluster mass 𝑀 decrease during the evolution. The concentration +parameter 𝑊0 grows with time and decreases slightly after core col- +lapse. The half-mass radius 𝑟h also increases with time and decreases +as the cluster loses most of its mass. +Motivated by the above results, here we examine possible correla- +tions between any pairs among the concentration parameter 𝑊0, the +truncation parameter 𝑔, the cluster mass 𝑀, the half-mass radius 𝑟h, +and the semimajor axis of the cluster orbit 𝑎. The values of 𝑎 were +taken as the average of the apogalactic and perigalactic distances in +Baumgardt et al. (2019a), and the rest are our best-fit values in Table +1. The Spearman rank-order correlation coefficients, 𝐶s, were then +calculated for all possible combinations; there were only two pairs +with an absolute value of 𝐶s greater than 0.5. The first pair is the +concentration parameter 𝑊0 and the truncation parameter 𝑔. Their +𝐶s = −0.65 indicates a strong anti-correlation between 𝑊0 and 𝑔. +The distribution is presented in Fig. 8. The second pair is the trunca- +tion parameter 𝑔 and the semimajor axis 𝑎 of the cluster orbit. The +corresponding correlation coefficient 𝐶s = 0.60 indicates a strong +correlation between 𝑔 and 𝑎; the result is presented in Fig. 9. +The anti-correlation between the concentration parameter 𝑊0 and +the truncation parameter 𝑔 is reasonable, as those with smaller trun- +cation parameters would have experienced stronger tidal fields and +evolve faster. It is likely that a certain fraction of them become +core-collapsed clusters and thus have larger concentrations. This +anti-correlation is also consistent with the simulations in Zocchi +et al. (2016). They showed that when the clusters form, the value of +concentration parameter 𝑊0 is nearly 4 and the value of truncation +parameter 𝑔 is nearly 2.5. During the evolution, the truncation pa- +rameter 𝑔 decreases, but the concentration parameter 𝑊0 increases. +MNRAS 000, 1–15 (20XX) + +Dynamical Properties of Globular Clusters +11 +0.0 +0.6 +1.2 +1.8 +2.4 +g +10 +0 +10 +20 +30 +log ra +27.5 +30.0 +32.5 +35.0 +M [105 M ] +7.5 +9.0 +10.5 +12.0 +rh [pc] +5.16 +5.22 +5.28 +5.34 +5.40 +D [kpc] +0 +10 +20 +30 +W0 +1.8 +2.4 +3.0 +3.6 +4.2 + [ +] +0.0 +0.6 +1.2 +1.8 +2.4 +g +10 +0 +10 +20 +30 +log ra +27.5 +30.0 +32.5 +35.0 +M [105 M ] +7.5 +9.0 +10.5 +12.0 +rh [pc] +5.16 +5.22 +5.28 +5.34 +5.40 +D [kpc] +1.8 +2.4 +3.0 +3.6 +4.2 + [ +] +Figure 5. The MCMC posterior parameter distributions of NGC 5139. +Table 3. The parameters of two models of NGC 5139. The first column indicates different models. The second to eighth columns show the fitting parameters. +The quantities in the last two columns are 𝜅 and 𝜒2r . +Model +𝑊0 +𝑔 +log ˆ𝑟a +𝑀 +𝑟h +𝐷 +Υ +𝜅 +𝜒2r +(105 M⊙) +(pc) +(kpc) +(Υ⊙) +A +4.02+0.48 +−0.65 +1.94+0.27 +−0.26 +0.41+0.08 +−0.10 +32.82+0.65 +−0.67 +8.82+0.19 +−0.17 +5.32 ± 0.03 +2.38 ± 0.09 +1.15 +3.86 +B +14.16+0.25 +−0.23 +1.28 ± 0.03 +11.38+5.83 +−6.00 +30.10 ± 0.59 +10.260.17 +0.16 +5.25 ± 0.03 +3.07 ± 0.09 +1.00 +5.64 +MNRAS 000, 1–15 (20XX) + +12 +Cheng and Jiang +101 +102 +103 +15 +20 +25 +μ [mag/arcsec2] +Model A +101 +102 +103 +15 +20 +25 +Model B +101 +102 +103 +0 +10 +20 +30 +σlos [km/s] +Model A +101 +102 +103 +0 +10 +20 +30 +Model B +101 +102 +103 +r [arcsec] +0 +10 +20 +30 +σpm [km/s] +Model A +101 +102 +103 +r [arcsec] +0 +10 +20 +30 +Model B +Figure 6. The comparison of the profiles from two models of NGC 5139. Left panels show the results from Model A and the right panels show those from +Model B. The open circles represent the data of Baumgardt (2017), Baumgardt & Hilker (2018), and Dalgleish et al. (2020) for line-of-sight velocity dispersions +and Watkins et al. (2015a) for proper motion velocity dispersions. The solid triangles are used for the data of Kamann et al. (2018) for line-of-sight velocity +dispersions and Vasiliev & Baumgardt (2021) for proper motion velocity dispersions. The models are expressed by grey lines. +Therefore, in Fig. 8, younger clusters are located at the top-left cor- +ner, and the older clusters are distributed at the bottom-right corner. +However, the exact relationships between these two parameters for +different clusters are still complicated and the strength of this anti- +correlation was not quantitatively investigated before. +On the other hand, the correlation between the truncation parame- +ter 𝑔 and the semimajor axis 𝑎 can be easily understood. The smaller +truncation parameter shows that a stronger tidal field influences the +cluster, and those clusters with smaller 𝑎 do experience stronger +tidal fields. However, the relation between the two above-mentioned +parameters shall also depends on the initial size and the orbital evolu- +tion of a cluster. The contribution from different Galactic components +make the exact behavior of the tidal field more complicated. It is rea- +sonable that this correlation has a correlation coefficient 𝐶s = 0.60. +The strong 𝑊0 − 𝑔 anti-correlation and 𝑔 − 𝑎 correlation shall +be regarded as observational results as the employed parameters are +obtained through our data-model fitting or from an observational +catalog in literature. In addition, these observational anti-correlation +and correlation agree with theoretical predictions. +6 SUMMARY AND CONCLUSIONS +In this work, we studied 18 clusters with the LIMEPY models, a unified +family of isothermal models. It can generate clusters with differ- +ent amounts of concentration, truncation, and anisotropy, which are +parametrized by continuous real numbers. Including some current +observational data, such as the MUSE survey and Gaia mission, the +fittings were carried out with a Markov Chain Monte Carlo ensemble +sampler EMCEE and the parameters were determined by minimizing +the 𝜒2 of the fittings. +The measurable physical properties such as masses and distances, +were compared with the values from the literature. Usually, Baum- +gardt & Hilker (2018) has larger masses, while Watkins et al. (2015b) +has smaller ones, and our results are in between. The smaller half- +mass radius in our results is consistent with the smaller mass esti- +mated compared with Baumgardt & Hilker (2018). Some differences +between the radius estimations might come from the effect of the +mass spectrum. For distance, our estimations are in agreement with +the literature. The mass-to-light ratios are also similar to the litera- +ture. +Generally, the models could produce profiles similar to the obser- +vational ones for most clusters. For NGC 5139, there are two groups +of parameters that correspond to a better fitting for the surface bright- +ness or the velocity-dispersion profiles. The anisotropic model gives a +smaller 𝜒2r and agrees with the best-fit results in Zocchi et al. (2017). +Some possible central dark objects, like an intermediate-mass black +hole or a group of stellar-mass black holes might improve the fitting. +NGC 6388 is also a candidate to host an intermediate-mass black +hole, with the actual central line-of-sight velocities being uncertain. +The data we used have the extension to nearly 5 arcsec with a velocity +dispersion ∼20 km/s. It could be fitted well with the LIMEPY model +except for the slope of proper-motion velocity dispersion. +For the anisotropy, NGC 5139 and NGC 7078 are anisotropic +MNRAS 000, 1–15 (20XX) + +Dynamical Properties of Globular Clusters +13 +101 +102 +103 +−1 +0 +1 +σT/σR − 1 +NGC 104 +101 +102 +−0.5 +0.0 +0.5 +NGC 288 +101 +102 +−0.5 +0.0 +0.5 +NGC 362 +101 +102 +−0.5 +0.0 +0.5 +σT/σR − 1 +NGC 1851 +101 +102 +−0.5 +0.0 +0.5 +NGC 2808 +101 +102 +−0.5 +0.0 +0.5 +NGC 3201 +101 +102 +103 +−0.5 +0.0 +0.5 +σT/σR − 1 +NGC 5139 +101 +102 +103 +−2 +−1 +0 +1 +2 +NGC 5904 +101 +102 +103 +−1 +0 +1 +NGC 6121 +101 +102 +−0.5 +0.0 +0.5 +σT/σR − 1 +NGC 6218 +101 +102 +−0.5 +0.0 +0.5 +NGC 6266 +101 +102 +−0.5 +0.0 +0.5 +NGC 6388 +101 +102 +103 +−1 +0 +1 +σT/σR − 1 +NGC 6397 +101 +102 +−0.5 +0.0 +0.5 +NGC 6441 +101 +102 +103 +−1 +0 +1 +NGC 6656 +101 +102 +r [arcsec] +−0.5 +0.0 +0.5 +σT/σR − 1 +NGC 6715 +101 +102 +103 +r [arcsec] +−0.5 +0.0 +0.5 +NGC 6752 +101 +102 +r [arcsec] +−1 +0 +1 +NGC 7078 +Figure 7. The anisotropy profiles of the clusters. The open circles represent the data of Libralato et al. (2022). The data of Jindal et al. (2019) are shown in +solid triangles. The crosses are used for Watkins et al. (2015a). The models are expressed by grey lines. The horizontal grey dashed lines represent zeros which +indicate isotropy. For each panel, the name of the cluster is mentioned at the top-left corner. +MNRAS 000, 1–15 (20XX) + +14 +Cheng and Jiang +4 +5 +6 +7 +8 +9 +W0 +0.5 +1.0 +1.5 +2.0 +2.5 +g +Figure 8. The truncation parameter versus the concentration parameter. The +vertical axis represents the truncation parameter and the horizontal axis ex- +presses the concentration parameter. Each point corresponds to a particular +cluster. +5 +10 +15 +20 +25 +a [kpc] +0.5 +1.0 +1.5 +2.0 +2.5 +g +Figure 9. The truncation parameter versus the semimajor axis of the cluster. +The vertical axis represents the truncation parameter and the horizontal axis +expresses the semimajor axis. Each data point corresponds to a particular +cluster. +with 𝜅 = 1.15 and 𝜅 = 1.16. The anisotropy leads to the rise in +central velocity dispersion in these clusters. Our estimations could +have some underestimations because the data are combined proper +motion dispersion profiles rather than separated radial and tangential +profiles. Nevertheless, the results are reasonable compared with some +literature, such as Watkins et al. (2015a) and Watkins et al. (2015b), +where the anisotropy in the studied clusters seem small. +From a theoretical aspect, the truncation parameter may render +the cluster to have a finite extension and isotropic profiles near the +edge. It is similar to the effect of the external tidal field. In addition, a +strong anti-correlation between the concentration parameter 𝑊0 and +the truncation parameter 𝑔 was confirmed, which gives the imprint +of the dynamical evolution of clusters. Finally, a strong correlation +between the truncation parameter 𝑔 and the semimajor axis 𝑎 was +also found, which could result from the influence of the Galactic tidal +field. +ACKNOWLEDGEMENTS +We are grateful to the reviewer, Holger Baumgardt, for the useful sug- +gestions which improved this paper significantly. We acknowledge +the financial support from the Ministry of Science and Technology, +Taiwan, (MOST grant 110-2112-M-007-035). We are grateful to the +authors of Trager et al. (1995), Harris (1996), McLaughlin & van der +Marel (2005), Baumgardt (2017), Baumgardt & Hilker (2018), Dal- +gleish et al. (2020), Kamann et al. (2018), McLaughlin et al. (2006), +Watkins et al. (2015a), Vasiliev & Baumgardt (2021), Häberle et al. +(2021), McNamara et al. (2003), McNamara et al. (2012), Zloczewski +et al. (2012), Watkins et al. (2015b), Baumgardt & Vasiliev (2021), +Baumgardt et al. (2020), Libralato et al. (2022), Jindal et al. (2019), +Baumgardt et al. 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L., 2016, MNRAS, 462, +696 +Zocchi A., Gieles M., Hénault-Brunet V., 2017, MNRAS, 468, 4429 +den Brok M., van de Ven G., van den Bosch R., Watkins L., 2014, MNRAS, +438, 487 +van Leeuwen F., 2009, A&A, 497, 209 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–15 (20XX) + diff --git a/9tE4T4oBgHgl3EQfDQub/content/tmp_files/load_file.txt b/9tE4T4oBgHgl3EQfDQub/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..439dc93f5c85483a8697f5d7faa9e586e2ca164d --- /dev/null +++ b/9tE4T4oBgHgl3EQfDQub/content/tmp_files/load_file.txt @@ -0,0 +1,1821 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf,len=1820 +page_content='MNRAS 000, 1–15 (20XX) Preprint 13 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 Investigating Dynamical Properties of Globular Clusters through a Family of Lowered Isothermal Models Chia-Hsuan Cheng1 and Ing-Guey Jiang1,2 1Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan 2Institute of Astronomy, National Tsing-Hua University, Hsinchu, Taiwan Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' in original form ZZZ ABSTRACT To investigate the dynamical properties of globular clusters, the surface brightness and kinematic data were collected and fitted to a family of lowered isothermal models called LIMEPY models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For 18 studied globular clusters, the amounts of concentration, truncation, and anisotropy were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, the cluster mass, half-mass radius, distance, and mass-to-light ratio were also obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In general, LIMEPY models could describe these clusters well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Among these 18 clusters, NGC 5139, NGC 6388, and NGC 7078 were claimed to be candidates to host intermediate-mass black holes in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The models could not appropriately fit the central proper-motion velocity dispersion of NGC 5139 and the slope of proper-motion velocity-dispersion profile of NGC 6388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Thus, more dedicated models with intermediate-mass black holes or a group of stellar-mass black holes at cluster centers may need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Considering NGC 7078, our model with some degree of anisotropy can fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Finally, the strong concentration-truncation anti-correlation and truncation-semimajor-axis correlation were revealed, which could be the observational imprint of the dynamical evolution of globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Key words: methods: numerical – stars: kinematics and dynamics – globular clusters: general – globular clusters: individual – galaxies: star clusters: general 1 INTRODUCTION Globular clusters are one of the oldest objects in the universe (Van- denberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' They extend spherically in several or tens of parsecs with hundreds of thousands of stars (Harris 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The high stellar densities make them the primary venue for hosting exotic objects like millisecond pulsars (Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 1991) and blue stragglers (Bailyn 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Globular clusters have been proposed to possibly also host intermediate-mass black holes (Ebisuzaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' With higher density, the core of a globular cluster relaxes faster than the halo and the relaxation time is short compared to the age of the cluster (Oort & van Herk 1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Thus, the center of globular clusters is expected to be isothermal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Having theoretical models describing globular clusters is help- ful in obtaining the physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The isothermal sphere is a model with isothermal cores, so it could be considered a suitable simple model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' However, this model extends to the infinite and has an unrealistic infinite mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This problem can be solved by introducing some cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For example, energy truncation can limit the velocity, so the stars with larger velocities escape from the cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' this results in a cluster model with finite mass and range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The truncation can be regarded as the effect of the external tidal field on star clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Different truncations lead to different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For example, subtract- ing a constant from the energy leads to the Woolley model (Woolley 1954), and further subtraction from the distribution function gives the King model (King 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The velocity distributions of clusters in the above models are isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' However, for realistic models, the possible anisotropy shall be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The diffusion caused by stellar encounters facilitates the entry of some stars into the cluster halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These stars diffuse to the halo along radial orbits and increase the radial anisotropy in the halo (Spitzer & Shapiro 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The violent relaxation in the stage of cluster formation can also contribute to some radial anisotropy in the cluster halo (Lynden-Bell 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' To include anisotropy in a model, one can add the angular momentum into the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The distribution function now depends on both the energy and the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For example, the Michie-King model (Michie 1963) includes the angular momentum in an exponential term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This model possesses the expected properties which contain an isothermal core with some anisotropy at the outer parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' A model with multi-mass components is another aspect of im- provement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Da Costa & Freeman (1976) made the extension from the King model by assuming that each component has the same form of distribution function with different constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Later, an anisotropic multi-mass model was introduced by Gunn & Griffin (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Re- cently, some extensions and unification of these isothermal models have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Considering the Woolley and the King model as different schemes of energy truncation characterized by some in- tegers, Gomez-Leyton & Velazquez (2014) established an extended model which parametrized the truncation by a non-negative real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This was further generalized by Gieles & Zocchi (2015) to include the radial anisotropy and multi-mass components in a fam- ily of lowered isothermal models, which can cover more properties of star clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' They also provided a fast model solver written as a Python code, LIMEPY, for this family of lowered isothermal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' © 20XX The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='04868v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='GA] 12 Jan 2023 2 Cheng and Jiang Thus, these models proposed by Gieles & Zocchi (2015) are called LIMEPY models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' As presented by Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2016), LIMEPY models could capture the main properties of the globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Moreover, Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2017) applied LIMEPY models in the study of NGC 5139 and found that part of the observed large central velocity dispersion could be produced by anisotropic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Thus, their results could provide some constraints on the previously proposed central intermediate- mass black hole in NGC 5139 (Noyola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This globular cluster, also named 𝜔 Centauri, is the most complex one which has many sub-populations (Sanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2020) and was heavily investi- gated with many controversial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' On the other hand, the central kinematics of NGC 6093 was studied by employing new integral- field spectrograph data, and the existence of an intermediate-mass black hole was supported (Göttgens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, NGC 6388 is also a candidate residence of the intermediate-mass black hole (Lützgendorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Moreover, with Gaia data, Vasiliev & Baumgardt (2021) per- formed a comprehensive study on the kinematic properties of many Galactic globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The proper motions were measured and the corresponding proper-motion dispersion profiles of 100 clusters were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Combining with HST and other literature data, Baum- gardt & Vasiliev (2021) also accurately derived the distances to these Galactic globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Therefore, motivated by the development of LIMEPY models, the controversial results of the central kinematics and intermediate-mass black holes, and the availability of new data derived from the Gaia mission, herein, we investigated the properties of 18 globular clusters with the LIMEPY models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Including data from recent observations such as the MUSE survey (Kamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2018) and Gaia mission (Vasiliev & Baumgardt 2021), the physical parameters of these clusters were obtained through the data-model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Our results could lead to updated and accurate descriptions of the dynamical states of these clusters for the cases in which the data could be well fitted by the LIMEPY models which can be isotropic or anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Our results might also imply the possible existence of intermediate-mass black holes for some globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For the rest of this paper, in Section 2, we introduce the model’s distribution function and essential properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The observational data are described in Section 3, and the parameter determination method is shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The results and discussions are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In Section 6, some conclusions are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2 THE MODEL The LIMEPY models were employed as the standard model in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' As presented (Gieles & Zocchi 2015), there are single-mass and multi-mass cases in LIMEPY models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Considering the single-mass models, the distribution functions have the following form: 𝑓 (𝐸, 𝐽) = 𝐴 exp � −𝐽2 2𝑟2a 𝑠2 � 𝐸𝛾 � 𝑔, 𝜙(𝑟t) − 𝐸 𝑠2 � , (1) for 𝐸 ≤ 𝜙(𝑟t) and 𝑓 (𝐸, 𝐽) = 0 for 𝐸 > 𝜙(𝑟t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The function 𝐸𝛾(𝑔, 𝑥) represents 𝑒𝑥 for 𝑔 = 0 and 𝑒𝑥𝛾(𝑔, 𝑥)/Γ(𝑔) for 𝑔 > 0, where 𝛾(𝑔, 𝑥) is the lower incomplete gamma function and Γ(𝑔) stands for the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This distribution function depends on the specific energy 𝐸 and the specific angular momentum 𝐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The function 𝜙 is the gravitational potential and 𝑟t is the truncation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The parameter 𝑔 is called the truncation parameter, and it regulates the energy truncation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The parameter 𝑟a is the anisotropic radius, and it determines how anisotropic a system is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' When 𝑟a grows, the model is less anisotropic, and 𝑟a → ∞ corresponds to an isotropic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The constants 𝐴 and 𝑠 are used to set the physical scale of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The density can be obtained by integrating the distribution function 𝑓 (𝐸, 𝐽) over the velocity space: 𝜌 = ∫ 𝑓 (𝐸, 𝐽) d3𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2) Since 𝐸 = 𝑣2/2 + 𝜙(𝑟) and the distribution function is zero for 𝐸 > 𝜙(𝑟t), it can be just integrated from 0 to 𝑣max = [2𝜙(𝑟t) − 2𝜙(𝑟)]1/2 at each 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This 𝑣max becomes zero when 𝑟 = 𝑟t and the density vanishes for 𝑟 ≥ 𝑟t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Hence, the truncation radius 𝑟t represents the distance where the density comes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The gravitational potential 𝜙 is subjected to the Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For spherical systems such as globular clusters, the equation results in the following form: d2𝜙 d𝑟2 + 2 𝑟 d𝜙 d𝑟 = 4𝜋𝐺𝜌, (3) where 𝑟 is the radial coordinate and 𝐺 is the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The relevant quantities were first turned into dimensionless ones for solving the Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The dimensionless potential is defined as ˆ𝜙 = [𝜙(𝑟t) − 𝜙]/𝑠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The dimensionless density and radius are ˆ𝜌 = 𝜌/𝜌0 and ˆ𝑟 = 𝑟/𝑟0, where 𝜌0 and 𝑟0 satisfy 4𝜋𝐺𝑟2 0𝜌0/𝑠2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Then, the Poisson equation becomes d2 ˆ𝜙 dˆ𝑟2 + 2 ˆ𝑟 d ˆ𝜙 dˆ𝑟 = −9 ˆ𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (4) The equation is solved with the boundary conditions that, at ˆ𝑟 = 0, d ˆ𝜙/dˆ𝑟 = 0 and ˆ𝜙 = 𝑊0, where 𝑊0 is a constant that specifies a particular solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Hence,𝑊0 is also a parameter of the LIMEPY model, called the concentration parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It characterizes the concentration of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' As previously mentioned, LIMEPY models provide an extended fam- ily of isothermal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Those famous models are included as sub- families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For example, the Woolley model (Woolley 1954) can be produced by setting 𝑔 = 0, 𝑟a → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' When 𝑔 = 1 and 𝑟a → ∞, the King model (King 1966) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The Wilson model (Wilson 1975), which is more extended, corresponds to 𝑔 = 2 and 𝑟a → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Models with 𝑊0 → ∞ or 𝑔 → ∞ become the isothermal spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, the polytrope can be represented as 𝑊0 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It includes the Plummer model (Plummer 1911) which corresponds to the model with 𝑔 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It has a finite mass but infinite extents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In general, the model with appropriate 𝑊0 and 𝑟a can be finite in extent if 𝑔 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 and conversely infinite in extent with 𝑔 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, Gieles & Zocchi (2015) also showed that one kind of finite model is unsuitable for star clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These systems have an upturn in the density far from the center, so there is a large amount of mass in the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The ratio of the virial radius and half-mass radius 𝑟v/𝑟h is a crucial parameter for these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' They suggested that the models with 𝑟v/𝑟h ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='64 can adequately describe star clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The LIMEPY models describe spherical systems with different con- centrations, truncation, and radial anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In general, the model is isotropic near the center but could be anisotropic in the middle part of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The energy truncation limits the contribution of anisotropy to radial orbits with 𝐸 ≈ 𝜙(𝑟t) and thus suppresses the degree of radial anisotropy near the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The corresponding physi- cal picture is that a cluster under the interaction of an external tidal field has a preferential mass loss on stars with radial orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This re- duces the amount of anisotropy in the outer region (Oh & Lin 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Simulations of star clusters in the tidal field confirmed this isotropic behavior near the edge (Tiongco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Thus, the energy truncation acts as a role of the tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In fact, MNRAS 000, 1–15 (20XX) Dynamical Properties of Globular Clusters 3 the tidal field can also make the outer region profiles tangentially anisotropic (Baumgardt & Makino 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition to the anisotropic radius 𝑟a, there is a convenient anisotropic parameter 𝜅 ≡ 2𝐾r/𝐾t, where 𝐾r is the total radial ki- netic energy and 𝐾t is the total tangential kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' If 𝜅 > 1, the system is radially anisotropic, and if 𝜅 < 1, the system is tangen- tially anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' When 𝜅 = 1, it is an isotropic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Therefore, 𝜅 represents a simple and global measure of the anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' We mainly used 𝜅 to determine the amount of the anisotropy of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2016), the comparisons with N-body simulations illustrated the variation of model parameters of a cluster during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The cluster started with the Plummer model and the sim- ulation snapshots at different time were fitted with LIMEPY models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The concentration parameter tended to increase with time, which was also suggested previously by King (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The truncation pa- rameter 𝑔 decreased roughly from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It corresponded to an increased truncation by the tidal field as a cluster gradually filled the Roche volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Thus, a cluster tends to become more concentrated and truncated with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, the degree of radial anisotropy increased due to radial diffusion but decreased later during the core collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 3 THE OBSERVATIONAL DATA One of our primary goals is to provide updated results with a complete inclusion of all available observational data for globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The observational data of 𝑉-band surface brightness 𝜇 were taken from Trager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (1995), which provided a catalog of surface brightness profiles for over a hundred Galactic globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Some proce- dures were needed before the data were ready for the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' There was a correction related to extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The method is based on the global mean curve discussed in Fitzpatrick (1999), which uses the mean value for the ratio of the extinction 𝐴𝑉 and the reddening 𝐸(𝐵 − 𝑉) so that 𝐴𝑉 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1𝐸(𝐵 − 𝑉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' We took the reddening in the catalog of Harris (1996) (2010 version) and then computed the corrected surface brightness by 𝜇𝑖 = 𝜇𝑖,0 − 𝐴𝑉 , where 𝜇𝑖,0 denotes the data before the correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The data with 𝑤𝑖 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 were not adopted according to McLaughlin & van der Marel (2005), where 𝑤𝑖 is the weight of each data given in Trager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Because the data number was large, which might make the surface brightness dominate the fitting, we sliced the radial range with equal logarithmic width and averaged the surface brightness and the weight in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The bin number was 55 which equaled the largest data number of the velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' To compute the uncertainty for each data, we followed the method in McLaughlin & van der Marel (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The uncertainty of the data was obtained by 𝜖𝜇,𝑖 = 𝜖𝜇,b/𝑤𝑖, where 𝜖𝜇,b is the base error bar for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For line-of-sight velocity dispersion, we used the profiles derived from the collected literature (Baumgardt 2017), the data from un- published spectra of stars in the ESO and Keck Science archives (Baumgardt & Hilker 2018), and the dispersion from the integral- field-unit data from the WAGGS project (Dalgleish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The above data are expressed by open circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The data from the MUSE survey (Kamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2018) were also used and denoted by solid triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Some additional data were supplemented and marked as crosses, such as those from McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2006) for NGC 104 and Larson & Seth (2015, private communication) for NGC 1851 and NGC 2808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (The data of McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2006) and Larson & Seth (2015, private communication) were collected from the compi- lation in Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b) and others were collected from the compilation in the updated web catalog (third version) of Baumgardt & Hilker (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=') For proper-motion velocity dispersion, we mainly took the data from the Hubble Space Telescope from Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015a) and the Gaia data from Vasiliev & Baumgardt (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Open circles expressed the former, and solid triangles expressed the latter in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Some additional data were supplemented and denoted by crosses, which include Häberle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2021) for NGC 6441, McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2006) for NGC 104, McNamara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2003) for NGC 7078, McNamara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2012) for NGC 6266, and Zloczewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2012) for NGC 6656 and NGC 6752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (The data of Vasiliev & Baumgardt (2021) and Häberle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2021) were collected from the updated web catalog of Baumgardt & Hilker (2018), and the data of McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2006), McNamara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2003), McNamara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2012), and Zloczewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2012) were collected from Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=') Some proper motion data were downloaded in units of km/s, which depends on the cluster distance written in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These data were transformed into mas/yr as the observational val- ues for our work here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The transformation is 𝑣 = 𝑣0/𝐷𝐶, where 𝑣 and 𝑣0 are the velocity in mas/yr and km/s, 𝐷 is the distance and 𝐶 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='74047 km yr kpc−1 mas−1 s−1 which is a factor for the unit conversion (van Leeuwen 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The values of cluster distances were taken from the corresponding literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' By taking the root mean square of the upper and lower error bars from the literature, we obtained a symmetric uncertainty for each data for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Finally, to focus on the systems with enough observational information, we studied 18 clusters with more than five data points in each type of the above observational profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 4 THE DETERMINATION OF PHYSICAL PARAMETERS It was shown in Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2017) that models with different amounts of anisotropy could give the same surface brightness but different kinematic profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Thus, using the surface brightness data alone can lead to some degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Therefore, here we included the surface brightness, the light-of-sight velocity dispersion, and the proper-motion velocity dispersion data to obtain complete pictures of the physical structures and kinematic properties of globular clusters by determining related physical parameters through the data-model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Following the method in Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2017), we employed the one-step fitting procedure with the single-mass LIMEPY models in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' With all three considered types of observational data, a single step of the fitting was performed to determine all cluster parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The fitting was done through the minimization of the 𝜒2 function: 𝜒2 = 𝜒2 sb + 𝜒2 los + 𝜒2 pm, (5) where 𝜒2 sb, 𝜒2 los, 𝜒2pm are the contributions from surface brightness, line-of-sight velocity dispersion, and proper-motion velocity disper- sion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' They are defined by 𝜒2 sb = 𝑛sb ∑︁ 𝑖=1 [𝜇𝑖 − ¯𝜇(𝑟𝑖)]2 𝜖2 𝜇,𝑖 , (6) 𝜒2 los = 𝑛los ∑︁ 𝑖=1 [𝜎los,𝑖 − ¯𝜎los(𝑟𝑖)]2 𝜖2 los,𝑖 , (7) and 𝜒2 pm = 𝑛pm ∑︁ 𝑖=1 [𝜎pm,𝑖 − ¯𝜎pm(𝑟𝑖)]2 𝜖2 pm,𝑖 , (8) MNRAS 000, 1–15 (20XX) 4 Cheng and Jiang where 𝜇𝑖 is the 𝑖-th observational data of a surface brightness profile, ¯𝜇(𝑟𝑖) is the theoretical surface brightness at that radial coordinate 𝑟𝑖, and 𝜖𝜇,𝑖 is the error bar of the data 𝜇𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Similarly, 𝜎los,𝑖, ¯𝜎los(𝑟𝑖), 𝜖los,𝑖 are the corresponding quantities for line-of-sight velocity dis- persion, and 𝜎pm,𝑖, ¯𝜎pm(𝑟𝑖), 𝜖pm,𝑖 are the observational data, the- oretical value, and error bar for proper-motion velocity dispersion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The numbers of observational data are 𝑛sb, 𝑛los, 𝑛pm, respectively, for the surface brightness, line-of-sight velocity disper- sion, and proper-motion velocity dispersion, individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The LIMEPY code was employed to obtain the above theoretical pro- files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This code needed five input parameters, including the concen- tration parameter 𝑊0, the truncation parameter 𝑔, the dimensionless anisotropy radius ˆ𝑟a, the cluster mass 𝑀, and the half-mass radius 𝑟h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The LIMEPY code generated several profiles, such as the surface mass density Σ(𝑟𝑖), line-of-sight mean-square velocity 𝑢2 L(𝑟𝑖), radial and tangential mean-square velocity on the projected plane 𝑢2 R(𝑟𝑖) and 𝑢2 T(𝑟𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Thus, the value of ¯𝜎los(𝑟𝑖) is simply the square root of 𝑢2 L(𝑟𝑖), and ¯𝜎pm(𝑟𝑖) is the square root of [𝑢2 R(𝑟𝑖) + 𝑢2 T(𝑟𝑖)]/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' To complete the data-model fitting, two more parameters were needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The cluster distance 𝐷 is a parameter that converts the radial coordinate of the theoretical profile from pc to arcsec and the ob- servational proper-motion velocity dispersion from mas/yr to km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The V-band mass-to-light ratio Υ is a parameter for producing the luminosity density Σ(𝑟𝑖)/Υ, and the surface brightness ¯𝜇(𝑟𝑖) can be obtained by ¯𝜇(𝑟𝑖) = 𝑀V,⊙ − 5(1 + log 𝑐) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 log(Σ(𝑟𝑖)/Υ), (9) where 𝑀V,⊙ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='83 mag is the V-band absolute magnitude of the Sun and 𝑐 = 𝜋/648000 rad/arcsec is a factor for the unit conversion (Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Through the minimization of the 𝜒2 function, the best-fit values of seven parameters 𝑊0, 𝑔, ˆ𝑟a, 𝑀, 𝑟h, 𝐷, Υ can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' We used the code EMCEE (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2013) to perform the 𝜒2 minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It is an affine-invariant ensemble sampler that employs the Markov chain Monte Carlo (MCMC) process (Goodman & Weare 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' One has to decide the initial distribution and the parameters range for the EMCEE samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For the concentration parameter 𝑊0, the range was set to 1 < 𝑊0 < 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It covers a similar range in Table II of King (1966) and represents various degrees of concentration of star clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Figure 4 in Gieles & Zocchi (2015) showed the relevant models for star clusters and the corresponding parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' hence we set 0 < 𝑔 < 3 for the truncation parameter accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The dimensionless anisotropy radius ˆ𝑟a needs a wide range to include the isotropic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Therefore, we set a large range for log ˆ𝑟a as −1 < log ˆ𝑟a < 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For the remained parameters, we checked the literature values and considered wider ranges to include more possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The ranges of these parameters were set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 < 𝑀 < 50 (105 M⊙), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 < 𝑟h < 15 (pc), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 < 𝐷 < 35 (kpc), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 < Υ < 5 (Υ⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Finally, the initial distributions of all parameters are set to be uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 5 RESULTS AND DISCUSSION The best-fit results are displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The first column shows the names of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Seven fitting parameters are listed from the second to eighth columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The second column presents the concen- tration parameter 𝑊0 and the values range roughly from 3 to 9 for these clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The third and the fourth columns show the truncation parameter 𝑔 and the logarithm of the dimensionless anisotropy radius log ˆ𝑟a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The fifth and sixth columns list the cluster mass 𝑀 and the half-mass radius 𝑟h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These clusters have 𝑟h ≲ 10 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Among them, NGC 5139 has the largest mass and radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The heliocentric distance 𝐷 is shown in the seventh column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Most clusters have 𝐷 ≲ 12 kpc except for NGC 6715, which is roughly two times distant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The eighth column reveals the V-band mass-to-light ratio Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' To understand the anisotropy conveniently, the quantity 𝜅 is shown in the ninth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 5139 and NGC 7078 have 𝜅 > 1, indicating the anisotropic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The quantity in the last column is the reduced chi-square 𝜒2r defined by 𝜒2 r = 𝜒2 𝑛 − 𝑛p , (10) where 𝑛 is the total number of data and 𝑛p is the number of parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 Comparison with Previous Work To compare our results with the previous work, we used the mea- surable physical properties estimated in the published literature, as listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' We first considered the comparison of the cluster’s total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In general, the masses estimated by Baumgardt & Hilker (2018) are larger than those estimated by Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b), and our results are usually between their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Almost all of our results are very close to the masses estimated in Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' We also compared our half-mass radius with the one in the catalog of Baumgardt & Hilker (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Generally, our results are smaller, consistent with the results of total mass, since our masses are lower than those in Baumgardt & Hilker (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Therefore, the radii of the clusters tend to be smaller to fit the line-of-sight velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Some differences between the radius might come from the mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The radial distributions of different species may introduce additional variation between the half-mass radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Nevertheless, the mass-to-light ratios obtained in our work are consistent with the values in Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2020) and Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For distance comparison, we compared with the values in Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b), Baumgardt & Vasiliev (2021), and Harris (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b) derived the distance by comparing their proper motion velocity dispersion with the line-of-sight velocity dispersion from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Baumgardt & Vasiliev (2021) calculated the mean distance from several methods, such as the Gaia EDR3 parallaxes, the method by fitting nearby subdwarfs to globular cluster main sequences, the color-magnitude diagram fitting, and the distances from the period-luminosity relation of RR Lyrae stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The distances in Harris (1996) are a compilation of the distance measurements from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 1 shows the ratio of our distance 𝐷 and the one published in literature 𝐷lit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=', 𝐷/𝐷lit, for each considered cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For each panel, the compared literature is labeled at the top-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Each point represents a particular cluster studied in the compared literature and this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The dashed line represents the unity, and the solid line is the average value of the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Two numbers are shown in the bottom-right of the panels, the left number is the averaged 𝐷/𝐷lit, and the right one is the averaged |𝐷/𝐷lit−1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These numbers indicate that our results are closer to Harris (1996) and Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b), and slightly lower than Baumgardt & Vasiliev (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In general, our results agree with the values from these studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='2 The Profiles Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2 to 4 show the profiles of surface brightness, line-of-sight veloc- ity dispersion, and proper-motion velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The horizontal axis is the distance from the cluster’s center in arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The vertical axis gives the surface brightness in mag/arcsec2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2, and ve- locity dispersion in km/s from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 3 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It can be seen that LIMEPY MNRAS 000, 1–15 (20XX) Dynamical Properties of Globular Clusters 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The properties of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The first column lists the names of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Columns two to eight show the fitting parameters, which are concentration parameter 𝑊0, truncation parameter 𝑔, the logarithm of the dimensionless anisotropy radius log ˆ𝑟a, cluster mass 𝑀, half-mass radius 𝑟h, distance 𝐷, and V-band mass-to-light ratio Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Column nine presents the quantity 𝜅 which measures the amount of anisotropy, and the final column gives 𝜒2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' cluster 𝑊0 𝑔 log ˆ𝑟a 𝑀 𝑟h 𝐷 Υ 𝜅 𝜒2r (105 M⊙) (pc) (kpc) (Υ⊙) NGC 104 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='13+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='02 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 NGC 288 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='46+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='38 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='40+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='44 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='32 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='80+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='18 NGC 362 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='83 NGC 6752 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='35+0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 NGC 7078 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='86+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='17 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='46 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The literature parameters of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The number in the parentheses represents the literature, (1) stands for Baumgardt & Hilker (2018), (2) refers to Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b), (3) corresponds to Baumgardt & Vasiliev (2021), (4) represents Harris (1996), and (5) is Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The updated values for (1) and (5) are picked from the web catalog of Baumgardt & Hilker (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' cluster 𝑀 𝑀 𝑟h 𝐷 𝐷 𝐷 Υ Υ (105 M⊙) (105 M⊙) (pc) (kpc) (kpc) (kpc) (Υ⊙) (Υ⊙) (1) (2) (1) (2) (3) (4) (5) (2) NGC 104 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='031 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 NGC 288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='934 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='11 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='37 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='56 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='988+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='089 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='088 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 1851 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='78+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='133 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 NGC 2808 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='89 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='45+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='060+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='112 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='111 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='02 NGC 3201 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='78 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='737+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='043 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='042 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='09 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 5139 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='52+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='45 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='43 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='19+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='426 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='047 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='68 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='61 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='479 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='060 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 NGC 6121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='871 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='011 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='69 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 6218 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='43 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='412+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='105 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='04 NGC 6388 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='27+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='95 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='34 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='45 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='171+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='162 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='161 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='07 NGC 6397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='966 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='70+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='90 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='09 NGC 6441 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='30 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='709+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='096 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='095 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='02 MNRAS 000, 1–15 (20XX) 6 Cheng and Jiang 5 10 15 20 25 D [kpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='2 D / Dlit Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='05 5 10 15 20 25 D [kpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='2 Baumgardt & Vasiliev (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='96, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='05 5 10 15 20 25 D [kpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='2 Harris (1996) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='98, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='05 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The comparison of the cluster distance with the values mentioned in earlier studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The horizontal axis is the cluster distance obtained in this work, and the vertical axis shows the ratio of our value to the distance given in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The dashed line and the solid line represent the unity and the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Each panel is for comparison with the particular publication, as labeled at the top-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' At the bottom-right corner, the left number is the averaged 𝐷/𝐷lit, and the right number is the averaged |𝐷/𝐷lit − 1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' models can produce similar profiles as observational ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' To ex- amine these clusters more quantitatively, we classified the results by 𝜒2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Many clusters were found to have 𝜒2r < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These clusters have suitable fittings for all three profiles, as shown in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 362 has the largest 𝜒2r , and the model profiles agree with the observations in surface brightness and line-of-sight velocity disper- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' However, the central part of the modeled proper-motion velocity dispersion is slightly larger than the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Data with a small error bar in the outer part located much higher than the profile, mak- ing the fitting worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 6441 also has a larger 𝜒2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The model agrees well with the surface brightness and the outer part of the proper motion velocity dispersion but predicts larger values for the inner part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The model can also fit the rough trend of the line-of-sight velocity dispersion, but some points lie below the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For NGC 3201, the model has smaller line-of-sight velocity dis- persion for radius above 100 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' There are also some under estimations for the proper motions in the outermost region, where the observational profile tends to level off rather than continue to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Some scenarios were proposed to explain the higher ve- locity dispersion in the outer part, such as the orbital history with accretion and the embedding by a dark matter halo (Bianchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It was also found that binary stars could contribute to part of the effect (Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For NGC 6715, the model agrees with the observations, except for the outermost region of the line-of-sight velocity dispersion, where the observational profile grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This rise is probably caused by the stars in the nucleus of the Sagittarius dwarf galaxy, where NGC 6715 inhabits (Bellazzini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 5139 has large central velocity dispersions, which the model cannot explain well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For NGC 6388, the model has a steeper proper- motion velocity dispersion profile than the observational one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Further discussions of these two clusters will be made in the following sub- section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='3 Possible Intermediate-Mass Black Hole ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Stellar black holes exist in astrophysical systems such as X-ray bina- ries (Mikolajewska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, supermassive black holes are also confirmed to exist at the centers of our Milky Way (GRAV- ITY Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2019) and other galaxies (Blandford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Whether there are any intermediate-mass black holes in the universe is one of the most important questions in astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Globu- lar clusters are considered good candidates to host intermediate-mass black holes and thus attract much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Among 18 globular clus- ters in the present work, NGC 5139 was discussed previously as a likely candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For our work here, the data-model fitting of NGC 5139 led to two groups of model parameters, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These groups have very different concentration parameters 𝑊0 and logarithm of the dimensionless anisotropy radius log ˆ𝑟a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' One has smaller 𝑊0 and log ˆ𝑟a, and the other has larger values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Hence, we do further fittings with narrower ranges as 1 < 𝑊0 < 8, −1 < log ˆ𝑟a < 2, and 8 < 𝑊0 < 15, 2 < log ˆ𝑟a < 20, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The results are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' We denote the one with lower 𝜒2r as Model A, the result previously listed in Table 1 and presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Model A has a low concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It also has a small dimensionless anisotropy radius with 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15, making it more anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In contrast, Model B is isotropic with a high concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The best-fit profiles are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Model A fits the surface brightness well but predicts lower central velocity dispersion, espe- cially for the proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' On the other hand, Model B has good fittings on both velocity dispersion but a poor fitting on the surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The deviation in surface brightness leads to a larger 𝜒2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Although the data and radial range of the observational kinematic profiles differs, the parameters from Model A agree with those in the best-fit model in Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These results obtained with two models show that it is difficult to perfectly and simultaneously fit all profiles of NGC 5139 with the current considered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This could indicate the existence of central dark objects which can cause an increase in central velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These objects could be an intermediate-mass black hole (Noyola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Baumgardt 2017) or a group of stellar-mass black holes at the cluster center (Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Both can also suppress the mass segregation of the stars (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Peuten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2016) and render the cluster to have a larger core (Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Peuten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The main difference is that the intermediate- mass black hole could produce some stars faster than 60 km/s in the central 20 arcsec of NGC 5139, which was not confirmed in current observations (Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 6388 is another candidate cluster that may host a central intermediate-mass black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The study of the integrated light spec- tra revealed a high central LOS velocity dispersion ∼25 km/s within 2 arcsec (Lützgendorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' However, there was also a result that suggests a dispersion ∼15 km/s in the same region derived from stars’ radial velocities (Lanzoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Hence, the actual kinematic behavior of the cluster center is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The data we used have the extension to nearly 5 arcsec with a velocity dispersion ∼20 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Our results show that the surface brightness and line-of-sight velocity MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 1–15 (20XX) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='Dynamical Properties of Globular Clusters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='μ [mag/arcsec2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 288 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 362 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='μ [mag/arcsec2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 1851 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 2808 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 3201 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='μ [mag/arcsec2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 5139 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 5904 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 6121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='μ [mag/arcsec2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 6218 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 6266 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 6388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='μ [mag/arcsec2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='NGC 6397 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 NGC 6441 100 101 102 103 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 NGC 6656 100 101 102 103 r [arcsec] 15 20 25 μ [mag/arcsec2] NGC 6715 100 101 102 103 r [arcsec] 15 20 25 NGC 6752 100 101 102 103 r [arcsec] 15 20 25 NGC 7078 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The surface brightness profiles of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The observations are shown as crosses, and the models are expressed by grey lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For each panel, the name of the cluster is mentioned at the top-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' MNRAS 000, 1–15 (20XX) 8 Cheng and Jiang 101 102 103 0 5 10 15 σlos [km/s] NGC 104 102 103 0 1 2 3 4 NGC 288 101 102 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 NGC 362 101 102 103 0 5 10 σlos [km/s] NGC 1851 101 102 0 5 10 15 NGC 2808 101 102 103 0 2 4 NGC 3201 101 102 103 0 10 20 30 σlos [km/s] NGC 5139 101 102 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 NGC 5904 102 103 0 2 4 6 NGC 6121 101 102 0 2 4 σlos [km/s] NGC 6218 101 102 0 5 10 15 20 NGC 6266 101 102 0 10 20 NGC 6388 101 102 103 0 2 4 6 σlos [km/s] NGC 6397 101 102 0 10 20 NGC 6441 101 102 103 0 5 10 NGC 6656 101 102 103 r [arcsec] 0 5 10 15 20 σlos [km/s] NGC 6715 101 102 103 r [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 NGC 6752 101 102 103 r [arcsec] 0 5 10 15 NGC 7078 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The line-of-sight velocity dispersion profiles of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The open circles represent the data of Baumgardt (2017), Baumgardt & Hilker (2018), and Dalgleish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The data of Kamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2018) are shown in solid triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The crosses are used for additional data of some clusters mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The models are expressed by grey lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For each panel, the name of the cluster is mentioned at the top-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' MNRAS 000, 1–15 (20XX) Dynamical Properties of Globular Clusters 9 101 102 103 0 5 10 15 σpm [km/s] NGC 104 101 102 0 2 4 NGC 288 101 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 NGC 362 101 102 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 σpm [km/s] NGC 1851 101 102 103 0 5 10 15 NGC 2808 102 103 0 1 2 3 4 NGC 3201 101 102 103 0 10 20 30 σpm [km/s] NGC 5139 100 101 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 NGC 5904 102 103 0 2 4 NGC 6121 102 103 0 1 2 3 4 σpm [km/s] NGC 6218 100 101 102 0 10 20 NGC 6266 101 102 0 5 10 15 20 NGC 6388 101 102 103 0 2 4 6 σpm [km/s] NGC 6397 100 101 102 0 10 20 NGC 6441 101 102 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 NGC 6656 101 102 r [arcsec] 0 5 10 15 20 σpm [km/s] NGC 6715 101 102 103 r [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 NGC 6752 100 101 102 r [arcsec] 0 5 10 15 20 NGC 7078 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The proper-motion velocity dispersion profiles of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The open circles represent the data of Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The data of Vasiliev & Baumgardt (2021) is shown in solid triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The crosses are used for additional data of some clusters mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The models are expressed by grey lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For each panel, the name of the cluster is mentioned at the top-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' MNRAS 000, 1–15 (20XX) 10 Cheng and Jiang dispersion can be fitted well without the central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' However, the model predicts a steeper proper-motion velocity-dispersion pro- file than the observations, being higher inside but lower outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This behavior can also be seen in Figure 9 of Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 7078 is also a candidate cluster that could host an intermediate-mass black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The increase in central velocity dis- persion found in Hubble Space Telescope was explained by an intermediate-mass black hole (Gerssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' However, the cluster can also be fitted with a group of dark stellar remnants (den Brok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2014) or N-body simulations without intermediate-mass black holes (Baumgardt 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In our results, the cluster could be fit- ted well without central black holes, and some degree of anisotropy was observed, which can raise the central velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, although there are raised velocity dispersions in observation, the un- certainties of the data are also large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Therefore, we obtain a better fitting than NGC 5139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='4 The Anisotropy Two clusters, NGC 5139 and NGC 7078, possess small dimensionless anisotropy radius and reveal some degree of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The former has 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 and the latter has 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Other clusters have isotropic behavior with 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='00 and a large anisotropy radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' One effect of radial anisotropy is that it can increase the central velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The rise in central velocity dispersions can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' On the other hand, the amount of anisotropy estimated from our fittings could be underestimated, since the difference between tangential and radial proper motions will be averaged out in the combined proper motion velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The results are reasonable compared with some previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For example, the parameters of NGC 5139 are similar to those es- timated in Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2017) which the fittings were carried out with both radial and tangential proper motion velocity dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The weak anisotropy in many clusters were also reported by Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015a) and Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b), in which most of our samples were also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b) showed that their distance estimation had good agreement with Harris (1996) and con- cluded that the assumption of isotropy for their samples is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015a) examined the ratio 𝜎T/𝜎R, which compared the tangential and radial components of the proper motion velocity dispersion at different radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' They found that the cluster centers are relatively isotropic, and the behavior of the increasing anisotropy with the radius was very moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' From their figures, it can be seen that the decreasing of 𝜎T/𝜎R with a growing radius is more evident for NGC 5139 and NGC 7078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In recent years, Gaia has provided the proper motion data in the outer parts of globular clusters, and the behavior of 𝜎T/𝜎R reveals more evidence of anisotropy (Jindal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Vasiliev & Baum- gardt 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In both studies, NGC 5904 appears to be isotropic, and NGC 104, NGC 5139, and NGC 7078 show radial anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Some clusters are anisotropic in one study but are isotropic or uncertain in another;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' these include NGC 2808, NGC 6121, NGC 6397, NGC 6656, and NGC 6752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, the anisotropy profiles 𝜎T/𝜎R−1 from the observations and our models are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The observational data was mainly from a recent report on the globular-cluster survey through Hubble Space Telescope (Libralato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It includes 16 clusters of our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The remaining two clusters were supplemented with the data from Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The data from Gaia (Jindal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2019) which contains half of our samples were also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 7, the data of the above-discussed literature are expressed by open circles, crosses, and solid triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' the profiles are roughly isotropic or slightly radial anisotropic within 𝑟 ≲ 100 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The larger anisotropy appears mainly in the outer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The radial anisotropy of NGC 5139 tends to increase from near 100 arcsec and later decrease to isotropy in 𝑟 ≳ 1000 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Our model predicts the decrease in radial anisotropy at a larger radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For NGC 7078, the model shows a similar and milder profile to the observational one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 6121 shows isotropy inside but grows to tangential anisotropy at a larger radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The cluster was also found to be tangential anisotropy in Vasiliev & Baumgardt (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It could imply a more substantial influence from the tidal field, which is consistent with our results that this cluster has a smaller truncation parameter than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 The Imprint of Galactic Tidal Field As mentioned earlier, the truncation parameter has the effect of mak- ing the extent of the system finite, and also drives the profile to be isotropic near the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' These make the truncation parameter play a similar role as the external tidal field for the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The exter- nal field generally becomes weaker for a larger distance from the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Thus, clusters at larger distances from the Galactic center might be more extended and have larger values of truncation parameter 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, Chernoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (1986) found that the tidal field can increase the evolution rate of the cluster through relaxation and shock heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Therefore, clusters closer to the Galactic center tend to evolve faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' They also suggested that inner regions of the Galaxy could be good places to look for the core-collapsed clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This agreed with Djorgovski & King (1986) who found that the mean and median distances of core-collapsed clusters from the Galactic center are smaller than 5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Moreover, the simulation in Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2016) showed some related properties during the evolution of a globular cluster in an external tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For example, the truncation parameter 𝑔 and the cluster mass 𝑀 decrease during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The concentration parameter 𝑊0 grows with time and decreases slightly after core col- lapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The half-mass radius 𝑟h also increases with time and decreases as the cluster loses most of its mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Motivated by the above results, here we examine possible correla- tions between any pairs among the concentration parameter 𝑊0, the truncation parameter 𝑔, the cluster mass 𝑀, the half-mass radius 𝑟h, and the semimajor axis of the cluster orbit 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The values of 𝑎 were taken as the average of the apogalactic and perigalactic distances in Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2019a), and the rest are our best-fit values in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The Spearman rank-order correlation coefficients, 𝐶s, were then calculated for all possible combinations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' there were only two pairs with an absolute value of 𝐶s greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The first pair is the concentration parameter 𝑊0 and the truncation parameter 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Their 𝐶s = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='65 indicates a strong anti-correlation between 𝑊0 and 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The distribution is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The second pair is the trunca- tion parameter 𝑔 and the semimajor axis 𝑎 of the cluster orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The corresponding correlation coefficient 𝐶s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='60 indicates a strong correlation between 𝑔 and 𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' the result is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The anti-correlation between the concentration parameter 𝑊0 and the truncation parameter 𝑔 is reasonable, as those with smaller trun- cation parameters would have experienced stronger tidal fields and evolve faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It is likely that a certain fraction of them become core-collapsed clusters and thus have larger concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This anti-correlation is also consistent with the simulations in Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' They showed that when the clusters form, the value of concentration parameter 𝑊0 is nearly 4 and the value of truncation parameter 𝑔 is nearly 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' During the evolution, the truncation pa- rameter 𝑔 decreases, but the concentration parameter 𝑊0 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' MNRAS 000, 1–15 (20XX) Dynamical Properties of Globular Clusters 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='2 1.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='40 D [kpc] 0 10 20 30 W0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='2 [ ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='4 g 10 0 10 20 30 log ra 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 M [105 M ] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 rh [pc] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='40 D [kpc] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='2 [ ] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The MCMC posterior parameter distributions of NGC 5139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The parameters of two models of NGC 5139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The first column indicates different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The second to eighth columns show the fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The quantities in the last two columns are 𝜅 and 𝜒2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Model 𝑊0 𝑔 log ˆ𝑟a 𝑀 𝑟h 𝐷 Υ 𝜅 𝜒2r (105 M⊙) (pc) (kpc) (Υ⊙) A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='41+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='82+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='67 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='82+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='86 B 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='38+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='83 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='00 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='59 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='64 MNRAS 000, 1–15 (20XX) 12 Cheng and Jiang 101 102 103 15 20 25 μ [mag/arcsec2] Model A 101 102 103 15 20 25 Model B 101 102 103 0 10 20 30 σlos [km/s] Model A 101 102 103 0 10 20 30 Model B 101 102 103 r [arcsec] 0 10 20 30 σpm [km/s] Model A 101 102 103 r [arcsec] 0 10 20 30 Model B Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The comparison of the profiles from two models of NGC 5139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Left panels show the results from Model A and the right panels show those from Model B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The open circles represent the data of Baumgardt (2017), Baumgardt & Hilker (2018), and Dalgleish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2020) for line-of-sight velocity dispersions and Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015a) for proper motion velocity dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The solid triangles are used for the data of Kamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2018) for line-of-sight velocity dispersions and Vasiliev & Baumgardt (2021) for proper motion velocity dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The models are expressed by grey lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Therefore, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 8, younger clusters are located at the top-left cor- ner, and the older clusters are distributed at the bottom-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' However, the exact relationships between these two parameters for different clusters are still complicated and the strength of this anti- correlation was not quantitatively investigated before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' On the other hand, the correlation between the truncation parame- ter 𝑔 and the semimajor axis 𝑎 can be easily understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The smaller truncation parameter shows that a stronger tidal field influences the cluster, and those clusters with smaller 𝑎 do experience stronger tidal fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' However, the relation between the two above-mentioned parameters shall also depends on the initial size and the orbital evolu- tion of a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The contribution from different Galactic components make the exact behavior of the tidal field more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It is rea- sonable that this correlation has a correlation coefficient 𝐶s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The strong 𝑊0 − 𝑔 anti-correlation and 𝑔 − 𝑎 correlation shall be regarded as observational results as the employed parameters are obtained through our data-model fitting or from an observational catalog in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, these observational anti-correlation and correlation agree with theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 6 SUMMARY AND CONCLUSIONS In this work, we studied 18 clusters with the LIMEPY models, a unified family of isothermal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It can generate clusters with differ- ent amounts of concentration, truncation, and anisotropy, which are parametrized by continuous real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Including some current observational data, such as the MUSE survey and Gaia mission, the fittings were carried out with a Markov Chain Monte Carlo ensemble sampler EMCEE and the parameters were determined by minimizing the 𝜒2 of the fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The measurable physical properties such as masses and distances, were compared with the values from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Usually, Baum- gardt & Hilker (2018) has larger masses, while Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b) has smaller ones, and our results are in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The smaller half- mass radius in our results is consistent with the smaller mass esti- mated compared with Baumgardt & Hilker (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Some differences between the radius estimations might come from the effect of the mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For distance, our estimations are in agreement with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The mass-to-light ratios are also similar to the litera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Generally, the models could produce profiles similar to the obser- vational ones for most clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For NGC 5139, there are two groups of parameters that correspond to a better fitting for the surface bright- ness or the velocity-dispersion profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The anisotropic model gives a smaller 𝜒2r and agrees with the best-fit results in Zocchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Some possible central dark objects, like an intermediate-mass black hole or a group of stellar-mass black holes might improve the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' NGC 6388 is also a candidate to host an intermediate-mass black hole, with the actual central line-of-sight velocities being uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The data we used have the extension to nearly 5 arcsec with a velocity dispersion ∼20 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It could be fitted well with the LIMEPY model except for the slope of proper-motion velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For the anisotropy, NGC 5139 and NGC 7078 are anisotropic MNRAS 000, 1–15 (20XX) Dynamical Properties of Globular Clusters 13 101 102 103 −1 0 1 σT/σR − 1 NGC 104 101 102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 NGC 288 101 102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 NGC 362 101 102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 σT/σR − 1 NGC 1851 101 102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 NGC 2808 101 102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 NGC 3201 101 102 103 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 σT/σR − 1 NGC 5139 101 102 103 −2 −1 0 1 2 NGC 5904 101 102 103 −1 0 1 NGC 6121 101 102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 σT/σR − 1 NGC 6218 101 102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 NGC 6266 101 102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 NGC 6388 101 102 103 −1 0 1 σT/σR − 1 NGC 6397 101 102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 NGC 6441 101 102 103 −1 0 1 NGC 6656 101 102 r [arcsec] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 σT/σR − 1 NGC 6715 101 102 103 r [arcsec] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 NGC 6752 101 102 r [arcsec] −1 0 1 NGC 7078 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The anisotropy profiles of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The open circles represent the data of Libralato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The data of Jindal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2019) are shown in solid triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The crosses are used for Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The models are expressed by grey lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The horizontal grey dashed lines represent zeros which indicate isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' For each panel, the name of the cluster is mentioned at the top-left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' MNRAS 000, 1–15 (20XX) 14 Cheng and Jiang 4 5 6 7 8 9 W0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 g Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The truncation parameter versus the concentration parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The vertical axis represents the truncation parameter and the horizontal axis ex- presses the concentration parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Each point corresponds to a particular cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 5 10 15 20 25 a [kpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='5 g Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The truncation parameter versus the semimajor axis of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The vertical axis represents the truncation parameter and the horizontal axis expresses the semimajor axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Each data point corresponds to a particular cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' with 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='15 and 𝜅 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' The anisotropy leads to the rise in central velocity dispersion in these clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Our estimations could have some underestimations because the data are combined proper motion dispersion profiles rather than separated radial and tangential profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Nevertheless, the results are reasonable compared with some literature, such as Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015a) and Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b), where the anisotropy in the studied clusters seem small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' From a theoretical aspect, the truncation parameter may render the cluster to have a finite extension and isotropic profiles near the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' It is similar to the effect of the external tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' In addition, a strong anti-correlation between the concentration parameter 𝑊0 and the truncation parameter 𝑔 was confirmed, which gives the imprint of the dynamical evolution of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Finally, a strong correlation between the truncation parameter 𝑔 and the semimajor axis 𝑎 was also found, which could result from the influence of the Galactic tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We are grateful to the reviewer, Holger Baumgardt, for the useful sug- gestions which improved this paper significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' We acknowledge the financial support from the Ministry of Science and Technology, Taiwan, (MOST grant 110-2112-M-007-035).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' We are grateful to the authors of Trager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (1995), Harris (1996), McLaughlin & van der Marel (2005), Baumgardt (2017), Baumgardt & Hilker (2018), Dal- gleish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2020), Kamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2018), McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2006), Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015a), Vasiliev & Baumgardt (2021), Häberle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2021), McNamara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2003), McNamara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2012), Zloczewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2012), Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2015b), Baumgardt & Vasiliev (2021), Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2020), Libralato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2022), Jindal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2019), Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' (2019a), for making their data publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' This paper used the VizieR catalogue access tool, operated at CDS, Strasbourg, France, and the Astrophysics Data System Bibliographic Services of National Aeronautics Space and Administration, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' Software: LIMEPY (Gieles & Zocchi 2015), EMCEE (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' 2013), corner, NumPy, and SciPy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' DATA AVAILABILITY The electronic file of Table 1 is available in machine-readable form at VizieR (vizier.' metadata={'source': 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typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} +page_content=' MNRAS 000, 1–15 (20XX)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfDQub/content/2301.04868v1.pdf'} diff --git a/ANFKT4oBgHgl3EQfVi5k/content/tmp_files/2301.11788v1.pdf.txt b/ANFKT4oBgHgl3EQfVi5k/content/tmp_files/2301.11788v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e722465402c14282571894ffd7a12cb96ea51a7 --- /dev/null +++ b/ANFKT4oBgHgl3EQfVi5k/content/tmp_files/2301.11788v1.pdf.txt @@ -0,0 +1,679 @@ +Prepared for submission to JHEP +Higgs Inflation: constraining the top quark mass +and breaking the H0-σ8 correlation +Jamerson G. Rodrigues,a Micol Benetti,b,c Rayff de Souzaa and Jailson Alcaniza +aObservatório Nacional, 20921-400, Rio de Janeiro, RJ, Brazil +bScuola Superiore Meridionale, Largo San Marcellino 10, 80138, Napoli, Italy +cIstituto Nazionale di Fisica Nucleare (INFN) Sezione di Napoli, Complesso Universitario di Monte +Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy +E-mail: jamersoncg@gmail.com, micol.benetti@unina.it, +souzarayff@gmail.com, alcaniz@on.br +Abstract: Extending previous results [JHEP 11 (2021) 091], we explore aspects of the +reheating mechanism for non-minimal Higgs inflation in the strong coupling regime. We +constrain the radiative corrections for the inflaton’s potential by considering the Coleman- +Weinberg approximation and use the Renormalization Group Equations for the Higgs field +to derive an upper limit on the quark top mass, mt. Using the current Cosmic Microwave +Background, Barion Acoustic Oscillation, and Supernova data, we obtain mt ≤ 170.44 GeV, +confirming the observational compatibility of the model with recent mt estimates reported +by the CMS collaboration. We also analyze the breakdown of the well-known correlation +involving the Hubble constant H0 and the clustering parameter σ8, which makes the model +interesting in light of the cosmological tensions discussed over the last decade. +Keywords: Cosmology, Primordial Universe, Cosmic Microwave Background, Higgs Field, +Cosmological Parameters. +arXiv:2301.11788v1 [astro-ph.CO] 27 Jan 2023 + +Contents +1 +Introduction +1 +2 +Non-minimal Inflation and Slow-Roll Analysis +3 +3 +Reheating analysis and results +4 +4 +Physical and cosmological consequences +7 +4.1 +Constraints on the top quark mass +7 +4.2 +The H0 − σ8 correlation +7 +5 +Conclusions +9 +1 +Introduction +The fundamental theory behind the initial conditions that led to the temperature fluctua- +tions in the Cosmic Microwave Background (CMB) [1, 2] and the formation of Large-Scale +Structure (LSS) of the universe [3–5] remains an open question in modern cosmology. In +this context, the paradigm of inflation rises as the most elegant description of the primordial +Universe [6–10]. In order to induce cosmic acceleration, the dynamical equations for the +inflaton field must enable a slowly varying solution, leading to a quasi-de Sitter Universe. +In the well-known slow-roll mechanism this is achieved in an approximately flat direction +of the inflaton’s scalar potential. +One particularly appealing approach is to induce a non-minimal coupling between the +inflaton and gravity, which results in a plateau at the large field regime [11–13] and drives +the model predictions to the sweet-spot of CMB observations [14]. From the phenomeno- +logical perspective, one specially interesting model was introduced by Berzrukov and Sha- +poshnikov [15], where the standard Higgs field rules the inflationary period at early times. +Such configuration allows one to compare the predictions of the model for the cosmological +observables with the phenomenology of the related particles at electroweak scale of energy. +Such analysis was explored in a number of interesting papers, see e.g. [16–20]. +Although robust, the analysis of inflationary models rely on a set of assumptions about +the evolution of cosmological quantities. In particular, the evolution of cosmological scales +from the moment they cross the Hubble radius during inflation up to the their re-entrance +at later times must be matched to all the eras of the cosmological expansion in order to +solve the horizon problem [21]. The matching condition can be written in the form +ln +� +k +a0H0 +� += −Nk − Nrh − NRD + ln +�aeqHeq +a0H0 +� ++ ln +� Hk +Heq +� +, +(1.1) +– 1 – + +where Nk is the number of e-folds the universe expanded between the horizon crossing +moment of the pivot scale k and the end of inflation and Nrh is the number of e-folds counted +from the end of inflation to the onset of the radiation dominance in the early Universe +(reheating). Also, NRD gives the amount of expansion between the end of reheating and +the end of radiation dominated era, while the subscript “eq" and “0" represent quantities +evaluated at matter-radiation equality and the present, respectively. One is not able to set +the amount of expansion the universe experienced in the inflationary period, Nk, without +further information about the subsequent periods of the expansion. This is particularly +problematic for the reheating period. +In a previous communication [20], we performed a Monte-Carlo Markov Chain (MCMC) +analysis of CMB and clustering data to check the observational viability of non-minimally +coupled φ4 models for a fixed inflationary e-fold number. In particular, we considered the +first order correction to the perturbative expansion of the inflationary potential, also known +as Coleman-Weinberg approximation [22], and constrained possible radiative corrections +coming from the underlying field theory supporting this cosmological scenario. In addition, +we used the two-loop Renormalization Group Equations to connect the model’s predictions +at inflationary energy scales to the electroweak observables and derived an estimate of the +top quark mass mt, indicating a possible tension with the Monte-Carlo Tevatron and LHC +reconstruction [23]. +In this work, we extend and complement the analysis reported in [20] by exploring the +predictions of non-minimal Higgs inflation for a wide range of the inflationary e-fold number +Nk and, consequently, of Nrh. Following the procedure developed in [20, 24], we employ a +MCMC analysis to compare the predictions of this inflationary scenario with the most recent +Cosmic Microwave Background (CMB), Baryon Acoustic Oscillation (BAO), and Supernova +(SN) data [1–5]. +In particular, we obtain new constraints on the radiative corrections +coming from the underlying field theory supporting this cosmological scenario and derive +an upper limit for the top quark mass, which is compared with recent mt measurements +from different experiments. Furthermore, we also explore whether this model could shed +some light on the so-called cosmological tensions, which include the well-known H0 tension, +a ∼ 4σ-discrepancy between direct measurements of H0 using low-z SN (H0 = 73.48 ± 1.66 +km/s/Mpc [25]) and the H0 estimate from current CMB data assuming the standard model +(H0 = 67.72 ± 0.41 km/s/Mpc [14]) [26, 27]. It is worth mentioning that most of the usual +mechanisms to solve this problem have failed so far, as alleviating the H0 discrepancy +worsens the agreement of other parameters with the data. In particular, the clustering +parameter, σ8, is constrained at σ8 = 0.766+0.024 +−0.021 by the Kilo-Degree Survey (KiDS-1000) +lensing estimation [28] and its correlation with the Hubble constant leads to significantly +too high values as the value of H0 increases. Breaking such a correlation is not only tricky +but also challenging for many cosmological scenarios. +This work is organized as follows. +In Sec. 2, we briefly introduce the non-minimal +inflationary scenario and present the results of the slow-roll analysis. In Sec. 3, we discuss +aspects of the reheating stage following the Higgs inflation and present the main results of +our statistical analysis of the cosmological data. Sec. 4 discusses the constraints derived on +the top quark mass and some implications on the current cosmological tensions. The main +– 2 – + +conclusions of this work are presented in Sec. 5. +2 +Non-minimal Inflation and Slow-Roll Analysis +As mentioned earlier, a common method to achieve slow-roll inflation is to induce a non- +minimal coupling between the inflaton field and gravity. Such procedure yields non-canonical +terms for the original scalar field and the metric, suggesting the use of a set of conformal +transformations in order to obtain the theory description in the familiar Einstein-Hilbert +formalism. A more detailed exposition of this approach can be found in [20]. +The Einstein frame lagrangian reads +LE = −M2 +P ˜R +2 ++ 1 +2(∂µχ)†(∂µχ) − VE(χ) , +(2.1) +and the subsequent time evolution is dictated by the inflaton’s potential +VE(χ) = λM4 +P +4ξ2 +� +1 − e− +� +2 +3 +χ +MP +�2 +� +�1 + a′ ln +� +1 +ξ e +� +2 +3 +χ +MP − 1 +ξ +� +� +(2.2) +where the large field regime is assumed for the inflaton, χ ≫ +√ +6MP , and a large coupling +regime is assumed for the non-minimal coupling, ξ ≫ 1. Note that the deviation from +the tree level potential is quantified by the parameter a′ ≡ βλ/λ, where βλ is the running +equation for the quartic coupling λ. The above potential was obtained by adopting the +prescription II procedure to compute the radiative corrections in the Jordan frame and all +couplings are computed at the scale M = MP , where MP is the reduced Planck mass [20]. +Once with the effective potential in the Einstein frame, the relevant slow-roll inflation- +ary parameters can be readily computed, which can be related to the spectral index and +tensor-to-scalar ratio, characteristic of the power spectrum of CMB perturbations probed +by Planck [1]. Although the field strength χ∗, necessary to compute the relevant inflation- +ary parameters, cannot be measured directly, we can infer its value from the duration of +inflation from horizon crossing up to the end of inflationary expansion, characterized by the +number of e-folds, which is also dependent on the form of the potential (2.2). +However, the inflationary number of e-folds is not a free parameter entirely, as it is +tied to the subsequent evolution of the universe, given its association with the horizon exit +of relevant cosmological scales. Therefore, the relevant scales probed by Planck seem to +correspond to an interval of 50-60 e-folds [21], which guides our range of exploration of the +parameter Nk. +In Fig. 1 we present our results for the spectral index and tensor-to-scalar ratio in the +nS × r plane, with a′ ranging from −0.1 (lower limit) to 1.0 (upper limit)1. Note that there +is a significant dependence of the inflationary predictions with the amount of expansion +during inflation, achieving compatibility with the Planck result2. It is also important to +1The values of a′ varying between [-0.010, 0.053], [-0.020, 0.036] and [-0.027, 0.023], corresponding to +Nk = 50, 55 and 60, respectively, are in agreement with the 95% C.L. Planck result. +2This agreement relies on the slow-roll approximations for the inflationary parameters and the phe- +nomenological power-law expansion of the primordial power spectrum. +– 3 – + +Figure 1. ns vs. r for Nk = 50, 55 & 60. The points in each curve indicate the parameters for +a null resultant of the radiative corrections (a′ = 0). The blue areas show the favored regions by +Planck 2018, with 68% and 95% confidence level (Planck TT, TE, EE + lowE + lensing + BK15 ++ BAO data set) [14]. +mention that the results obtained for the prediction of inflationary parameters are highly +independent of the coupling parameter ξ. +3 +Reheating analysis and results +Between the end of inflation and the onset of a radiation-dominated universe, the universe +undergoes a reheating period. Even though there are a number of proposals for the dy- +namics of the cosmos in this period [29–36], the reheating era is exceptionally difficult to +be constrained by observations, given the small length scales characteristic of this micro- +physical process. For previous works exploring the impact of reheating to the cosmological +observables see e.g. [37–40] and references therein. +In order to understand the influence of the reheating period on the inflationary predic- +tions, one can follow the steps developed in [38] and resume the matching condition (1.1) +to the expression: +Nk = −1 + 3ωrh +4 +Nrh − ln +� +V 1/4 +end +Hk +� ++ 61.55 , +(3.1) +where the amount of expansion through the inflationary period is explicitly related to the +reheating characteristics of the proposed model. Here, ωrh represents the effective equation- +of-state parameter of the cosmological fluid during reheating, Vend is the amplitude of the +inflaton’s potential energy at the end of inflation, Hk is the Hubble parameter evaluated at +horizon crossing and k = 0.05 Mpc−1 is the pivot scale. We also consider grh ∼ 100 for the +relativistic degrees of freedom to obtain the numerical factor above. +– 4 – + +0.08 +Nx = 50 +Nx = 55 +0.06 +Nx = 60 +0.05 +0.03 +0.D2 +0.D1 +0.00 +0.95 +0.96 +L60 +0.98 +0.99 +fha′ +r0.02 +H0 +σ8 +Nk=50 +0.179 ± 0.072 +0.032 ± 0.013 +68.82 ± 0.38 +0.841 ± 0.005 +Nk=52 +0.040 ± 0.015 +0.007 ± 0.002 +68.31 ± 0.41 +0.835 ± 0.005 +Nk=54 +0.011 ± 0.014 +0.004 ± 0.001 +67.71 ± 0.45 +0.817 ± 0.003 +Nk=54.5 +0.009 ± 0.013 +0.004 ± 0.001 +67.68 ± 0.43 +0.811 ± 0.003 +Nk=55 +0.010 ± 0.013 +0.004 ± 0.001 +67.71 ± 0.44 +0.804 ± 0.003 +Nk=56 +0.022 ± 0.015 +0.005 ± 0.001 +67.94 ± 0.45 +0.793 ± 0.003 +Nk=58 +0.283 ± 0.169 +0.044 ± 0.019 +68.37 ± 0.39 +0.779 ± 0.004 +Nk=60 +0.243 ± 0.088 +0.042 ± 0.015 +68.46 ± 0.38 +0.766 ± 0.005 +Table 1. Constraints for fixed Nk at 68% C.L. using the Planck TT, TE, EE + lowE + lensing + +BICEP2/Keck + BAO + Pantheon combination. +In what concerns non-minimal inflationary models, it is possible to show that the +inflaton condensate starts the reheating process oscillating with an effective matter-like +equation of state (ω1 = 0) and, after crossing a critical value χcr, finishes the process as a +radiation-like component of energy (ω2 = 1/3) [41, 42]. After some algebraic manipulations +and using the approximation Hk ∼ +� +V∗/3, valid during inflation, one obtains: +Nk = −1 +4N1 − ln +� +V 1/4 +end (a′) +� +V∗(a′)/3 +� ++ 61.55 +(3.2) +where we highlight the a′ dependence of the inflationary potential. +We analyze the present model for fixed values of Nk and compute the values of Vend +and Hk following the slow-roll approximations. +In our analysis we assume a standard +cosmological model with a modified primordial spectrum in which the radiative correction +parameter, a′, is free to vary. +For the parameter estimation we use the free available +CosmoMC code [43]3 and a combination of early and late data4 (for more details we refer +the reader to [20]). Table 1 shows the derived constraints on the most significant parameters +of our analysis. +Note that by computing the values of Vend and Hk, we can obtain the corresponding +values for N1, i.e, the amount of expansion that the universe went through, as matter- +like dominated, during the reheating process. The corresponding values are presented in +Figure 2. Note also that, for an expansion of ∼ 56 e-folds or greater during inflation, N1 +would have to assume negative values to satisfy the matching equation (3.2). By definition, +this condition would imply in a contraction of the universe between the end of inflation +3This is a MCMC code interfaced with the Boltzmann solver Code for Anisotropies in the Microwave +Background (CAMB) [44]. We modified CAMB following the indications of ModeCode [45, 46] in order to +analyse the specific form of the potential V (φ). +4We use the CMB Planck (2018) likelihood [1], using Plik temperature power spectrum, TT, and HFI +polarization EE likelihood at ℓ ≤ 29; BICEP2 and Keck Array experiments B-mode polarization data [2]; +BAO measurements from 6dFGS +[3], SDSS-MGS [47], and BOSS DR12 [4] surveys, and the Pantheon +sample of Type Ia supernovae [5]. +– 5 – + +Figure 2. Nk vs. N1 for each inflationary number of e-folds taken into consideration. N1 is given +by the matching equation (3.2), with a′ coming from the MCMC analysis (highlighted beside each +point). Through a linear regression between the points (solid blue line), we estimate a maximum +number Nk - where the transition to a radiation-dominated Universe happens instantaneously. +and the onset of the radiation-dominated epoch5. Thus, following the standard approach, +we discard these possibilities as non-physical. Therefore, we can tighten the bounds on +the maximum value for the inflationary number of e-folds, which yields an instantaneous +transition to the radiation-dominated expansion. +The results presented above are insensitive to the specific physical process that leads to +the transition between matter and radiation-like expansion in the reheating. As pointed out +in [17, 41], non-perturbative processes may occur before the perturbative decays become +viable (preheating), displacing the transition between the two expansion behaviors, which +is particularly true in the model of Higgs Inflation. In this context, a specially interesting +result was obtained in [48], where the authors discussed the resonant production of Higgs +and gauge degrees of freedom in the linear regime of the Higgs Inflation scenario. +For +100 < ξ < 1000, the preheating dominant process is the Higgs self-resonance, leading to +N1 ≃ 3. For higher values of the non-minimal coupling, ξ > 1000, it was pointed out that a +substantial amount of energy stored in the inflaton condensate is transferred to relativistic +gauge bosons already at the very first oscillation of the background (instant preheating), +leading to N1 = 0. Note that these results are in agreement with our analysis for Nk ≃ 55 +and Nk ≃ 56, respectively, which is also in agreeement with the MCMC result for the +5It is also possible to obtain N1 > 0 even for Nk > 56 if one considers exotic scenarios for the transition to +radiation dominance, including intermediary phase transitions of the reheating fluid to an exotic component +of energy ω′ > 1/3. +– 6 – + +0.179 +20 +*: Nk.max~55.98 +10.04 +10 +M +N0.011 +10.009 +10.01 +0 +10.022 +0.283 +-10 +0.243 +50 +52 +54 +55 +56 +58 +60 +Nkradiative corrections in the interval a′ ≃ [−0.003, 0.037] at 68% (C.L.). +4 +Physical and cosmological consequences +4.1 +Constraints on the top quark mass +It is helpful to recall that the result mentioned above is obtained in the framework of the +Higgs Inflation scenario, where a′ is associated with the β-function of the Higgs quartic +coupling λ. Once the renormalization group equations for the standard Higgs couplings +are considered, it is possible to link the cosmological constraints to the phenomenology +of the associated particles at the electroweak scale of energy6. In this context, following +the approach developed in [20], one shall infer an upper limit on the top quark pole mass, +mt ≤ 170.44 GeV, to reproduce the values of a′ above. Also, it is worth emphasizing that +this limit on mt is relatively insensitive to the amplitude of the non-minimal coupling once +the strong limit (ξ ≫ 1) is assumed. +The most precise constraints on the top quark mass are extracted from the kinematic +reconstruction of the t¯t events where mt is employed in the Monte-Carlo generator in order +to fit the data [49, 50]. This MC top quark mass is usually assumed to be the pole mass +even though the theoretical uncertainties inherent to this association are hard to quantify +[51]. From [52], the average value for the top quark mass is set to mt = 172.69 ± 0.30 GeV, +obtained from LHC and Tevatron data. If contrasted with the limit on mt obtained from +the cosmological analysis, this represents a significant discrepancy of 7.5σ. +Instead, one may consider theoretically cleaner the inference of the top quark pole +mass from the measurements of the cross-section of the top quark production, since the +theoretical computation of σ(t¯t) is explicitly performed in a renormalization scheme (e.g., +MS) [53]. In this case, the average value obtained from the Tevatron and LHC runs is +172.5 ± 0.7 GeV [52], lowering the discrepancy with our cosmological estimate of mt to +≈ 3σ. More recently, the CMS collaboration reported mt = 170.5±0.8 GeV, obtained from +the differential cross-section of the top production [54]. Such result perfectly agrees with +the results of our cosmological analysis of the Higgs Inflation. +4.2 +The H0 − σ8 correlation +The accuracy of cosmological and astrophysical measurements has significantly improved +in recent decades. While this has led to increasingly evident confirmation of the validity of +the standard cosmological model, it has also exposed some critical issues that have given +rise to heated debate. The well-known H0 tension has been extensively explored without +concluding so far (we refer the reader to [26, 27] and references therein). +It has also been widely pointed out that some of the current attempts to solve the +H0 tension have failed because as they alleviate the discrepancy on H0, they worsen the +agreement of other parameters with the data. In particular, the clustering parameter, σ8, is +constrained at σ8 = 0.766+0.024 +−0.021 by the Kilo-Degree Survey (KiDS-1000) lensing estimation +6The parameters considered in the definition of a′ are evaluated at the renormalization scale M = MP . +– 7 – + +0.75 +0.80 +0.85 +8 +66 +67 +68 +69 +70 +H0 +0.75 +0.8 +0.85 +8 +N=50 +N=52 +N=54 +N=54.5 +N=55 +N=56 +N=58 +N=60 +Figure 3. Confidence levels and posterior distributions for the H0 and σ8 parameters using the +joint data set CMB Planck (2018) + BICEP2 and Keck Array + BAO + Pantheon SNe Ia sample +and considering several values of Nk. +[28] and its correlation with the Hubble constant leads to values that are significantly too +high as the value of H0 increases. +It is generally agreed that a model that manages to resolve both tensions is a model +that breaks this degeneracy, but building such a model is proving difficult. So far, only +a handful of scenarios seem to succeed, such as the conjecture of a universe transition +from anti-de Sitter vacua to de Sitter vacua [55–57], some early interacting models [58] or +specific parametrizations of dark energy equation of state [59]. The model studied here is +promising, as it breaks the degeneracy between the two parameters. In particular, Table 1 +shows that as Nk increases between the values of 50 and 54.5, the values of the radiative +parameter, a′, H0, and σ8 decrease. Nevertheless, at the turning value of Nk = 54.5, there +is a behavior change, i.e., as Nk increases, the values of a′ and H0 also increase. In contrast, +the value of the clustering parameter, σ8, does not seem to be affected by this turning point +and continues to decrease. It means that, for values of Nk ∈ [54.5, 60]7, the correlation +between H0 and σ8 breaks down, as also shown in Figure 3. In particular, for the limiting +value Nk = 56, i.e., an instantaneous transition to the radiation-dominated expansion, the +degeneracy H0 − σ8 is such that it reduces the H0 tension, constraining H0 = 67.94 ± 0.45 +7As discussed earlier, we consider the cases Nk > 56 to be non-physical since they predict negative values +of N1. +– 8 – + +Km/s/Mpc, which is ≈ 3σ off from the SNe Ia measurements [25] and allowing a value of +σ8 = 0.793 ± 0.003, that is in full agreement with KiDS-1000 results [28]. +5 +Conclusions +In this work, we revisited the non-minimal inflationary scenario subject to radiative cor- +rections. +By performing an observational analysis of the φ4 primordial potential, non- +minimally coupled to the Ricci scalar, in light of the most recent CMB, clustering and +Supernova data and considering the allowed range for the observable inflationary e-folds, +we constrained the possible values of the radiative corrections of the inflaton potential, +encoded in the parameter a′, and the usual set of cosmological parameters. +From this analysis, we presented two main results. First, we set an upper limit to the +number of e-folds from the horizon crossing moment up to the end of inflation, Nk ≲ 56, +relative to instantaneous reheating, by considering the matching equation for the pivot scale +k = 0.05 Mpc−1. An even more stringent limit is imposed once considered the preheating +structure of the Higgs Inflation, yielding 55 ≲ Nk ≲ 56. Accordingly, the MCMC analysis +of the model translates into an upper bound for the top quark pole mass, mt ≤ 170.44 GeV, +which raises two possible interpretations for the consistency of the model at low-energies. +For example, considering the value of the top quark mass reconstructed from the analysis of +LHC and Tevatron data, Mt = 172.69 ± 0.30 GeV [23], implies a significant tension of 7.5σ +between the observed low-energy value and the amount inferred by the cosmological MCMC +analysis. On the other hand, assuming the top quark mass extracted from differential cross- +section of the top production, Mt = 170.5 ± 0.8 GeV, obtained by the CMS collaboration +[54], we found a perfect agreement between the cosmological analysis of the Higgs field and +its electroweak behaviour. +Second, the MCMC analysis of current observational data confirms the observational +viability of the model and shows that for the interval Nk ∈ [54.5, 60], it can break down the +well-known H0 − σ8 correlation (see Table 1). In particular, considering an instantaneous +transition to the radiation-dominated expansion, which occurs for Nk = 56, the H0 tension +is reduced to ≈ 3σ whereas the value of σ8 shows a complete agreement with KiDS-1000 +results. +These results reinforce the need to investigate Higgs inflation and its extensions from +both theoretical and observational sides and show that perspectives for a complete coherence +of the scenario may converge once data from future collider experiments [60, 61] improve +our understanding of the physics at the eletroweak scale. +Acknowledgments +We thank André Sznajder for helpful conversations. JGR acknowledges financial support +from the Programa de Capacitação Institucional (PCI) do Observatório Nacional/MCTI. +MB acknowledges Istituto Nazionale di Fisica Nucleare (INFN), sezione di Napoli, iniziativa +specifica QGSKY. RdS is supported by the Coordenação de Aperfeiçoamento de Pessoal +de Nível Superior (CAPES). 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Plus 137 (2022), no. 1 39, +[arXiv:2107.05003]. +– 13 – + diff --git a/ANFKT4oBgHgl3EQfVi5k/content/tmp_files/load_file.txt b/ANFKT4oBgHgl3EQfVi5k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..66322ac82c3d72ed68e2b7ac8cbdf8572b18bea9 --- /dev/null +++ b/ANFKT4oBgHgl3EQfVi5k/content/tmp_files/load_file.txt @@ -0,0 +1,807 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf,len=806 +page_content='Prepared for submission to JHEP Higgs Inflation: constraining the top quark mass and breaking the H0-σ8 correlation Jamerson G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Rodrigues,a Micol Benetti,b,c Rayff de Souzaa and Jailson Alcaniza aObservatório Nacional, 20921-400, Rio de Janeiro, RJ, Brazil bScuola Superiore Meridionale, Largo San Marcellino 10, 80138, Napoli, Italy cIstituto Nazionale di Fisica Nucleare (INFN) Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy E-mail: jamersoncg@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='com, micol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='benetti@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='it, souzarayff@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='com, alcaniz@on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='br Abstract: Extending previous results [JHEP 11 (2021) 091], we explore aspects of the reheating mechanism for non-minimal Higgs inflation in the strong coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' We constrain the radiative corrections for the inflaton’s potential by considering the Coleman- Weinberg approximation and use the Renormalization Group Equations for the Higgs field to derive an upper limit on the quark top mass, mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Using the current Cosmic Microwave Background, Barion Acoustic Oscillation, and Supernova data, we obtain mt ≤ 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='44 GeV, confirming the observational compatibility of the model with recent mt estimates reported by the CMS collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' We also analyze the breakdown of the well-known correlation involving the Hubble constant H0 and the clustering parameter σ8, which makes the model interesting in light of the cosmological tensions discussed over the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Keywords: Cosmology, Primordial Universe, Cosmic Microwave Background, Higgs Field, Cosmological Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='11788v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='CO] 27 Jan 2023 Contents 1 Introduction 1 2 Non-minimal Inflation and Slow-Roll Analysis 3 3 Reheating analysis and results 4 4 Physical and cosmological consequences 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='1 Constraints on the top quark mass 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='2 The H0 − σ8 correlation 7 5 Conclusions 9 1 Introduction The fundamental theory behind the initial conditions that led to the temperature fluctua- tions in the Cosmic Microwave Background (CMB) [1, 2] and the formation of Large-Scale Structure (LSS) of the universe [3–5] remains an open question in modern cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In this context, the paradigm of inflation rises as the most elegant description of the primordial Universe [6–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In order to induce cosmic acceleration, the dynamical equations for the inflaton field must enable a slowly varying solution, leading to a quasi-de Sitter Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In the well-known slow-roll mechanism this is achieved in an approximately flat direction of the inflaton’s scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' One particularly appealing approach is to induce a non-minimal coupling between the inflaton and gravity, which results in a plateau at the large field regime [11–13] and drives the model predictions to the sweet-spot of CMB observations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' From the phenomeno- logical perspective, one specially interesting model was introduced by Berzrukov and Sha- poshnikov [15], where the standard Higgs field rules the inflationary period at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Such configuration allows one to compare the predictions of the model for the cosmological observables with the phenomenology of the related particles at electroweak scale of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Such analysis was explored in a number of interesting papers, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' [16–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Although robust, the analysis of inflationary models rely on a set of assumptions about the evolution of cosmological quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In particular, the evolution of cosmological scales from the moment they cross the Hubble radius during inflation up to the their re-entrance at later times must be matched to all the eras of the cosmological expansion in order to solve the horizon problem [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The matching condition can be written in the form ln � k a0H0 � = −Nk − Nrh − NRD + ln �aeqHeq a0H0 � + ln � Hk Heq � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='1) – 1 – where Nk is the number of e-folds the universe expanded between the horizon crossing moment of the pivot scale k and the end of inflation and Nrh is the number of e-folds counted from the end of inflation to the onset of the radiation dominance in the early Universe (reheating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Also, NRD gives the amount of expansion between the end of reheating and the end of radiation dominated era, while the subscript “eq" and “0" represent quantities evaluated at matter-radiation equality and the present, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' One is not able to set the amount of expansion the universe experienced in the inflationary period, Nk, without further information about the subsequent periods of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' This is particularly problematic for the reheating period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In a previous communication [20], we performed a Monte-Carlo Markov Chain (MCMC) analysis of CMB and clustering data to check the observational viability of non-minimally coupled φ4 models for a fixed inflationary e-fold number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In particular, we considered the first order correction to the perturbative expansion of the inflationary potential, also known as Coleman-Weinberg approximation [22], and constrained possible radiative corrections coming from the underlying field theory supporting this cosmological scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In addition, we used the two-loop Renormalization Group Equations to connect the model’s predictions at inflationary energy scales to the electroweak observables and derived an estimate of the top quark mass mt, indicating a possible tension with the Monte-Carlo Tevatron and LHC reconstruction [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In this work, we extend and complement the analysis reported in [20] by exploring the predictions of non-minimal Higgs inflation for a wide range of the inflationary e-fold number Nk and, consequently, of Nrh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Following the procedure developed in [20, 24], we employ a MCMC analysis to compare the predictions of this inflationary scenario with the most recent Cosmic Microwave Background (CMB), Baryon Acoustic Oscillation (BAO), and Supernova (SN) data [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In particular, we obtain new constraints on the radiative corrections coming from the underlying field theory supporting this cosmological scenario and derive an upper limit for the top quark mass, which is compared with recent mt measurements from different experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Furthermore, we also explore whether this model could shed some light on the so-called cosmological tensions, which include the well-known H0 tension, a ∼ 4σ-discrepancy between direct measurements of H0 using low-z SN (H0 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='48 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='66 km/s/Mpc [25]) and the H0 estimate from current CMB data assuming the standard model (H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='41 km/s/Mpc [14]) [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' It is worth mentioning that most of the usual mechanisms to solve this problem have failed so far, as alleviating the H0 discrepancy worsens the agreement of other parameters with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In particular, the clustering parameter, σ8, is constrained at σ8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='766+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='024 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='021 by the Kilo-Degree Survey (KiDS-1000) lensing estimation [28] and its correlation with the Hubble constant leads to significantly too high values as the value of H0 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Breaking such a correlation is not only tricky but also challenging for many cosmological scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' This work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 2, we briefly introduce the non-minimal inflationary scenario and present the results of the slow-roll analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 3, we discuss aspects of the reheating stage following the Higgs inflation and present the main results of our statistical analysis of the cosmological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 4 discusses the constraints derived on the top quark mass and some implications on the current cosmological tensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The main – 2 – conclusions of this work are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 2 Non-minimal Inflation and Slow-Roll Analysis As mentioned earlier, a common method to achieve slow-roll inflation is to induce a non- minimal coupling between the inflaton field and gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Such procedure yields non-canonical terms for the original scalar field and the metric, suggesting the use of a set of conformal transformations in order to obtain the theory description in the familiar Einstein-Hilbert formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' A more detailed exposition of this approach can be found in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The Einstein frame lagrangian reads LE = −M2 P ˜R 2 + 1 2(∂µχ)†(∂µχ) − VE(χ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='1) and the subsequent time evolution is dictated by the inflaton’s potential VE(χ) = λM4 P 4ξ2 � 1 − e− � 2 3 χ MP �2 � �1 + a′ ln � 1 ξ e � 2 3 χ MP − 1 ξ � � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='2) where the large field regime is assumed for the inflaton, χ ≫ √ 6MP , and a large coupling regime is assumed for the non-minimal coupling, ξ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Note that the deviation from the tree level potential is quantified by the parameter a′ ≡ βλ/λ, where βλ is the running equation for the quartic coupling λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The above potential was obtained by adopting the prescription II procedure to compute the radiative corrections in the Jordan frame and all couplings are computed at the scale M = MP , where MP is the reduced Planck mass [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Once with the effective potential in the Einstein frame, the relevant slow-roll inflation- ary parameters can be readily computed, which can be related to the spectral index and tensor-to-scalar ratio, characteristic of the power spectrum of CMB perturbations probed by Planck [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Although the field strength χ∗, necessary to compute the relevant inflation- ary parameters, cannot be measured directly, we can infer its value from the duration of inflation from horizon crossing up to the end of inflationary expansion, characterized by the number of e-folds, which is also dependent on the form of the potential (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' However, the inflationary number of e-folds is not a free parameter entirely, as it is tied to the subsequent evolution of the universe, given its association with the horizon exit of relevant cosmological scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Therefore, the relevant scales probed by Planck seem to correspond to an interval of 50-60 e-folds [21], which guides our range of exploration of the parameter Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 1 we present our results for the spectral index and tensor-to-scalar ratio in the nS × r plane, with a′ ranging from −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='1 (lower limit) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='0 (upper limit)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Note that there is a significant dependence of the inflationary predictions with the amount of expansion during inflation, achieving compatibility with the Planck result2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' It is also important to 1The values of a′ varying between [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='010, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='053], [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='020, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='036] and [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='027, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='023], corresponding to Nk = 50, 55 and 60, respectively, are in agreement with the 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Planck result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 2This agreement relies on the slow-roll approximations for the inflationary parameters and the phe- nomenological power-law expansion of the primordial power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' – 3 – Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' ns vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' r for Nk = 50, 55 & 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The points in each curve indicate the parameters for a null resultant of the radiative corrections (a′ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The blue areas show the favored regions by Planck 2018, with 68% and 95% confidence level (Planck TT, TE, EE + lowE + lensing + BK15 + BAO data set) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' mention that the results obtained for the prediction of inflationary parameters are highly independent of the coupling parameter ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 3 Reheating analysis and results Between the end of inflation and the onset of a radiation-dominated universe, the universe undergoes a reheating period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Even though there are a number of proposals for the dy- namics of the cosmos in this period [29–36], the reheating era is exceptionally difficult to be constrained by observations, given the small length scales characteristic of this micro- physical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' For previous works exploring the impact of reheating to the cosmological observables see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' [37–40] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In order to understand the influence of the reheating period on the inflationary predic- tions, one can follow the steps developed in [38] and resume the matching condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='1) to the expression: Nk = −1 + 3ωrh 4 Nrh − ln � V 1/4 end Hk � + 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='55 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='1) where the amount of expansion through the inflationary period is explicitly related to the reheating characteristics of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Here, ωrh represents the effective equation- of-state parameter of the cosmological fluid during reheating, Vend is the amplitude of the inflaton’s potential energy at the end of inflation, Hk is the Hubble parameter evaluated at horizon crossing and k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='05 Mpc−1 is the pivot scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' We also consider grh ∼ 100 for the relativistic degrees of freedom to obtain the numerical factor above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' – 4 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='08 Nx = 50 Nx = 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='06 Nx = 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='D2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='D1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='96 L60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='99 fha′ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='02 H0 σ8 Nk=50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='179 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='013 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='841 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='005 Nk=52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='040 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='002 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='835 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='005 Nk=54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='004 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='001 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='817 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='003 Nk=54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='009 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='004 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='001 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='811 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='003 Nk=55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='010 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='004 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='001 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='804 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='003 Nk=56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='022 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='001 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='793 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='003 Nk=58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='283 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='044 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='019 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='779 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='004 Nk=60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='243 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='042 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='015 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='766 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='005 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Constraints for fixed Nk at 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' using the Planck TT, TE, EE + lowE + lensing + BICEP2/Keck + BAO + Pantheon combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In what concerns non-minimal inflationary models, it is possible to show that the inflaton condensate starts the reheating process oscillating with an effective matter-like equation of state (ω1 = 0) and, after crossing a critical value χcr, finishes the process as a radiation-like component of energy (ω2 = 1/3) [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' After some algebraic manipulations and using the approximation Hk ∼ � V∗/3, valid during inflation, one obtains: Nk = −1 4N1 − ln � V 1/4 end (a′) � V∗(a′)/3 � + 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='55 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='2) where we highlight the a′ dependence of the inflationary potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' We analyze the present model for fixed values of Nk and compute the values of Vend and Hk following the slow-roll approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In our analysis we assume a standard cosmological model with a modified primordial spectrum in which the radiative correction parameter, a′, is free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' For the parameter estimation we use the free available CosmoMC code [43]3 and a combination of early and late data4 (for more details we refer the reader to [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Table 1 shows the derived constraints on the most significant parameters of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Note that by computing the values of Vend and Hk, we can obtain the corresponding values for N1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='e, the amount of expansion that the universe went through, as matter- like dominated, during the reheating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The corresponding values are presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Note also that, for an expansion of ∼ 56 e-folds or greater during inflation, N1 would have to assume negative values to satisfy the matching equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' By definition, this condition would imply in a contraction of the universe between the end of inflation 3This is a MCMC code interfaced with the Boltzmann solver Code for Anisotropies in the Microwave Background (CAMB) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' We modified CAMB following the indications of ModeCode [45, 46] in order to analyse the specific form of the potential V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 4We use the CMB Planck (2018) likelihood [1], using Plik temperature power spectrum, TT, and HFI polarization EE likelihood at ℓ ≤ 29;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' BICEP2 and Keck Array experiments B-mode polarization data [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' BAO measurements from 6dFGS [3], SDSS-MGS [47], and BOSS DR12 [4] surveys, and the Pantheon sample of Type Ia supernovae [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' – 5 – Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Nk vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' N1 for each inflationary number of e-folds taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' N1 is given by the matching equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='2), with a′ coming from the MCMC analysis (highlighted beside each point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Through a linear regression between the points (solid blue line), we estimate a maximum number Nk - where the transition to a radiation-dominated Universe happens instantaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' and the onset of the radiation-dominated epoch5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Thus, following the standard approach, we discard these possibilities as non-physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Therefore, we can tighten the bounds on the maximum value for the inflationary number of e-folds, which yields an instantaneous transition to the radiation-dominated expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The results presented above are insensitive to the specific physical process that leads to the transition between matter and radiation-like expansion in the reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' As pointed out in [17, 41], non-perturbative processes may occur before the perturbative decays become viable (preheating), displacing the transition between the two expansion behaviors, which is particularly true in the model of Higgs Inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In this context, a specially interesting result was obtained in [48], where the authors discussed the resonant production of Higgs and gauge degrees of freedom in the linear regime of the Higgs Inflation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' For 100 < ξ < 1000, the preheating dominant process is the Higgs self-resonance, leading to N1 ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' For higher values of the non-minimal coupling, ξ > 1000, it was pointed out that a substantial amount of energy stored in the inflaton condensate is transferred to relativistic gauge bosons already at the very first oscillation of the background (instant preheating), leading to N1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Note that these results are in agreement with our analysis for Nk ≃ 55 and Nk ≃ 56, respectively, which is also in agreeement with the MCMC result for the 5It is also possible to obtain N1 > 0 even for Nk > 56 if one considers exotic scenarios for the transition to radiation dominance, including intermediary phase transitions of the reheating fluid to an exotic component of energy ω′ > 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' – 6 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='179 20 : Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='max~55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='98 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='04 10 M N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='011 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='009 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='01 0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='283 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='243 50 52 54 55 56 58 60 Nkradiative corrections in the interval a′ ≃ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='037] at 68% (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 4 Physical and cosmological consequences 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='1 Constraints on the top quark mass It is helpful to recall that the result mentioned above is obtained in the framework of the Higgs Inflation scenario, where a′ is associated with the β-function of the Higgs quartic coupling λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Once the renormalization group equations for the standard Higgs couplings are considered, it is possible to link the cosmological constraints to the phenomenology of the associated particles at the electroweak scale of energy6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In this context, following the approach developed in [20], one shall infer an upper limit on the top quark pole mass, mt ≤ 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='44 GeV, to reproduce the values of a′ above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Also, it is worth emphasizing that this limit on mt is relatively insensitive to the amplitude of the non-minimal coupling once the strong limit (ξ ≫ 1) is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The most precise constraints on the top quark mass are extracted from the kinematic reconstruction of the t¯t events where mt is employed in the Monte-Carlo generator in order to fit the data [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' This MC top quark mass is usually assumed to be the pole mass even though the theoretical uncertainties inherent to this association are hard to quantify [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' From [52], the average value for the top quark mass is set to mt = 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='30 GeV, obtained from LHC and Tevatron data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' If contrasted with the limit on mt obtained from the cosmological analysis, this represents a significant discrepancy of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Instead, one may consider theoretically cleaner the inference of the top quark pole mass from the measurements of the cross-section of the top quark production, since the theoretical computation of σ(t¯t) is explicitly performed in a renormalization scheme (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=', MS) [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In this case, the average value obtained from the Tevatron and LHC runs is 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='7 GeV [52], lowering the discrepancy with our cosmological estimate of mt to ≈ 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' More recently, the CMS collaboration reported mt = 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='8 GeV, obtained from the differential cross-section of the top production [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Such result perfectly agrees with the results of our cosmological analysis of the Higgs Inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='2 The H0 − σ8 correlation The accuracy of cosmological and astrophysical measurements has significantly improved in recent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' While this has led to increasingly evident confirmation of the validity of the standard cosmological model, it has also exposed some critical issues that have given rise to heated debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The well-known H0 tension has been extensively explored without concluding so far (we refer the reader to [26, 27] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' It has also been widely pointed out that some of the current attempts to solve the H0 tension have failed because as they alleviate the discrepancy on H0, they worsen the agreement of other parameters with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In particular, the clustering parameter, σ8, is constrained at σ8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='766+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='024 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='021 by the Kilo-Degree Survey (KiDS-1000) lensing estimation 6The parameters considered in the definition of a′ are evaluated at the renormalization scale M = MP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' – 7 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='85 8 66 67 68 69 70 H0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='85 8 N=50 N=52 N=54 N=54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5 N=55 N=56 N=58 N=60 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Confidence levels and posterior distributions for the H0 and σ8 parameters using the joint data set CMB Planck (2018) + BICEP2 and Keck Array + BAO + Pantheon SNe Ia sample and considering several values of Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' [28] and its correlation with the Hubble constant leads to values that are significantly too high as the value of H0 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' It is generally agreed that a model that manages to resolve both tensions is a model that breaks this degeneracy, but building such a model is proving difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' So far, only a handful of scenarios seem to succeed, such as the conjecture of a universe transition from anti-de Sitter vacua to de Sitter vacua [55–57], some early interacting models [58] or specific parametrizations of dark energy equation of state [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' The model studied here is promising, as it breaks the degeneracy between the two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In particular, Table 1 shows that as Nk increases between the values of 50 and 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5, the values of the radiative parameter, a′, H0, and σ8 decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Nevertheless, at the turning value of Nk = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5, there is a behavior change, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=', as Nk increases, the values of a′ and H0 also increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In contrast, the value of the clustering parameter, σ8, does not seem to be affected by this turning point and continues to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' It means that, for values of Nk ∈ [54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5, 60]7, the correlation between H0 and σ8 breaks down, as also shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In particular, for the limiting value Nk = 56, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=', an instantaneous transition to the radiation-dominated expansion, the degeneracy H0 − σ8 is such that it reduces the H0 tension, constraining H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='45 7As discussed earlier, we consider the cases Nk > 56 to be non-physical since they predict negative values of N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' – 8 – Km/s/Mpc, which is ≈ 3σ off from the SNe Ia measurements [25] and allowing a value of σ8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='793 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='003, that is in full agreement with KiDS-1000 results [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 5 Conclusions In this work, we revisited the non-minimal inflationary scenario subject to radiative cor- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' By performing an observational analysis of the φ4 primordial potential, non- minimally coupled to the Ricci scalar, in light of the most recent CMB, clustering and Supernova data and considering the allowed range for the observable inflationary e-folds, we constrained the possible values of the radiative corrections of the inflaton potential, encoded in the parameter a′, and the usual set of cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' From this analysis, we presented two main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' First, we set an upper limit to the number of e-folds from the horizon crossing moment up to the end of inflation, Nk ≲ 56, relative to instantaneous reheating, by considering the matching equation for the pivot scale k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='05 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' An even more stringent limit is imposed once considered the preheating structure of the Higgs Inflation, yielding 55 ≲ Nk ≲ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Accordingly, the MCMC analysis of the model translates into an upper bound for the top quark pole mass, mt ≤ 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='44 GeV, which raises two possible interpretations for the consistency of the model at low-energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' For example, considering the value of the top quark mass reconstructed from the analysis of LHC and Tevatron data, Mt = 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='30 GeV [23], implies a significant tension of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5σ between the observed low-energy value and the amount inferred by the cosmological MCMC analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' On the other hand, assuming the top quark mass extracted from differential cross- section of the top production, Mt = 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='8 GeV, obtained by the CMS collaboration [54], we found a perfect agreement between the cosmological analysis of the Higgs field and its electroweak behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Second, the MCMC analysis of current observational data confirms the observational viability of the model and shows that for the interval Nk ∈ [54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content='5, 60], it can break down the well-known H0 − σ8 correlation (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' In particular, considering an instantaneous transition to the radiation-dominated expansion, which occurs for Nk = 56, the H0 tension is reduced to ≈ 3σ whereas the value of σ8 shows a complete agreement with KiDS-1000 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' These results reinforce the need to investigate Higgs inflation and its extensions from both theoretical and observational sides and show that perspectives for a complete coherence of the scenario may converge once data from future collider experiments [60, 61] improve our understanding of the physics at the eletroweak scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' Acknowledgments We thank André Sznajder for helpful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' JGR acknowledges financial support from the Programa de Capacitação Institucional (PCI) do Observatório Nacional/MCTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' MB acknowledges Istituto Nazionale di Fisica Nucleare (INFN), sezione di Napoli, iniziativa specifica QGSKY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' RdS is supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' JSA is supported by CNPq (Grants no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 310790/2014-0 and – 9 – 400471/2014-0) and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro FAPERJ (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' 233906).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' We also acknowledge the use of CosmoMC and ModeCode packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' This work was developed thanks to the use of the National Observatory Data Center (CP- DON).' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} +page_content=' – 13 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf'} diff --git a/AtAzT4oBgHgl3EQfF_uW/content/tmp_files/2301.01021v1.pdf.txt b/AtAzT4oBgHgl3EQfF_uW/content/tmp_files/2301.01021v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..827b310a9c93d918d38afecd8c76f411aae7c122 --- /dev/null +++ b/AtAzT4oBgHgl3EQfF_uW/content/tmp_files/2301.01021v1.pdf.txt @@ -0,0 +1,1024 @@ +A fixed point can hide another one: the nonperturbative behavior of +the tetracritical fixed point of the O(N) models at large N +Shunsuke Yabunaka1, ∗ and Bertrand Delamotte2 +1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan +2Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, LPTMC, F-75005 Paris, France. +(Dated: January 4, 2023) +We show that at N = ∞ and below its upper critical dimension, d < dup, the critical and +tetracritical behaviors of the O(N) models are associated with the same renormalization group fixed +point (FP) potential. Only their derivatives make them different with the subtleties that taking +their N → ∞ limit and deriving them do not commute and that two relevant eigenperturbations +show singularities. This invalidates both the ϵ− and the 1/N− expansions. We also show how the +Bardeen-Moshe-Bander line of tetracritical FPs at N = ∞ and d = dup can be understood from a +finite-N analysis. +Field theories sometimes exhibit nonperturbative fea- +tures such as confinement [1], presence of bound states [2] +or exotic excitations [3], fixed points (FPs) of the renor- +malization group (RG) flows that are nonperturbative as +in the Kardar-Parisi-Zhang equation [4], divergence of +the perturbative RG flow at a finite RG scale [5], pres- +ence of a cusp in the FP potential as in the random field +Ising model [6], to cite but a few. Very often, these non- +perturbative effects are assumed either to occur in rather +complicated theories such as gauge and string theories or +in highly nontrivial statistical models. +O(N) models, which are the simplest scalar field theo- +ries, are often implicitly considered to be immune to these +complex phenomena. +Perturbative methods are there- +fore assumed to work almost all the time for these mod- +els, the exception to the rule being the Bardeen-Moshe- +Bander (BMB) phenomenon [7], related to the existence +of a line of tricritical FPs at N = ∞ and d = 3, which re- +quires nonperturbative FPs to be fully understood from a +large-N analysis [8]. From this viewpoint, the enormous +success of the ϵ = 4 − d expansion for the perturbative +calculation of the critical exponents associated with the +Wilson-Fisher (WF) FP [11] could let us believe that the +critical physics of the O(N) models is fully understood +for any N and d, especially since it is corroborated by +the 1/N and ϵ = d − 2 expansions [11]. +Our goal in this Letter is to show instead that although +the critical physics of the O(N) models, described by the +WF FP, is fully under perturbative control at both finite +and infinite N, the tetracritical physics of these models +at N = ∞ –and probably of infinitely many multicritical +behaviors– is not. We show below (i) that at N = ∞, it +is also associated with the WF FP, which is unexpected, +and (ii) that it nonetheless shows non-perturbative fea- +tures that are beyond the reach of the standard imple- +mentation of both the large-N and ϵ- expansions. We +show in particular a very intriguing phenomenon related +to the large-N limit of the tetracritical FP of the O(N) +models: from the second order, the derivatives of the +∗ yabunaka123@gmail.com +N = ∞ tetracritical FP potential, that is, of the WF FP +potential, are not identical to the limit of the derivatives +of the finite-N tetracritical FP potentials when N → ∞. +This turns out to be crucial for understanding the large- +N limit of tetracritical phenomena and shows that this +limit is much less trivial than what is usually said [9–11]. +The perturbative tetracritical FP corresponds to the +massless (ϕ2)4 theory, the upper critical dimension of +which is dup = 8/3. It is found in perturbation theory in +ϵ = 8/3 − d for all N ≥ 1 and it is three times infrared +unstable [12]. Calling λ/(384N 3) the coupling in front of +the dimensionless (ϕ2)4 term, the large-N perturbative +flow equation for λ reads [13]: +∂tλ = −3ϵλ + 9λ2 +4N + O(N −2). +(1) +From Eq. (1), we find that at leading order in N, the +nontrivial FP solution is λ∗ = 4ϵN/3 from which follows +that perturbation theory does not allow for a control of +the large-N limit of the tetracritical FP at fixed ϵ. Only +the double limit N → ∞ and ϵ → 0 such that the product +ϵN remains finite can possibly be under control. We come +back on this point in the following. +Let us recall that in generic dimensions d < 4, the +only nontrivial FP found in the standard large-N anal- +ysis of the O(N) models is the WF FP [14]. Thus, no +tetracritical FP is found at N = ∞ and d < 8/3 which +is paradoxical considering that it is perturbatively found +for all N < ∞ and ϵ > 0. +We show below that the solution to the paradox above +lies in the field dependence of the tetracritical FP poten- +tial whereas it cannot be obtained from its field expansion +and in particular from λ∗. The recourse to functional RG +methods is therefore mandatory. +The best way to implement functional RG is to con- +sider Wilson’s RG, as it is inherently functional [15]. We +recall below the take-away philosophy of the modern ver- +sion of Wilson’s RG known as the nonperturbative – or +functional – renormalization group (NPRG). +NPRG is based on the idea of integrating fluctuations +step by step [16]. It is implemented on the Gibbs free +energy Γ [17–23] of a model defined by an Hamiltonian +arXiv:2301.01021v1 [cond-mat.stat-mech] 3 Jan 2023 + +2 +(or euclidean action) H and a partition function Z. To +this model is associated a one-parameter family of models +with Hamiltonians Hk = H + ∆Hk and partition func- +tions Zk, where k is a momentum scale. In Hk, ∆Hk is +chosen such that only the rapid fluctuations in the origi- +nal model, those with wavenumbers |q| > k, are summed +over in the partition function Zk. Thus, the slow modes +(|q| < k) need to be decoupled in Zk and this is achieved +by giving them a mass of order k, that is by taking for +∆Hk a quadratic (mass-like) term, which is nonvanishing +only for the slow modes: +Zk[J] = +� +Dϕi exp(−H[ϕ] − ∆Hk[ϕ] + J · ϕ) +(2) +with ∆Hk[ϕ] = +1 +2 +� +q Rk(q2)ϕi(q)ϕi(−q), where, for in- +stance, Rk(q2) = (k2 − q2)θ(k2 − q2) and J · ϕ = +� +x Ji(x)ϕi(x). The k-dependent Gibbs free energy Γk[φ] +is defined as the (slightly modified) Legendre transform +of log Zk[J]: +Γk[φ] + log Zk[J] = J · φ − 1 +2 +� +q +Rk(q2)φi(q)φi(−q) (3) +with +� +q = +� +ddq/(2π)d. +With the choice of regulator +function Rk above, Γk[φ] interpolates between the Hamil- +tonian H when k is of order of the ultraviolet cut-off Λ +of the theory: ΓΛ ∼ H, and the Gibbs free energy Γ of +the original model when k = 0: Γk=0 = Γ. The exact +RG flow equation of Γk gives the evolution of Γk with +k between these two limiting cases. It is known as the +Wetterich equation. It reads [18]: +∂tΓk[φ] = 1 +2Tr[∂tRk(q2)(Γ(2) +k [q, −q; φ] + Rk(q))−1], (4) +where t = log(k/Λ), Tr stands for an integral over q and +a trace over group indices and Γ(2) +k [q, −q; φ] is the matrix +of the Fourier transforms of δ2Γk/δφi(x)δφj(y). +In most cases, Eq. (4) cannot be solved exactly and +approximations are mandatory. The best known approx- +imation consists in expanding Γk in powers of the deriva- +tives of φi and to truncate the expansion at a given fi- +nite order[24–32]. The approximation at lowest order is +dubbed the local potential approximation (LPA). For the +O(N) model it consists in approximating Γk by: +Γk[φ] = +� +x +�1 +2(∇φi)2 + Uk(φ) +� +(5) +where φ = √φiφi. Fixed points are found only for di- +mensionless quantities and the standard large-N limit +by rescaling the field and the potential by factors N −1/2 +and N −1 respectively. +Thus, we define the dimen- +sionless and rescaled field ¯φ and potential ¯Uk as ¯φ = +v +− 1 +2 +d +k +2−d +2 N −1/2φ and ¯Uk(¯φ) = v−1 +d k−dN −1Uk (φ) with +v−1 +d += 2d−1dπd/2Γ( d +2). The LPA flow of ¯Uk then reads: +∂t ¯Uk(¯φ) = − d ¯Uk(¯φ) + 1 +2(d − 2)¯φ ¯U ′ +k(¯φ)+ +� +1 − 1 +N +� +¯φ +¯φ + ¯U ′ +k(¯φ) + 1 +N +1 +1 + ¯U ′′ +k (¯φ) +(6) +FIG. 1. d = 2.6: ¯U(¯φ) for the T3 FP of Eq. (6). Green, red, +blue and black curves correspond to N = 1500, 2250, 4500 +and 42000. The orange dashed curve corresponds to the WF +FP at N = ∞. Inset: Close view of ¯U(¯φ) around ¯φi. +with ∂t = k∂k. The standard large-N limit of the LPA +flow equation above is obtained by (i) replacing the fac- +tor 1 − 1/N by 1, (ii) dropping the last term in Eq. (6) +because it is assumed to be sub-leading [33]. As a con- +sequence of the two steps above, the explicit dependence +in N in Eq. (6) disappears in the large-N limit. +The crucial point of the large-N limit is that assuming +point (ii) above, the resulting LPA flow equation on ¯Uk +can be shown to be exact in the limit N → ∞ [34]. Under +this assumption, all FPs of the O(N) models have been +found exactly at N = ∞ [14, 33–36]. The result is the +following: In a generic dimension d < 4 there is only one +nongaussian FP at N = ∞ which is the usual Wilson- +Fisher FP (WF). The exceptions to the rule above are the +BMB lines of FPs [7, 14, 37–39] existing in dimensions +d = 2 + 2/p with p an integer larger than 1. +We now show that the procedure described above is too +restrictive to study the large-N limit of the tetracritical +FPs. As said above, the standard large-N analysis con- +sists in neglecting the last term in Eq. (6). However, this +term is negligible only if (1 + ¯U ′′ +k (¯φ))−1 does not coun- +terbalance at large N its 1/N prefactor for some finite +values of ¯φ. We now show that because of singularities in +the third derivative of ¯Uk(¯φ), the contribution of the last +term in Eq. (6) cannot be neglected in the FP equation +of ¯U ′′ +k (¯φ) obtained by differentiating twice Eq. (6) (see +footnote below Eq. (8) for more detail). This turns out +to be sufficient to invalidate the standard large-N limit +in the tetracritical case. +We have numerically solved Eq. (6) and have found +for several values of N and d < 8/3 the perturbative +tetracritical FP that we call T3(N, d). As expected, T3 +bifurcates from the Gaussian FP in d = 8/3−. We have +followed it down to d = 2.6, see Fig. +1 and Fig. +3 +of the Suppl. Mat. The FP potential of T3, (i) shows +as expected two maxima, one of which being located at +¯φ = 0 and another one at ¯φ2 > 0, and two minima at +¯φ1 and ¯φ3 such that ¯φ3 > ¯φ2 > ¯φ1 > 0, see Fig. +1, +(ii) can be continuously followed up to arbitrarily large +values of N at fixed d < 8/3, (iii) has its three extrema + +T() +0.38466 +0.38464 +0.9 +0.38462 +0.8 +0.38460 +0.7 +1.75 +1.80 +1.85 +$1.90 +0.6 +0.5 +2.03 +¯φ1, ¯φ2, ¯φ3 approaching each other when N is increased at +fixed d. These extrema tend to a common value ¯φ0 when +N → ∞ which is the minimum of the FP potential, see +Fig. 1 and Fig. 4 of the Suppl. Mat. Point (ii) above is +paradoxical because it seems to contradict the standard +large-N approach where only the WF FP is found in a +generic dimension d < 8/3 at N = ∞. We now show +that the WF FP potential at N = ∞ is in fact the limit +when N → ∞ of the potential of T3 for d < 8/3. This +solves the above paradox because it explains why on one +hand there exists a nontrivial tetracritical FP at N = ∞ +and d < 8/3 and on the other hand that there is no +other nontrivial and smooth solution of Eq. (6) at N = +∞ than the WF FP potential. However, this creates a +new paradox since obviously the critical and tetracritical +universal behaviors cannot be the same since the two FPs +do not have the same number of unstable eigendirections. +We now explain in detail this new paradox. +We can see on Fig. 1 that the FP potentials found +in d = 2.6 for large values of N are extremely flat in +the region, ¯φ ∈ [¯φ1, ¯φ3] because the three extrema are +very close and the height of the barrier between the two +minima very small. We have numerically found that the +height of the barrier scales as N −1 and the distance be- +tween the two minima as N −1/2 so that the curvatures +¯U ′′(¯φi) at the three extrema approach constant values as +N → ∞, see Fig. 4 of the Suppl. Mat. This suggests +that ¯U ′′(¯φ) while being well-behaved everywhere but be- +tween the three extrema, changes very rapidly within a +boundary layer around ¯φ0 of typical width N −1/2, mak- +ing divergent ¯U ′′′(¯φ0) when N → ∞. +It is not common in physics to encounter this kind +of situation where a series of functions fn(x) tends to a +smooth function f∞(x) whereas from a certain order p, +their derivatives f (p) +n (x) do not tend to f (p) +∞ (x). However, +a simple toy model explains trivially how this can occur. +Consider the series of functions fn(x) = n−1 sin(n2x). +Obviously, f∞(x) ≡ 0 which implies that f ′ +∞(x) ≡ 0 +whereas limn→∞ f ′ +n(0) = ∞. +In our case, at fixed d < 8/3, the limit of the T3 po- +tentials when N → ∞ is a nontrivial and well-defined +function that therefore must be the WF FP potential. +We have checked that it is indeed the limit of T3 when +N → ∞, see Fig. 1. The difference between the critical +and tetracritical behaviors is therefore not visible on the +potentials themselves but only on their derivatives as we +now show. +Let us study the boundary layer around ¯φ0. It is con- +venient for what follows to change variables. Following +Ref. [40], we define: V (µ) = U(φ) + (φ − Φ)2/2 with +µ = Φ2 and φ − Φ = −2ΦV ′(µ). As above, it is conve- +nient to rescale µ and V (µ): ¯µ = µ/N, ¯V = V/N. In +terms of these quantities, the FP equation for ¯V (¯µ) reads +0 = 1 − d ¯V + (d − 2)¯µ ¯V ′ + 4¯µ ¯V ′2 − 2 ¯V ′ − 4 +N ¯µ ¯V ′′. (7) +Eq. (7) has two remarkable features: (i) it is much sim- +pler than Eq. (6) because the nonlinearity comes only +0 +2 +4 +6 +8 +10 +12 +14 μ +0.02 +0.04 +0.06 +0.08 +V''[μ] +N=6×103 +N=1.7×104 +N=3.2×106 +N=∞WF +FIG. 2. Second derivative of the WF and T3 FP potentials +for different values of N in d = 2.6. +from the ( ¯V ′)2 term, (ii) it is the LPA equation obtained +from the Wilson-Polchinski (WP) version of the NPRG +[15, 41, 42]. Thus, ¯V (¯µ) is related to the potential ¯U(¯φ) of +the Wetterich version of the RG by the Legendre trans- +form of Eq. +(3). +The standard large-N analysis per- +formed in this version of the NPRG consists here again +in neglecting the last term in Eq. (7) because it is sup- +pressed by a 1/N factor. Under the assumption that this +term is indeed negligible, the resulting equation can be +solved exactly in the large-N limit [14, 35]. However, at +large N, it is clear on Eq. (7) that we have to deal with +singular perturbation theory since the small parameter +used in the 1/N expansion is in front of the term of high- +est derivative, that is, ¯V ′′. In this case, it is well-known +that at large N a boundary layer can exist for a partic- +ular value of ¯µ that becomes a singularity at N = ∞, +making this term non negligible [43]. +The value of ¯µ corresponding to ¯φ0 is called ¯µ0 and +is the minimum of ¯V (¯µ) at N += ∞. +We find for +¯V (¯µ) the same features about its three extrema ¯µi as +for ¯U(¯φ) at ¯φi: The three extrema ¯µi approach each +other and to ¯µ0 as N → ∞, the distances between +them scale as N −1/2 and the curvatures ¯V ′′(¯µi) as N 0. +Taking into account the scaling around ¯µ0 inside the +boundary layer, we introduce another scaled variable +˜µ = N 1/2(¯µ − ¯µ0). +Since at N = ∞, ¯V ′(¯µ) vanishes +at ¯µ = ¯µ0, ¯V (¯µ0) should approach 1/d at leading order +in N −1/2. We therefore define a scaled boundary layer +by ˜VN(˜µ) = N +� ¯V (¯µ0 + N −1/2˜µ) − 1/d +� +which implies +˜V ′′ +N(˜µ) = ¯V ′′(¯µ0 + N −1/2˜µ). We plot ˜V ′′ +N(˜µ) for several +values of N in Fig. 5 of the Suppl. Mat. +By substituting ˜VN(˜µ) by its value in Eq. +(7) and +solving it at order O(N −1/2), we find that ¯µ0 = 2/(d−2). +At order O(N −1), Eq. (7) becomes +− 8 ˜V ′′ +∞(˜µ) +d − 2 + 8 ˜V ′ +∞(˜µ)2 +d − 2 ++(d−2)˜µ ˜V ′ +∞(˜µ)−d ˜V∞(˜µ) = 0 (8) +[44] which is clearly invariant under ˜µ → −˜µ from which +it follows that ˜V ′ +∞(0) = 0. At ˜µ = ∞, ˜V ′′ +∞(˜µ) should tend +to a finite value that matches with ¯V ′′(µ) at ¯µ+ +0 . This +implies that the solution of Eq. (8) should be quadratic +when ˜µ → ∞. Substituting ˜V∞(˜µ) by ˜V ′′ +∞(˜µ = ∞)˜µ2/2 +in Eq. (8) and balancing the leading terms as ˜µ → ∞, +we find that ˜V ′′ +∞(˜µ = ∞) = (−d2 + 6d − 8)/16. Imposing + +4 +the two boundary conditions found above at ˜µ = 0 and +˜µ = ∞ selects a unique and globally defined solution +˜V ′′ +∞(˜µ) of Eq. (8) shown in Fig. 5 of the Suppl. Mat. +We find ¯V ′′(¯µ+ +0 ) = ¯V ′′ +WF(¯µ0) = ˜V ′′ +∞(˜µ = ∞) which proves +the matching at N = ∞ between the boundary layer and +the potential outside of the layer, see Fig. 2. We have +shown in Fig. 6 of the Suppl. Mat. the boundary layer +for ¯U ′′(¯φ) analogous to that of ¯V ′′(¯µ). To conclude, we +have proven that for d < 8/3, a boundary layer develops +at large N for the second derivative of the T3 potential +that becomes a singularity when N → ∞. What remains +to be understood is its physical relevance. +At first sight, what we have obtained for T3 looks para- +doxical because we could think that its potential being +identical to the WF potential at N = ∞, the linearized +flow around these two FPs should also be identical and +thus the same for all critical exponents. We now show +that this naive argument is wrong. +We have computed in d < 8/3 the relevant eigenvalues +of the RG flow around T3 and WF at finite and large N +and as expected we have found three for T3 and one for +WF. When N → ∞, one of the three eigenvalues at T3 +tends as expected to d − 2 which is the relevant eigen- +value ν−1 of the critical WF FP at N = ∞ [11, 14]. The +nontrivial point is that the two other relevant eigenvalues +at T3 have a well-defined limit when N → ∞ although +they do not play any role for the critical behavior of the +O(N = ∞) model. The solution to this paradox is that +they are associated with eigenperturbations that become +singular when N → ∞. That these two eigenperturba- +tions become singular is clear for one of them, called δ ¯V2, +on Fig. 9 of the Suppl. Mat. As for the other one, δ ¯V1, +its slope at ¯µ0 diverges as N 1/3 which implies that at +N = ∞, it becomes discontinuous at ¯µ0, see Figs. 9 and +10 of the Suppl. Mat. For ordinary second order phase +transitions, these eigenperturbations are excluded which +explains that the associated relevant eigenvalues do not +play any role. This solves all the paradoxes associated +with the tetracritical FPs at N = ∞ and d < 8/3. +What remains to be studied is the particular case N = +∞ and d = 8/3 where a line, called the BMB line, of +smooth tetracritical FPs shows up. It is obtained in the +WP version of the RG by integrating Eq. (7) in which +the last term, proportional to 1/N, has been discarded. +It is given by the following implicit expression [14]: +¯µ± = +C +¯V ′ � +1 − 2 ¯V ′� +� ±2 ¯V ′ +1 − 2 ¯V ′ +�4/3 ++ 2f(4 ¯V ′), +(9) +where f(x), which is analytic for x < 2, is given by +f(x) = +3 +2 − x + +4x +(2 − x)7/3 +� 1 +0 +dz +�2 − xz +z +�1/3 +(10) +and ¯µ± correspond to the two branches ¯µ > 3 and ¯µ < 3, +respectively. The derivative of the potential ¯V ′ is positive +(negative) on the former (latter) branch and C is a non- +negative integration constant. +¯V (¯µ) is analytic at ¯µ = +¯µ0 = 3 and ¯V ′(¯µ = 3) = 0. +In Fig. +7 of the Suppl. +Mat. different ¯V ′(¯µ) corresponding to different FPs of +the BMB line are shown. All FPs along the BMB line +share the same critical exponents, that is, the exponents +of the Gaussian FP which is itself tetracritical. Notice +that the WF FP which corresponds to C = 0, is the end +point of this line and deserves special attention. We come +back on this point in the following. +From Eq. (1), we have seen that λ∗ remains constant at +leading order in 1/N along the hyperbola of constant ϵN +of the (d, N) plane. This suggests that when the double +limit d → 8/3 and N → ∞ is taken at fixed α = ϵN, T3 +converges in d = 8/3 to one of the FPs of the BMB line. +We have analytically and numerically checked this and +have derived analytically the relation between α and C: +α = 162/C3, see Suppl. Mat. and Fig. 11. +Two extreme cases are worth studying. +First, the +Gaussian FP corresponds to the limit N → ∞ at fixed +dimension d = 8/3, that is, at α = 0. It corresponds +to C = ∞ in Eq. (9). Second, α = ∞, which implies +C = 0, corresponds to taking the limit ϵ → 0 at fixed +N = ∞, that is, to following the WF FP at N = ∞ up +to d = 8/3. However, at finite ϵ and N = ∞, we know +from the analysis above that the last term in Eq. (7) can- +not be neglected. Consistently, the same occurs for the +BMB line: the WF FP potential is indeed the end point +of the BMB line obtained by taking the limit C → 0 in +Eq. (9) but the derivatives of this potential can only be +studied by retaining the last term in Eq. (7). Here again, +this explains why the T3 FP in the C → 0 limit is three +times unstable and not only once unstable. +To conclude, we have solved the paradox of the appar- +ent absence of a nontrivial tetracritical FP at N = ∞ +and d < 8/3 by showing that this FP does exist but is +nothing else than the WF FP up to the subtlety that +the derivatives of the tetracritical FP potential are not +the derivatives of the WF FP potential. This makes the +large-N limit of the O(N) model much less trivial than +is usually advocated at least for multicritical phenom- +ena. +The fact that the tetracritical FP has two more +unstable infrared directions than the WF FP is related +to this subtle point because they are associated with sin- +gular eigenperturbations, a possibility which is usually +not considered. 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Polchinski, Nucl. Phys. B 231 (1984) 269. +[42] A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B, 270, 687 +(1986). +[43] M. H. Holmes, Introduction to perturbation methods Sec- +ond edition, Springer (2012). +[44] Note that the first term in Eq. (8) comes from the last +term in Eq. (6) or Eq. (7), which is formally proportional +to N −1 and neglected in the usual large-N analysis. How- +ever this term is indispensable to describe the boundary +layer of ¯U ′′(¯φ) or ¯V ′′(¯µ). +[45] R. D. Pisarski, Phys. Rev. Lett. 48, 574 (1982). +[46] H. Osborn and A. Stergiou, J. High Energy Phys. 5 51 +(2018). +[47] S. Yabunaka and B. Delamotte, Phys. Rev. Lett. 119, +191602 (2017); Phys. Rev. Lett. 121, 231601 (2018). +[48] S. Yabunaka, C. Fleming, and B. Delamotte, Phys. Rev. +E 106, 054105 (2022). + +6 +SUPPLEMENTAL MATERIALS +I. +T3 FP POTENTIALS IN d < 8/3 +We show in Fig. 3 the tetracritical FP potential ¯U(¯φ) +obtained with the LPA and solution of Eq. (6) for small +values of N. They have the typical shape of a tetracritical +potential showing two nontrivial minima. +0 +2 +4 +6 +8 ϕ +0.375 +0.38 +0.385 +0.39 +U(ϕ) +N=4.5 +N=1 +FIG. 3. ¯U(¯φ) for the T3 FP for different values of N in d = 2.6. +II. +LARGE-N BEHAVIOR OF THE EXTREMA +OF THE TETRACRITICAL POTENTIAL +The three nontrivial extrema of the T3 FP potential +in either the WP or Wetterich version of the RG, shown +in Fig. 1 of the main text, behave the same way when +N → ∞. We show on Fig. 4 the scaling in N of the height +of the barrier between the extrema ¯φi of the rescaled +potential ¯U(¯φ) of the Wetterich version of the RG, as +well as the distance between them. These extrema are +shown in Fig. 1 of the main text. +2500 +3500 +4500N +6.0×10-6 +8.0×10-6 +1.0×10-5 +1.2×10-5 +U[ϕ2]-U[ϕ3] +2500 +3500 +4500N +0.0325 +0.0350 +0.0375 +0.0400 +0.0425 +0.0450 +0.0475 +ϕ3-ϕ2 +FIG. 4. Left: Height of the potential barrier for the T3 FP +of Eq. (6) for large values of N in d = 2.6 (blue dots). The +equation of the full line is y = 0.0257/N. Right: Distance +between the maximum ¯φ2 and the minimum ¯φ3 for the T3 FP +of Eq. (6) for large values of N in d = 2.6 (blue dots). The +equation of the full line is y = 2.12506/N 1/2. +Since the height of the barrier, ∆ ¯U, scales as N −1 +and the distance between the extrema, ∆¯φ, as N −1/2, a +simple dimensional argument shows that the curvatures +at these extrema that goes as ∆ ¯U/(∆¯φ)2, do not scale +with N, that is, tend to constants when N → ∞, a fact +that we have numerically checked. +Thus, for d < 8/3 +and at large and finite N, the curvature of ¯U(¯φ) varies +between a positive value at ¯φ1, a negative value at ¯φ2 +and again a positive value at ¯φ3 on a distance of order +N −1/2. +III. +THE SCALED BOUNDARY LAYER ˜V ′′(˜µ) +By translating and rescaling by a factor N 1/2 the po- +sition and the width of the boundary layer of the second +derivative of the potential ¯V , it is possible to obtain a +finite limit for this scaled boundary layer when N → ∞. +We thus define the scaled variable ˜µ = N 1/2(¯µ − ¯µ0) +where ¯µ0 is the location of the boundary layer and the +scaled potential by ˜VN(˜µ) = N +� ¯V (N −1/2˜µ + ¯µ0) − 1/d +� +. +It follows from the definitions above that ˜V ′′ +N(˜µ) = +¯V ′′(¯µ0+N 1/2˜µ). We show in Fig. 5 this scaled boundary +layer for different values of N at large N as well as its +limit ˜V ′′ +∞(˜µ) at N = ∞. +-1000 +-500 +500 +1000 +μ˜ +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +N=2.5×104 +N=1.7×105 +N=3.2×106 +˜V''[μ] +˜ +FIG. 5. The scaled boundary layer for the second derivative +of the T3 FP potential ˜V ′′ +N(˜µ), Eqs. (7) for large values of N in +d = 2.6. The dashed curve is the global solution ˜V ′′ +∞(˜µ) of Eq. +(8) at N = ∞. The red horizontal line is y = (−d2+6d−8)/16 +for d = 2.6. It coincides with ˜V ′′ +∞(˜µ = ∞). +Notice that a finite difference ¯µ − ¯µ0 translates into an +infinite ˜µ when N → ∞. The matching at N = ∞ be- +tween the scaled boundary layer and the value of ¯V ′′(¯µ) +outside of the layer therefore requires that ˜V ′′ +∞(∞) = +¯V ′′(¯µ+ +0 ) = (−d2 + 6d − 8)/16 which is the case with the +solution for the scaled boundary layer given in the main +text, see also Fig. 5. +IV. +THE BOUNDARY LAYER OF ¯U ′′(¯φ) +The boundary layer has been derived in the main text +in WP version of the RG because it is simpler in this +version than in Wetterich version. However, it can also +be derived directly in this latter version or, once it is + +7 +obtained in one version, it can be translated in the other +by performing the Legendre transform given in Eq. (3). +0.5 +1.0 +1.5 +2.0 +2.5 +ϕ +5 +10 +15 +U''[ϕ] +N=1.5⨯104 +N=4.2⨯105 +N=4.2⨯106 +WF N=∞ +FIG. 6. The second derivative of the T3 FP potential ¯U ′′(¯φ) in +the Wetterich version of the RG, Eq. (6), for different values +of N in d = 2.6. +We show in Fig. 6 the boundary layer of ¯U ′′(¯φ) for +different values of N at large N. +V. +DIFFERENT FP POTENTIALS OF THE +BMB LINE +We show in Fig. 7 the first derivative of different FP +potentials of the BMB line at N = ∞ and in d = 8/3. +These FP potentials, implicitly given by the exact ex- +pression given in Eqs. (9) and (10) of the main text, are +indexed by the nonnegative constant C. +The WF FP +potential corresponds to C = 0. +1 +2 +3 +4 +5 +6 +7 +μ +-0.20 +-0.15 +-0.10 +-0.05 +0.05 +0.10 +0.15 +V'[μ] +C=2 +C=0.625 +C=0 +FIG. 7. ¯V ′(¯µ) for different FPs indexed by the constant C on +the BMB line given by Eqs. (9) and (10) of the main text. +We emphasize that the limit of ¯V ′′(¯µ0 = 3), when +C → 0 is not given by the second derivative of the WF +FP potential which is however the limit when C → 0 of +¯V (¯µ) along the BMB line. This is consistent with what +happens at fixed d < 8/3 when N → ∞ since the limit +d → 8/3 at fixed α = ∞ consists in following the WF FP +at N = ∞ up to d = 8/3, the derivatives of which are +not the limit of the derivatives of the T3 potential. +VI. +EIGENPERTURBATIONS AT THE +TETRACRITICAL FP +FIG. 8. +d = 2.6: Eigenperturbation δ ¯V3(¯µ) at the T3 FP +corresponding, when N → ∞, to the relevant eigenvalue λ3 = +d − 2. +We show in Figs. 8 and 9 the relevant eigenperturba- +tions δ ¯Vi of the T3 FP in d = 2.6 for different values of +N. Whereas δ ¯V3 tends to the relevant eigenperturbation +of the critical WF FP –with eigenvalue d − 2 which is +the inverse of the critical exponent νWF–, the two others +become singular in the N → ∞ limit. This is the rea- +son why they play no role for the critical behavior of the +O(N) model at N = ∞. +1 +2 +3 +4 +5 +6 +7 μ +-0.03 +-0.02 +-0.01 +0.00 +0.01 +0.02 +0.03 +δV1[μ] +N=5×102 +N=3×103 +N=1×106 +0 +1 +2 +3 +4 +5 +6 +7 μ +0.00 +0.01 +0.02 +0.03 +0.04 +δV2[μ] +N=5×102 +N=3×103 +N=1.4×105 +FIG. 9. d = 2.6: Eigenperturbations δ ¯Vn(¯µ) for n = 1, 2 at +the T3 FP corresponding respectively to the relevant eigen- +values λ1 ≃ 2.00 and λ2 ≃ 1.326 for different values of N. +These eigenperturbations tend to singular functions of ¯µ when +N → ∞. +We show in Fig. 10 that the slope of δ ¯V1(¯µ) at ¯µ0 in- +creases as N 1/3 which proves that this eigenperturbation +becomes discontinuous at infinite N. + +[μ] +V[] +0.05 +0.005 +0.04 +0.03 +0.000 +3.0 +35 +4.0 +N=3x103 +0.02 +0.005 +N=3x104 +0.01 +WF N=8 +0.00 +μ +1 +2 +3 +5 +6 +0.01 +-0.028 +1000 +104 +105 +106 +N +0.02 +0.05 +0.10 +0.20 +-δV 1'[10/3] +FIG. 10. d = 2.6: Slope of the eigenperturbation δ ¯V1(¯µ) of +the T3 FP at its minimum ¯µ0 = 10/3 for different values of +N. The equation of the full line is y = 0.00175N 1/3. +VII. +BMB LINE AND THE JOINED LIMIT ϵ → 0 +AND N → ∞ AT FIXED α = ϵN +When a T3 FP is followed along the hyperbola d = +8/3 − α/N, α ≥ 0, of the (d, N) plane, its potential con- +verges when N → ∞ to the potential of one of the FPs +of the BMB line, see Fig. 11. We derive below the re- +lationship α = 162/C3 between the parameter α of the +hyperbola and the parameter C that indexes the FPs +along the BMB line, see Eqs. (9) and (10) of the main +text. +This relationship can be derived as follows. The FP +potential of T3 is expanded as +¯V (¯µ) = +∞ +� +n=0 +an(¯µ − 3)n +(1) +around the minimum ¯µ0 = 3 of the N = ∞ potential. +Then, the coefficients an are expanded as +an = a(0) +n ++ N −1a(1) +n ++ O(N −2) +(2) +in power of 1/N. At order O(N 0), Eq. (7) yields a(0) +n += 0 +for n = 1, 2 and 3 and recursively determines a(0) +n +for n +larger than 5 in terms of a(0) +4 . +Now, �∞ +n=0 a(0) +n (¯µ−3)n is the expansion of a FP poten- +tial of the BMB line. For this potential, ¯µ± behaves from +Eq. (9) as ¯µ± ≃ 3 ± 24/3C| ¯V ′|1/3 for C ̸= 0 and | ¯V ′| ≪ 1. +This implies that a(0) +4 +is related to C by a(0) +4 += 3/(192C3). +At order O(N −1), it can be shown that a(1) +n +for all n but +n = 4 can be recursively determined in terms of a(0) +4 +or C, +if and only if the condition α = 162/C3 is satisfied which +proves the relationship between these two parameters. +We show in Fig. 11 different T3 FP potentials along +an hyperbola d = 8/3 − α/N with increasing values +of N. +These FP potentials converge to a potential +corresponding to the FP on the BMB line indexed by +C = (162/α)1/3. +0 +1 +2 +3 +4 +5 +6 +7 +μ +-0.10 +-0.05 +0.05 +0.10 +V'[μ] +C=0.672, N=∞ +N=4000 α=1600/3 +N=8000 α=1600/3 +N=24000 α=1600/3 +FIG. 11. ¯V ′(¯µ) of the T3 FP followed on the hyperbola d = +8/3 − α/N with fixed α = 1600/3 for increasing values of +N. In the double limit d → 8/3 and N → ∞, it converges +to the FP potential of the BMB line corresponding to C = +(162 × 3/1600)1/3 ≃ 0.672. + diff --git a/AtAzT4oBgHgl3EQfF_uW/content/tmp_files/load_file.txt b/AtAzT4oBgHgl3EQfF_uW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b65d341d6f3cf52a5d403dbb38f0dc0c6a11673d --- /dev/null +++ b/AtAzT4oBgHgl3EQfF_uW/content/tmp_files/load_file.txt @@ -0,0 +1,792 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf,len=791 +page_content='A fixed point can hide another one: the nonperturbative behavior of the tetracritical fixed point of the O(N) models at large N Shunsuke Yabunaka1, ∗ and Bertrand Delamotte2 1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan 2Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, LPTMC, F-75005 Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (Dated: January 4, 2023) We show that at N = ∞ and below its upper critical dimension, d < dup, the critical and tetracritical behaviors of the O(N) models are associated with the same renormalization group fixed point (FP) potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Only their derivatives make them different with the subtleties that taking their N → ∞ limit and deriving them do not commute and that two relevant eigenperturbations show singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This invalidates both the ϵ− and the 1/N− expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at N = ∞ and d = dup can be understood from a finite-N analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Field theories sometimes exhibit nonperturbative fea- tures such as confinement [1], presence of bound states [2] or exotic excitations [3], fixed points (FPs) of the renor- malization group (RG) flows that are nonperturbative as in the Kardar-Parisi-Zhang equation [4], divergence of the perturbative RG flow at a finite RG scale [5], pres- ence of a cusp in the FP potential as in the random field Ising model [6], to cite but a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Very often, these non- perturbative effects are assumed either to occur in rather complicated theories such as gauge and string theories or in highly nontrivial statistical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' O(N) models, which are the simplest scalar field theo- ries, are often implicitly considered to be immune to these complex phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Perturbative methods are there- fore assumed to work almost all the time for these mod- els, the exception to the rule being the Bardeen-Moshe- Bander (BMB) phenomenon [7], related to the existence of a line of tricritical FPs at N = ∞ and d = 3, which re- quires nonperturbative FPs to be fully understood from a large-N analysis [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' From this viewpoint, the enormous success of the ϵ = 4 − d expansion for the perturbative calculation of the critical exponents associated with the Wilson-Fisher (WF) FP [11] could let us believe that the critical physics of the O(N) models is fully understood for any N and d, especially since it is corroborated by the 1/N and ϵ = d − 2 expansions [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Our goal in this Letter is to show instead that although the critical physics of the O(N) models, described by the WF FP, is fully under perturbative control at both finite and infinite N, the tetracritical physics of these models at N = ∞ –and probably of infinitely many multicritical behaviors– is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show below (i) that at N = ∞, it is also associated with the WF FP, which is unexpected, and (ii) that it nonetheless shows non-perturbative fea- tures that are beyond the reach of the standard imple- mentation of both the large-N and ϵ- expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in particular a very intriguing phenomenon related to the large-N limit of the tetracritical FP of the O(N) models: from the second order, the derivatives of the ∗ yabunaka123@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='com N = ∞ tetracritical FP potential, that is, of the WF FP potential, are not identical to the limit of the derivatives of the finite-N tetracritical FP potentials when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This turns out to be crucial for understanding the large- N limit of tetracritical phenomena and shows that this limit is much less trivial than what is usually said [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The perturbative tetracritical FP corresponds to the massless (ϕ2)4 theory, the upper critical dimension of which is dup = 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is found in perturbation theory in ϵ = 8/3 − d for all N ≥ 1 and it is three times infrared unstable [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Calling λ/(384N 3) the coupling in front of the dimensionless (ϕ2)4 term, the large-N perturbative flow equation for λ reads [13]: ∂tλ = −3ϵλ + 9λ2 4N + O(N −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (1) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (1), we find that at leading order in N, the nontrivial FP solution is λ∗ = 4ϵN/3 from which follows that perturbation theory does not allow for a control of the large-N limit of the tetracritical FP at fixed ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Only the double limit N → ∞ and ϵ → 0 such that the product ϵN remains finite can possibly be under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We come back on this point in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Let us recall that in generic dimensions d < 4, the only nontrivial FP found in the standard large-N anal- ysis of the O(N) models is the WF FP [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, no tetracritical FP is found at N = ∞ and d < 8/3 which is paradoxical considering that it is perturbatively found for all N < ∞ and ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show below that the solution to the paradox above lies in the field dependence of the tetracritical FP poten- tial whereas it cannot be obtained from its field expansion and in particular from λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The recourse to functional RG methods is therefore mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The best way to implement functional RG is to con- sider Wilson’s RG, as it is inherently functional [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We recall below the take-away philosophy of the modern ver- sion of Wilson’s RG known as the nonperturbative – or functional – renormalization group (NPRG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' NPRG is based on the idea of integrating fluctuations step by step [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is implemented on the Gibbs free energy Γ [17–23] of a model defined by an Hamiltonian arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01021v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='stat-mech] 3 Jan 2023 2 (or euclidean action) H and a partition function Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' To this model is associated a one-parameter family of models with Hamiltonians Hk = H + ∆Hk and partition func- tions Zk, where k is a momentum scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In Hk, ∆Hk is chosen such that only the rapid fluctuations in the origi- nal model, those with wavenumbers |q| > k, are summed over in the partition function Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, the slow modes (|q| < k) need to be decoupled in Zk and this is achieved by giving them a mass of order k, that is by taking for ∆Hk a quadratic (mass-like) term, which is nonvanishing only for the slow modes: Zk[J] = � Dϕi exp(−H[ϕ] − ∆Hk[ϕ] + J · ϕ) (2) with ∆Hk[ϕ] = 1 2 � q Rk(q2)ϕi(q)ϕi(−q), where, for in- stance, Rk(q2) = (k2 − q2)θ(k2 − q2) and J · ϕ = � x Ji(x)ϕi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The k-dependent Gibbs free energy Γk[φ] is defined as the (slightly modified) Legendre transform of log Zk[J]: Γk[φ] + log Zk[J] = J · φ − 1 2 � q Rk(q2)φi(q)φi(−q) (3) with � q = � ddq/(2π)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' With the choice of regulator function Rk above, Γk[φ] interpolates between the Hamil- tonian H when k is of order of the ultraviolet cut-off Λ of the theory: ΓΛ ∼ H, and the Gibbs free energy Γ of the original model when k = 0: Γk=0 = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The exact RG flow equation of Γk gives the evolution of Γk with k between these two limiting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is known as the Wetterich equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It reads [18]: ∂tΓk[φ] = 1 2Tr[∂tRk(q2)(Γ(2) k [q, −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' φ] + Rk(q))−1], (4) where t = log(k/Λ), Tr stands for an integral over q and a trace over group indices and Γ(2) k [q, −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' φ] is the matrix of the Fourier transforms of δ2Γk/δφi(x)δφj(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In most cases, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (4) cannot be solved exactly and approximations are mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The best known approx- imation consists in expanding Γk in powers of the deriva- tives of φi and to truncate the expansion at a given fi- nite order[24–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The approximation at lowest order is dubbed the local potential approximation (LPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' For the O(N) model it consists in approximating Γk by: Γk[φ] = � x �1 2(∇φi)2 + Uk(φ) � (5) where φ = √φiφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Fixed points are found only for di- mensionless quantities and the standard large-N limit by rescaling the field and the potential by factors N −1/2 and N −1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, we define the dimen- sionless and rescaled field ¯φ and potential ¯Uk as ¯φ = v − 1 2 d k 2−d 2 N −1/2φ and ¯Uk(¯φ) = v−1 d k−dN −1Uk (φ) with v−1 d = 2d−1dπd/2Γ( d 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The LPA flow of ¯Uk then reads: ∂t ¯Uk(¯φ) = − d ¯Uk(¯φ) + 1 2(d − 2)¯φ ¯U ′ k(¯φ)+ � 1 − 1 N � ¯φ ¯φ + ¯U ′ k(¯φ) + 1 N 1 1 + ¯U ′′ k (¯φ) (6) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6: ¯U(¯φ) for the T3 FP of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Green, red, blue and black curves correspond to N = 1500, 2250, 4500 and 42000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The orange dashed curve corresponds to the WF FP at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Inset: Close view of ¯U(¯φ) around ¯φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' with ∂t = k∂k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The standard large-N limit of the LPA flow equation above is obtained by (i) replacing the fac- tor 1 − 1/N by 1, (ii) dropping the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) because it is assumed to be sub-leading [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As a con- sequence of the two steps above, the explicit dependence in N in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) disappears in the large-N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The crucial point of the large-N limit is that assuming point (ii) above, the resulting LPA flow equation on ¯Uk can be shown to be exact in the limit N → ∞ [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Under this assumption, all FPs of the O(N) models have been found exactly at N = ∞ [14, 33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The result is the following: In a generic dimension d < 4 there is only one nongaussian FP at N = ∞ which is the usual Wilson- Fisher FP (WF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The exceptions to the rule above are the BMB lines of FPs [7, 14, 37–39] existing in dimensions d = 2 + 2/p with p an integer larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now show that the procedure described above is too restrictive to study the large-N limit of the tetracritical FPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As said above, the standard large-N analysis con- sists in neglecting the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, this term is negligible only if (1 + ¯U ′′ k (¯φ))−1 does not coun- terbalance at large N its 1/N prefactor for some finite values of ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now show that because of singularities in the third derivative of ¯Uk(¯φ), the contribution of the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) cannot be neglected in the FP equation of ¯U ′′ k (¯φ) obtained by differentiating twice Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) (see footnote below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) for more detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This turns out to be sufficient to invalidate the standard large-N limit in the tetracritical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have numerically solved Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) and have found for several values of N and d < 8/3 the perturbative tetracritical FP that we call T3(N, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As expected, T3 bifurcates from the Gaussian FP in d = 8/3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have followed it down to d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 3 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The FP potential of T3, (i) shows as expected two maxima, one of which being located at ¯φ = 0 and another one at ¯φ2 > 0, and two minima at ¯φ1 and ¯φ3 such that ¯φ3 > ¯φ2 > ¯φ1 > 0, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1, (ii) can be continuously followed up to arbitrarily large values of N at fixed d < 8/3, (iii) has its three extrema T() 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38464 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='85 $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 ¯φ1, ¯φ2, ¯φ3 approaching each other when N is increased at fixed d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These extrema tend to a common value ¯φ0 when N → ∞ which is the minimum of the FP potential, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 4 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Point (ii) above is paradoxical because it seems to contradict the standard large-N approach where only the WF FP is found in a generic dimension d < 8/3 at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now show that the WF FP potential at N = ∞ is in fact the limit when N → ∞ of the potential of T3 for d < 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This solves the above paradox because it explains why on one hand there exists a nontrivial tetracritical FP at N = ∞ and d < 8/3 and on the other hand that there is no other nontrivial and smooth solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) at N = ∞ than the WF FP potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, this creates a new paradox since obviously the critical and tetracritical universal behaviors cannot be the same since the two FPs do not have the same number of unstable eigendirections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now explain in detail this new paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We can see on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 that the FP potentials found in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 for large values of N are extremely flat in the region, ¯φ ∈ [¯φ1, ¯φ3] because the three extrema are very close and the height of the barrier between the two minima very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have numerically found that the height of the barrier scales as N −1 and the distance be- tween the two minima as N −1/2 so that the curvatures ¯U ′′(¯φi) at the three extrema approach constant values as N → ∞, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 4 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This suggests that ¯U ′′(¯φ) while being well-behaved everywhere but be- tween the three extrema, changes very rapidly within a boundary layer around ¯φ0 of typical width N −1/2, mak- ing divergent ¯U ′′′(¯φ0) when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is not common in physics to encounter this kind of situation where a series of functions fn(x) tends to a smooth function f∞(x) whereas from a certain order p, their derivatives f (p) n (x) do not tend to f (p) ∞ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, a simple toy model explains trivially how this can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Consider the series of functions fn(x) = n−1 sin(n2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Obviously, f∞(x) ≡ 0 which implies that f ′ ∞(x) ≡ 0 whereas limn→∞ f ′ n(0) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In our case, at fixed d < 8/3, the limit of the T3 po- tentials when N → ∞ is a nontrivial and well-defined function that therefore must be the WF FP potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have checked that it is indeed the limit of T3 when N → ∞, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The difference between the critical and tetracritical behaviors is therefore not visible on the potentials themselves but only on their derivatives as we now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Let us study the boundary layer around ¯φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is con- venient for what follows to change variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [40], we define: V (µ) = U(φ) + (φ − Φ)2/2 with µ = Φ2 and φ − Φ = −2ΦV ′(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As above, it is conve- nient to rescale µ and V (µ): ¯µ = µ/N, ¯V = V/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In terms of these quantities, the FP equation for ¯V (¯µ) reads 0 = 1 − d ¯V + (d − 2)¯µ ¯V ′ + 4¯µ ¯V ′2 − 2 ¯V ′ − 4 N ¯µ ¯V ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) has two remarkable features: (i) it is much sim- pler than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) because the nonlinearity comes only 0 2 4 6 8 10 12 14 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="08 V''[μ] N=6×103 N=1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='7×104 N=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='2×106 N=∞WF FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Second derivative of the WF and T3 FP potentials for different values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' from the ( ¯V ′)2 term, (ii) it is the LPA equation obtained from the Wilson-Polchinski (WP) version of the NPRG [15, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, ¯V (¯µ) is related to the potential ¯U(¯φ) of the Wetterich version of the RG by the Legendre trans- form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The standard large-N analysis per- formed in this version of the NPRG consists here again in neglecting the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) because it is sup- pressed by a 1/N factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Under the assumption that this term is indeed negligible, the resulting equation can be solved exactly in the large-N limit [14, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, at large N, it is clear on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) that we have to deal with singular perturbation theory since the small parameter used in the 1/N expansion is in front of the term of high- est derivative, that is, ¯V ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In this case, it is well-known that at large N a boundary layer can exist for a partic- ular value of ¯µ that becomes a singularity at N = ∞, making this term non negligible [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The value of ¯µ corresponding to ¯φ0 is called ¯µ0 and is the minimum of ¯V (¯µ) at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We find for ¯V (¯µ) the same features about its three extrema ¯µi as for ¯U(¯φ) at ¯φi: The three extrema ¯µi approach each other and to ¯µ0 as N → ∞, the distances between them scale as N −1/2 and the curvatures ¯V ′′(¯µi) as N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Taking into account the scaling around ¯µ0 inside the boundary layer, we introduce another scaled variable ˜µ = N 1/2(¯µ − ¯µ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Since at N = ∞, ¯V ′(¯µ) vanishes at ¯µ = ¯µ0, ¯V (¯µ0) should approach 1/d at leading order in N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We therefore define a scaled boundary layer by ˜VN(˜µ) = N � ¯V (¯µ0 + N −1/2˜µ) − 1/d � which implies ˜V ′′ N(˜µ) = ¯V ′′(¯µ0 + N −1/2˜µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We plot ˜V ′′ N(˜µ) for several values of N in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' By substituting ˜VN(˜µ) by its value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) and solving it at order O(N −1/2), we find that ¯µ0 = 2/(d−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At order O(N −1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) becomes − 8 ˜V ′′ ∞(˜µ) d − 2 + 8 ˜V ′ ∞(˜µ)2 d − 2 +(d−2)˜µ ˜V ′ ∞(˜µ)−d ˜V∞(˜µ) = 0 (8) [44] which is clearly invariant under ˜µ → −˜µ from which it follows that ˜V ′ ∞(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At ˜µ = ∞, ˜V ′′ ∞(˜µ) should tend to a finite value that matches with ¯V ′′(µ) at ¯µ+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This implies that the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) should be quadratic when ˜µ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Substituting ˜V∞(˜µ) by ˜V ′′ ∞(˜µ = ∞)˜µ2/2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) and balancing the leading terms as ˜µ → ∞, we find that ˜V ′′ ∞(˜µ = ∞) = (−d2 + 6d − 8)/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Imposing 4 the two boundary conditions found above at ˜µ = 0 and ˜µ = ∞ selects a unique and globally defined solution ˜V ′′ ∞(˜µ) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We find ¯V ′′(¯µ+ 0 ) = ¯V ′′ WF(¯µ0) = ˜V ′′ ∞(˜µ = ∞) which proves the matching at N = ∞ between the boundary layer and the potential outside of the layer, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 6 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' the boundary layer for ¯U ′′(¯φ) analogous to that of ¯V ′′(¯µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' To conclude, we have proven that for d < 8/3, a boundary layer develops at large N for the second derivative of the T3 potential that becomes a singularity when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' What remains to be understood is its physical relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At first sight, what we have obtained for T3 looks para- doxical because we could think that its potential being identical to the WF potential at N = ∞, the linearized flow around these two FPs should also be identical and thus the same for all critical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now show that this naive argument is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have computed in d < 8/3 the relevant eigenvalues of the RG flow around T3 and WF at finite and large N and as expected we have found three for T3 and one for WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' When N → ∞, one of the three eigenvalues at T3 tends as expected to d − 2 which is the relevant eigen- value ν−1 of the critical WF FP at N = ∞ [11, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The nontrivial point is that the two other relevant eigenvalues at T3 have a well-defined limit when N → ∞ although they do not play any role for the critical behavior of the O(N = ∞) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The solution to this paradox is that they are associated with eigenperturbations that become singular when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' That these two eigenperturba- tions become singular is clear for one of them, called δ ¯V2, on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 9 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As for the other one, δ ¯V1, its slope at ¯µ0 diverges as N 1/3 which implies that at N = ∞, it becomes discontinuous at ¯µ0, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 9 and 10 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' For ordinary second order phase transitions, these eigenperturbations are excluded which explains that the associated relevant eigenvalues do not play any role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This solves all the paradoxes associated with the tetracritical FPs at N = ∞ and d < 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' What remains to be studied is the particular case N = ∞ and d = 8/3 where a line, called the BMB line, of smooth tetracritical FPs shows up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is obtained in the WP version of the RG by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) in which the last term, proportional to 1/N, has been discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is given by the following implicit expression [14]: ¯µ± = C ¯V ′ � 1 − 2 ¯V ′� � ±2 ¯V ′ 1 − 2 ¯V ′ �4/3 + 2f(4 ¯V ′), (9) where f(x), which is analytic for x < 2, is given by f(x) = 3 2 − x + 4x (2 − x)7/3 � 1 0 dz �2 − xz z �1/3 (10) and ¯µ± correspond to the two branches ¯µ > 3 and ¯µ < 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The derivative of the potential ¯V ′ is positive (negative) on the former (latter) branch and C is a non- negative integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' ¯V (¯µ) is analytic at ¯µ = ¯µ0 = 3 and ¯V ′(¯µ = 3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 7 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' different ¯V ′(¯µ) corresponding to different FPs of the BMB line are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' All FPs along the BMB line share the same critical exponents, that is, the exponents of the Gaussian FP which is itself tetracritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Notice that the WF FP which corresponds to C = 0, is the end point of this line and deserves special attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We come back on this point in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (1), we have seen that λ∗ remains constant at leading order in 1/N along the hyperbola of constant ϵN of the (d, N) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This suggests that when the double limit d → 8/3 and N → ∞ is taken at fixed α = ϵN, T3 converges in d = 8/3 to one of the FPs of the BMB line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have analytically and numerically checked this and have derived analytically the relation between α and C: α = 162/C3, see Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Two extreme cases are worth studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' First, the Gaussian FP corresponds to the limit N → ∞ at fixed dimension d = 8/3, that is, at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It corresponds to C = ∞ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Second, α = ∞, which implies C = 0, corresponds to taking the limit ϵ → 0 at fixed N = ∞, that is, to following the WF FP at N = ∞ up to d = 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, at finite ϵ and N = ∞, we know from the analysis above that the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) can- not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Consistently, the same occurs for the BMB line: the WF FP potential is indeed the end point of the BMB line obtained by taking the limit C → 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) but the derivatives of this potential can only be studied by retaining the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Here again, this explains why the T3 FP in the C → 0 limit is three times unstable and not only once unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' To conclude, we have solved the paradox of the appar- ent absence of a nontrivial tetracritical FP at N = ∞ and d < 8/3 by showing that this FP does exist but is nothing else than the WF FP up to the subtlety that the derivatives of the tetracritical FP potential are not the derivatives of the WF FP potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This makes the large-N limit of the O(N) model much less trivial than is usually advocated at least for multicritical phenom- ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The fact that the tetracritical FP has two more unstable infrared directions than the WF FP is related to this subtle point because they are associated with sin- gular eigenperturbations, a possibility which is usually not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We conjecture that what has been found above at large N and for d ≤ 8/3 is valid for all mul- ticritical points with an odd number of eigendirections below or at their upper critical dimension because the BMB lines for all of them terminate at the WF FP [14], a fact that in itself is almost enough to imply everything else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Let us finally point out that what we have found for the tetracritical FP is very different from what was found around d = 3 at large-N in the tricritical case which re- quired the existence of new FPs to be fully understood at finite N [45–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We also conjecture that this phe- nomenon is not specific to the O(N) models but should rather be generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We acknowledge A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Codello and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Defenu and for correspondence and discussions at an early stage of this 5 work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' was supported by Grant-in-Aid for Young Scientists (18K13516).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Pel´aez, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Reinosa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Serreau, M.' metadata={'source': 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+page_content=' (6) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7), which is formally proportional to N −1 and neglected in the usual large-N analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' How- ever this term is indispensable to describe the boundary layer of ¯U ′′(¯φ) or ¯V ′′(¯µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [45] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Pisarski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 48, 574 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [46] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Osborn and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Stergiou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5 51 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Yabunaka and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Delamotte, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 119, 191602 (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 121, 231601 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [48] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Yabunaka, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Fleming, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Delamotte, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' E 106, 054105 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 6 SUPPLEMENTAL MATERIALS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' T3 FP POTENTIALS IN d < 8/3 We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 3 the tetracritical FP potential ¯U(¯φ) obtained with the LPA and solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) for small values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' They have the typical shape of a tetracritical potential showing two nontrivial minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 0 2 4 6 8 ϕ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='39 U(ϕ) N=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5 N=1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' ¯U(¯φ) for the T3 FP for different values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' LARGE-N BEHAVIOR OF THE EXTREMA OF THE TETRACRITICAL POTENTIAL The three nontrivial extrema of the T3 FP potential in either the WP or Wetterich version of the RG, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 of the main text, behave the same way when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 4 the scaling in N of the height of the barrier between the extrema ¯φi of the rescaled potential ¯U(¯φ) of the Wetterich version of the RG, as well as the distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These extrema are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 2500 3500 4500N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0×10-6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0×10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0×10-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='2×10-5 U[ϕ2]-U[ϕ3] 2500 3500 4500N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0475 ϕ3-ϕ2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Left: Height of the potential barrier for the T3 FP of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) for large values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 (blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The equation of the full line is y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0257/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Right: Distance between the maximum ¯φ2 and the minimum ¯φ3 for the T3 FP of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) for large values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 (blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The equation of the full line is y = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='12506/N 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Since the height of the barrier, ∆ ¯U, scales as N −1 and the distance between the extrema, ∆¯φ, as N −1/2, a simple dimensional argument shows that the curvatures at these extrema that goes as ∆ ¯U/(∆¯φ)2, do not scale with N, that is, tend to constants when N → ∞, a fact that we have numerically checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, for d < 8/3 and at large and finite N, the curvature of ¯U(¯φ) varies between a positive value at ¯φ1, a negative value at ¯φ2 and again a positive value at ¯φ3 on a distance of order N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' THE SCALED BOUNDARY LAYER ˜V ′′(˜µ) By translating and rescaling by a factor N 1/2 the po- sition and the width of the boundary layer of the second derivative of the potential ¯V , it is possible to obtain a finite limit for this scaled boundary layer when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We thus define the scaled variable ˜µ = N 1/2(¯µ − ¯µ0) where ¯µ0 is the location of the boundary layer and the scaled potential by ˜VN(˜µ) = N � ¯V (N −1/2˜µ + ¯µ0) − 1/d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It follows from the definitions above that ˜V ′′ N(˜µ) = ¯V ′′(¯µ0+N 1/2˜µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5 this scaled boundary layer for different values of N at large N as well as its limit ˜V ′′ ∞(˜µ) at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1000 500 500 1000 μ˜ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='14 N=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5×104 N=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='7×105 N=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="2×106 ˜V''[μ] ˜ FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The scaled boundary layer for the second derivative of the T3 FP potential ˜V ′′ N(˜µ), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) for large values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The dashed curve is the global solution ˜V ′′ ∞(˜µ) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The red horizontal line is y = (−d2+6d−8)/16 for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It coincides with ˜V ′′ ∞(˜µ = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Notice that a finite difference ¯µ − ¯µ0 translates into an infinite ˜µ when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The matching at N = ∞ be- tween the scaled boundary layer and the value of ¯V ′′(¯µ) outside of the layer therefore requires that ˜V ′′ ∞(∞) = ¯V ′′(¯µ+ 0 ) = (−d2 + 6d − 8)/16 which is the case with the solution for the scaled boundary layer given in the main text, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' THE BOUNDARY LAYER OF ¯U ′′(¯φ) The boundary layer has been derived in the main text in WP version of the RG because it is simpler in this version than in Wetterich version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, it can also be derived directly in this latter version or, once it is 7 obtained in one version, it can be translated in the other by performing the Legendre transform given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="5 ϕ 5 10 15 U''[ϕ] N=1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5⨯104 N=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='2⨯105 N=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='2⨯106 WF N=∞ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The second derivative of the T3 FP potential ¯U ′′(¯φ) in the Wetterich version of the RG, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6), for different values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 6 the boundary layer of ¯U ′′(¯φ) for different values of N at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' DIFFERENT FP POTENTIALS OF THE BMB LINE We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 7 the first derivative of different FP potentials of the BMB line at N = ∞ and in d = 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These FP potentials, implicitly given by the exact ex- pression given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) and (10) of the main text, are indexed by the nonnegative constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The WF FP potential corresponds to C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 2 3 4 5 6 7 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="15 V'[μ] C=2 C=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='625 C=0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' ¯V ′(¯µ) for different FPs indexed by the constant C on the BMB line given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) and (10) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We emphasize that the limit of ¯V ′′(¯µ0 = 3), when C → 0 is not given by the second derivative of the WF FP potential which is however the limit when C → 0 of ¯V (¯µ) along the BMB line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This is consistent with what happens at fixed d < 8/3 when N → ∞ since the limit d → 8/3 at fixed α = ∞ consists in following the WF FP at N = ∞ up to d = 8/3, the derivatives of which are not the limit of the derivatives of the T3 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' EIGENPERTURBATIONS AT THE TETRACRITICAL FP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6: Eigenperturbation δ ¯V3(¯µ) at the T3 FP corresponding, when N → ∞, to the relevant eigenvalue λ3 = d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 8 and 9 the relevant eigenperturba- tions δ ¯Vi of the T3 FP in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Whereas δ ¯V3 tends to the relevant eigenperturbation of the critical WF FP –with eigenvalue d − 2 which is the inverse of the critical exponent νWF–, the two others become singular in the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This is the rea- son why they play no role for the critical behavior of the O(N) model at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 2 3 4 5 6 7 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 δV1[μ] N=5×102 N=3×103 N=1×106 0 1 2 3 4 5 6 7 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='04 δV2[μ] N=5×102 N=3×103 N=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='4×105 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6: Eigenperturbations δ ¯Vn(¯µ) for n = 1, 2 at the T3 FP corresponding respectively to the relevant eigen- values λ1 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00 and λ2 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='326 for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These eigenperturbations tend to singular functions of ¯µ when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 10 that the slope of δ ¯V1(¯µ) at ¯µ0 in- creases as N 1/3 which proves that this eigenperturbation becomes discontinuous at infinite N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [μ] V[] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0 N=3x103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='005 N=3x104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 WF N=8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00 μ 1 2 3 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='028 1000 104 105 106 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="20 δV 1'[10/3] FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6: Slope of the eigenperturbation δ ¯V1(¯µ) of the T3 FP at its minimum ¯µ0 = 10/3 for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The equation of the full line is y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00175N 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' BMB LINE AND THE JOINED LIMIT ϵ → 0 AND N → ∞ AT FIXED α = ϵN When a T3 FP is followed along the hyperbola d = 8/3 − α/N, α ≥ 0, of the (d, N) plane, its potential con- verges when N → ∞ to the potential of one of the FPs of the BMB line, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We derive below the re- lationship α = 162/C3 between the parameter α of the hyperbola and the parameter C that indexes the FPs along the BMB line, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) and (10) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This relationship can be derived as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The FP potential of T3 is expanded as ¯V (¯µ) = ∞ � n=0 an(¯µ − 3)n (1) around the minimum ¯µ0 = 3 of the N = ∞ potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Then, the coefficients an are expanded as an = a(0) n + N −1a(1) n + O(N −2) (2) in power of 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At order O(N 0), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) yields a(0) n = 0 for n = 1, 2 and 3 and recursively determines a(0) n for n larger than 5 in terms of a(0) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Now, �∞ n=0 a(0) n (¯µ−3)n is the expansion of a FP poten- tial of the BMB line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' For this potential, ¯µ± behaves from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) as ¯µ± ≃ 3 ± 24/3C| ¯V ′|1/3 for C ̸= 0 and | ¯V ′| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This implies that a(0) 4 is related to C by a(0) 4 = 3/(192C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At order O(N −1), it can be shown that a(1) n for all n but n = 4 can be recursively determined in terms of a(0) 4 or C, if and only if the condition α = 162/C3 is satisfied which proves the relationship between these two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 11 different T3 FP potentials along an hyperbola d = 8/3 − α/N with increasing values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These FP potentials converge to a potential corresponding to the FP on the BMB line indexed by C = (162/α)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="10 V'[μ] C=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='672, N=∞ N=4000 α=1600/3 N=8000 α=1600/3 N=24000 α=1600/3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' ¯V ′(¯µ) of the T3 FP followed on the hyperbola d = 8/3 − α/N with fixed α = 1600/3 for increasing values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In the double limit d → 8/3 and N → ∞, it converges to the FP potential of the BMB line corresponding to C = (162 × 3/1600)1/3 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} diff --git a/AtFAT4oBgHgl3EQfrx4U/content/tmp_files/2301.08654v1.pdf.txt b/AtFAT4oBgHgl3EQfrx4U/content/tmp_files/2301.08654v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac7b732ebfc68189ba6e41e99c541d33e539b9c0 --- /dev/null +++ b/AtFAT4oBgHgl3EQfrx4U/content/tmp_files/2301.08654v1.pdf.txt @@ -0,0 +1,977 @@ +Automated extraction of capacitive coupling for quantum dot systems +Joshua Ziegler,1, ∗ Florian Luthi,2 Mick Ramsey,2 Felix Borjans,2 Guoji Zheng,2 and Justyna P. Zwolak1, † +1National Institute of Standards and Technology, Gaithersburg, MD 20899, USA +2Intel Components Research, Intel Corporation, 2501 NW 229th Avenue, Hillsboro, Oregon 97124, USA +(Dated: January 23, 2023) +Gate-defined quantum dots (QDs) have appealing attributes as a quantum computing platform. +However, near-term devices possess a range of possible imperfections that need to be accounted +for during the tuning and operation of QD devices. One such problem is the capacitive cross-talk +between the metallic gates that define and control QD qubits. A way to compensate for the capacitive +cross-talk and enable targeted control of specific QDs independent of coupling is by the use of virtual +gates. Here, we demonstrate a reliable automated capacitive coupling identification method that +combines machine learning with traditional fitting to take advantage of the desirable properties of +each. We also show how the cross-capacitance measurement may be used for the identification of +spurious QDs sometimes formed during tuning experimental devices. Our systems can autonomously +flag devices with spurious dots near the operating regime which is crucial information for reliable +tuning to a regime suitable for qubit operations. +I. +INTRODUCTION +Quantum dot (QD) arrays, in which charge carriers +are trapped in localized potential wells and qubits can +be made by use of the spin and permutation symmetries +of the carriers, are a promising quantum computing plat- +form [1–3]. In fact, last year has shown the first demon- +stration of QD two-qubit gates with fidelities exceeding +the thresholds for fault-tolerant computing [4–6]. How- +ever, because the individual charge carriers that make +up qubits have electrochemical sensitivity to minor im- +purities and imperfections, calibration and tuning of QD +devices is a nontrivial and time-consuming process, with +each QD requiring a careful adjustment of a gate voltage +to define charge number, and multiple gate voltages to +specify tunnel coupling between QDs for two-qubit gates +or to reservoirs for reset and measurement. While manual +calibration is achievable for small, few-QD devices, with +increasing size and complexity of QD arrays, the relevant +control parameter space grows quickly, necessitating the +development of autonomous tuning methods. +There have been numerous demonstrations of automa- +tion of the various phases of the tuning process for sin- +gle and double-QD devices [7]. Some approaches seek to +tackle tuning starting from device turn-on to coarse tun- +ing [8–11] while others assume that bootstrapping (cal- +ibration of measurement devices and identification of a +nominal regime for further investigation) and basic tun- +ing (confirmation of controllability and device character- +istics) have been completed and focus on a more tar- +geted automation of the coarse and charge tuning [12–16]. +While the initial auto-tuning approaches relied mainly +on the appealingly intuitive and relatively easy to imple- +ment conventional algorithms that typically involved a +∗ Current address: Intel Components Research, Intel Corporation, +2501 NW 229th Avenue, Hillsboro, Oregon 97124, USA +† jpzwolak@nist.gov +combination of techniques from regression analysis, pat- +tern matching, and quantum control theory, the more re- +cent algorithms take advantage of the modern computer +vision and machine learning [7]. +A typical accumulation-mode QD device consists of +two sets of gates—plungers and barriers—that collec- +tively control the overall potential profile, QD-specific +single-particle energy detuning of individual QDs, the +tunnel couplings between QDs, and tunnel rates between +the most outer QDs and reservoirs. Ideally, each plunger +gate would affect only the electrochemical potential of +a single targeted QD and each barrier gate only one in- +tended tunnel barrier. Due to the tight proximity, how- +ever, each gate capacitively couples to nearby potential +and tunnel barriers. This makes careful control of these +key parameters challenging. +One way to compensate for the capacitive cross-talk +between gates is to enable orthogonal control of the QDs +potential by implementing so-called virtual gates [17]. +Specifically, linear combinations of gate voltage changes +can be mapped onto onsite energy differences [17–20]. +These approaches have been key for the initialization and +control of larger QD arrays [21, 22]. +To autonomously identify capacitive couplings in a +device, various approaches have been demonstrated us- +ing both conventional fitting and machine learning (ML) +techniques [23–26]. However, these approaches, typically +relying on the Hough transform or conventional least- +squares fitting procedures, may be unreliable in the pres- +ence of data imperfections. +Hough transforms can ex- +tract slopes directly but may be sensitive to noise or be +excessively complex to analyze. The conventional fitting +can be more flexible but is susceptible to local minima +and can be time-consuming at inference time. +Convolutional neural networks (CNN) are well suited +for extracting high-level features from images and can +remain effective in the presence of noise or other imper- +fections [27]. However, ML methods can have difficulties +identifying data outside of the training distribution even +if it contains similar features [28]. Fortunately, given a +arXiv:2301.08654v1 [cond-mat.mes-hall] 20 Jan 2023 + +2 +simplified, high-level representation of the data, conven- +tional fitting approaches can be more targeted to extract +key information more effectively and quickly. +Here we develop a reliable automated capacitive cou- +pling identification method that combines ML with tra- +ditional fitting to take advantage of the desirable proper- +ties of each. We use an ML module for pixel classification +followed by linear regression for extracting targeted in- +formation and demonstrate effective performance across +noise levels and data variations. Testing each of these +methods on a set of eight simulated QD devices with +large variability and realistic noise variation mimicking +experimental conditions shows that the approach com- +bining ML and traditional fitting works well, with a root +mean square error (RMSE) of 0.034(14), corresponding +to a roughly 7 % error, for predicting virtual gate ma- +trix off-diagonal values (normalizing such that diagonal +values are one). +We also demonstrate how the cross-capacitance mea- +surement may be used for the identification of spurious +QDs formed during tuning experimental devices. Many +of the auto-tuning approaches proposed to date rely on +a series of small 2D scans capturing a relatively narrow +range of the voltage space [13, 14, 27, 29]. While such ap- +proaches improve the efficiency of tuning, they may result +in unexpected and difficult to assess failure modes when +the tuning algorithm terminates at an anti-crossing with +a spurious QD that may form in small potential wells due +to interface defects, surface roughness, or strain within +the device [30]. They are highly undesirable since they +may interfere with the QDs intended for use as qubits +and cannot themselves be used as qubits. To avoid de- +vice tuning failure, spurious QDs must be identified when +present and avoided. We test the utility of our approach +for capacitive coupling estimation by identifying spurious +QD in experimental measurements of QD devices [1]. +The manuscript is organized as follows: In Sec. II we +introduce the framework of combining traditional fitting +techniques with a pixel classifier to process the high-level +information extracted from experimental data. In Sec. III +we show the utility of the proposed framework to auto- +matically extract virtual gates as well as identify charge +transitions resulting from a formation of spurious QD. +Finally, in Sec. IV we summarize the results and discuss +the outlook. +II. +METHODS: MACHINE LEARNING AND +FIT +Capacitive couplings in a QD device can be measured +and, in a constant capacitance approximation, described +by a matrix that maps the physical gate voltages onto +the effect they each have on the QD’s chemical poten- +tials or barriers [17, 23, 24, 31–33]. +Measurement of +the elements of this matrix must be performed distinctly +for electrochemical potentials and tunnel barriers. Cou- +plings of the chemical potentials to each QD—which is +0.30 +0.32 +0.27 +0.29 +VP2(V) +(a) +0.8 +1.1 +1.4 +Current (arb. units) +0.30 +0.32 VP1(V) +(b) +NT LT CT RT PL +(c) +FIG. 1. An example 2D scan and corresponding pixel classifi- +cation, class clusters, and linear fits. (a) A simulated voltage +scan showing left and right transitions as well as a polariza- +tion line. (b) Pixel classification for the scan shown in (a). (c) +Regions of pixels and linear fits from the pixel classification. +The large dark points indicate the centers of pixel regions. +the focus of this work—can be extracted from shifts in +charge transition lines when each voltage is varied [17] +while the effect of each gate on tunnel barriers can be +assessed by measuring changes in the width of inter-dot +transitions, assuming the electron temperature is suffi- +ciently low [32]. Measured this way, the couplings are +relative, usually scaled with respect to the coupling of +the QD to the nearest gate. An absolute energy scale +can be obtained by measuring the gate lever arms with +photon-assisted tunneling, Coulomb diamonds, or bias +triangles [34]. However, for establishing the orthogonal +control the relative scale is sufficient [21]. +For a double QD, the virtualization matrix Gvirt re- +lating the physical plunger gates to virtual gates can be +represented by Eq. 1. Each row is normalized such that +the diagonal entries are 1 to reflect the relative nature of +our virtual gates. +Gvirt ≡ +�VP ′ +1 +VP ′ +2 +� += +� +1 +α12 +α21 +1 +� � +VP1 +VP2 +� +(1) +The relative cross-capacitances for chemical potentials +manifest themselves via the slopes of charge transition +lines, with the dominant terms of the cross-capacitance +matrix determined from a measurement in the space of +neighboring pairs of gates [21]. +We address the iden- +tification of the cross-capacitances as captured in two- +dimensional (2D) plunger-plunger gate scans, as shown in +Fig. 1(a). To translate the low-level QD data into high- +level information useful for automation we use a pixel +classifier, i.e., a CNN model with a structure similar to a +feature pyramid network [35]. The pixel classifier takes as + +3 +an input a small 2D plunger voltage scan obtained using +a charge sensor, as shown in Fig. 1(a). It then identi- +fies each pixel within the scan as belonging to one of the +charge transition classes—left, right, central, or inter-dot +(polarization line) transition, denoted as LT, RT, CT, or +PL, respectively—or to the no transition (NT) class. In +other words, the CNN provides a high-level classification +of the raw experimental data while keeping spatial infor- +mation about the relative location and orientation of the +detected features, which is useful for algorithmic process- +ing. Figure 1(b) shows the pixel classification of a scan +from Fig. 1(a). +To translate pixel classifications to capacitive cou- +plings, we identify contiguous regions within each class +of pixels in an image. A labeling algorithm from the mul- +tidimensional image processing package in SciPy is then +used to determine the relevant clusters of connected pix- +els [36]. This separates fragments of charge transitions +into distinct clusters so that each can be processed indi- +vidually. Each region of pixels classified as LT, CT, or +RT is then independently fitted using linear regression, as +shown in Fig. 1(c). When multiple segments for a given +class are present in an image, the capacitive coupling re- +turned is the average for all fitted lines weighted by the +standard deviations of each fit, yielding the solid lines in +Fig. 1(b) (offset arbitrarily for comparison with the pixel +regions). Standard deviations σ are computed from the +standard error of the fit, S, by σ = S/√n, where n is the +size of the pixel region, as in Student’s t-distribution [37]. +In addition, each region is tagged with its center in volt- +age space, shown by the large black points in Fig. 1(c), +which allows tracking the changes in charge transitions +and their slopes within the larger space. +A. +Data +The data used for training the ML tools and testing +the methods was generated using a simulation of QD de- +vices [12]. The simulation is composed of a calculation of +the electron density in the Thomas-Fermi approximation +and a capacitance matrix to determine the stable electron +configuration. To improve the robustness of the models, +the data is augmented with synthetic white, pink (1/f), +and telegraph noise [27]. The effect of a QD charge sen- +sor strongly coupled to the plunger gates is varied during +the scan to improve performance on this type of experi- +mental data. +The training dataset consists of 1.6 × 105 devices with +parameters varied over a uniform distribution with a +standard deviation equal to 1 % of each parameter’s +value. To train the ML models we randomly sample 10 +small scans per device and use charge state ground truth +to label each scan on a pixel level with the presence and +type of transition, yielding NT, LT, CT, RT, and PL la- +bels. Additionally, we extract the slopes of the transition +lines directly using the gradients of the device charge. +The test data is composed of eight simulated devices +with large variations in screening length and device pitch +1× +5× +10× +15× +20× +25× +30× +35× +0.0 +0.2 +0.4 +0.6 +RMSE +(a) +1× +15× +30× +0.0 +0.2 +0.4 +0.6 +(b) +1× +15× +30× +Noise Level +(c) +FIG. 2. (a) Root mean square error (RMSE) for all transition +classes (left, central, and right [LT, CT, RT]) as a function +of the synthetic noise level. (b) RMSE as a function of noise +level for the LT class. (c) RMSE as a function of noise level +for the RT class. +and with large shifts in the position of one of the plunger +gates. These changes lead to large variations in the slopes +of the charge transition lines, the capacitive coupling +between QDs, spacing between lines, and the relative +amount of left and right QD, making them largely dis- +tinct from the training data. To facilitate a controlled +study and track the performance of the pixel classifier +as data quality degrades, each large scan is randomly +sampled 50 times and the resulting small scans are aug- +mented with increasing levels of synthetic noise. +This +results in a set of 400 simulated test scans. Finally, sev- +eral large experimental measurements acquired using a +double-QD configuration on a three-QD Six/SiGe1−x de- +vice, fabricated on an industrial 300 mm process line [1], +are used to test the performance of the virtualization +algorithm. Experimental scans capturing spurious QDs +are used to demonstrate the algorithm for spurious QD +detection. +III. +RESULTS +We test the effectiveness of our automated approach to +extracting the cross-capacitance by first evaluating the +performance of each component, i.e., the pixel classifier +and the slope extractions, on each scan in the simulated +test set. The error of the pixel classifier in our frame- +work is defined as a fraction of pixels belonging to true +transitions that are not contained in line segments in the +CNN output. +This captures type-I errors without the + +4 +2.00 +2.20 +VP1(V) +2.20 +2.40 +VP2(V) +(a) +1.40 +1.55 VP ′ +1(V) +1.50 +1.60 +1.70 +VP ′ +2(V) +(b) +1.20 +1.35 VP ′′ +1 (V) +1.20 +1.30 +VP ′′ +2 (V) +(c) +−0.8 +−0.6 +−0.4 +−0.2 +Virt. gate +off diag. +FIG. 3. (a) Large experimentally measured charge stability diagram with a scatter plot of centers of pixel class regions overlaid. +The colors of the points indicate the virtual gate off-diagonal values identified by fits to the region. The sizes of the points +indicate the weights used when averaging. Only points with relative error less than 20 % are plotted. (a,b) Charge stability +diagram after applying virtual gates acquired near the (0, 0) − (1, 1) charge transition in (a) and near the (1, 3) − (2, 4) charge +transition in (b). In both (b) and (c) the virtualization is performed off-line, via an affine transform to the original scan shown +in (a) and the points are plotted using the same parameters as in (a). +effect of false type-II errors due to imperfect labels [38]. +Figure 2(a) shows the change in RMSE as a function of +the noise level in the simulated data. At the noise level of +1.0, i.e., the noise level estimated from experimental data +in Ref. [29], we observe an RMSE of 0.17(5). The RMSE +increases significantly to 0.50(11) at the noise level of 20. +For reference, a pixel classifier that always predicts the +NT class would have an RMSE of 0.62 ( +√ +0.4). For the +LT and RT classes relevant to cross-capacitances, shown +in Fig. 2(b) and (c), the pixel classifier for noise level 1.0 +has an RMSE of 0.20(8) and 0.11(8), respectively. +To verify that the slope extraction tool works as in- +tended, we test it across the eight large simulated test +devices. For these tests, we evaluate the pixel classifier +in windows of size roughly 1.5× the charging energy, as +estimated by the spacing of the first two charge transi- +tions. Outputs from the pixel classifier are cropped by +one pixel from the edge of the image before processing +due to missing context reducing CNN performance [39]. +The resulting classes of pixels are then grouped into dis- +tinct clusters. For each cluster consisting of more than +five pixels an independent linear fit is performed, return- +ing both the slope and the standard error of the fitted +line. This information can be used to find the orthog- +onal “virtual” control space or to flag transitions that +potentially belong to spurious dots, as described in the +following sections. +A. +Deriving virtual gates +As stated in Sec. II, in our approach the off-diagonal +elements, defining the virtual gates transformation, are +determined based on the slopes of the LT and RT cap- +tured in a given image, and the diagonal elements of the +capacitance matrix are set to 1.0. When multiple lines +belonging to the same class are detected, as in Fig. 1(a), +the capacitive coupling is calculated through a weighted +average, with the weight accounting for both the size +of the clusters and the standard deviations of respective +fits [37]. +The off-diagonal elements of the virtualization matrix +computed this way have an RMSE of 0.034(14) at the +noise level of 1.0 defined in Ref. [29], corresponding to a +roughly 8 % error compared to the ground truth values +derived from simulated data. We further test them on +a range of levels of synthetic noise and find the RMSE +rises by a factor of two at a level of noise of roughly 15× +the level of noise defined in Ref. [29], consistent with the +pixel classifier error. +To better understand the trends of the virtualization +matrix in the plunger-plunger space, we carry out a per- +formance analysis using the test set of eight large sim- +ulated charge stability diagrams and several experimen- +tally measured scans. +For each scan, we calculate the +fits to the pixel classification clusters based on a series of +small scans sampled at each point within the large scan +with the exclusion of a margin implemented to ensure +that all sampled scans fall within the full scan bound- +aries. The small scans and the margins are set to have a +size 1.5× the charging energy of a given simulated device. +Figure 3(a) shows the centers of the pixel region identi- +fied in each small scan [as in Fig. 1(c)] as the sampling +window is swept across a large experimentally measured +charge stability diagram. The regions identified by the +pixel classification are consistently placed correctly on +the charge transition lines regardless of the position of the + +5 +2.09 +2.18 VP1(V) +0.00 +0.25 +0.50 +Rel. virt. gate off-diag. +(a) +2.26 +2.34 VP2(V) +(b) +0 +1 +2 +3 +4 +Density +×10−2 +FIG. 4. Histograms of the off-diagonal elements of the virtu- +alization matrix for an experimentally measured scan shown +in Fig. 3(a) as a function of plunger gates, (a) VP1 and (b) +VP2. Off-diagonal values are normalized to the mean of the +virtual gates in the (1, 1) charge state for ease of comparison. +Virtual gate values are extracted from a strip of small scans +shifted by 6 mV (four pixels) to better visualize variation at +each plunger gate value. +line within a small scan. Region centers shift along the +charge transition lines as different portions of the line are +captured within the small scan and remain fixed when- +ever the same fragment of the charge transition is cap- +tured. The color of the points indicates the off-diagonal +values of the virtual gate matrix, α12 and α21. As ex- +pected, these coupling constants get larger in magnitude +as charges are added to each QD. Finally, the size of the +points in Fig. 3(a) indicates the 1/σ2 weight of the slopes +used when averaging multiple slopes from the same type +of transition within a small scan. As desired, the posi- +tions of the points with smaller sizes indicate that lines +that are smaller or less captured within a small scan have +fits with larger errors. Overall, this plot confirms that the +pixel classification and the fits are working as intended +at capturing charge transition lines and their slopes. +To demonstrate the spatial relevance of the virtual +gates derived from a set of fits across a device’s charge +landscape, in Fig. 3(b) and (c) we plot affine-transformed +charge stability diagrams, with points indicating fits +overlaid. The points plotted are the centers of pixel re- +gions with colors indicating the α12 and α21 values and +size indicating the inverse of the fit error squared (the +weight of the fit in the average). +The affine transfor- +mation applied in Fig. 3(b) corresponds to virtual gates +derived from an image near the (0, 0) − (1, 1) charge +transition with off-diagonal values α12 = −0.282(4), +α21 = −0.331(4). For Fig. 3(c), the affine transformation +applied has virtual gates from the (1, 3) − (2, 4) charge +transition, with off-diagonal values α12 = −0.363(4), +α21 = −0.480(4). As can be seen in the insets in Fig. 3(b) +and (c), these virtual gates are very effective at trans- +forming the target region to an orthogonal space, but the +difference between the extracted virtual gate off-diagonal +values are about 50 % higher for the latter case. This +highlights the importance of an efficient local method for +determining virtual gates. +To further understand how capacitive coupling varies +across a charge stability diagram, we can calculate varia- +tion as each plunger gate is adjusted. Figure 4(a) and (b) +show how virtual gates extracted from small scans change +as VP1 and VP2 are varied. To better show the trend, vir- +tual gates from small scans shifted by 3 mV (two pixels) +in the opposing direction are included. This shows that +the virtual gates extracted from small scans effectively +capture variation across charge stability diagrams. +B. +Detection of spurious dots +Visually, spurious QDs are recognized in large 2D scans +as charge transitions with slopes diverging from a mono- +tonic trend, see Fig. 5(a). In this framework, they may +be identified as transition lines with anomalous capaci- +tive couplings relative to the transitions around them. +As a demonstration, we use the pixel classification and +fit tools to analyze five experimental charge stability di- +agrams: two capturing properly formed QD, shown in +Fig. 5(a) and (b), and three capturing spurious QDs, +shown in Fig. 5(a), (b), and (c). While for extraction +of the virtualization matrix small scans are sufficient, de- +tection of spurious QD requires somewhat bigger scans +to ensure that the neighboring charge transitions are ad- +equately captured. In our analysis, we rely on 2D scans +of a size roughly three times the charging energy (four +times the area of scans typically used in auto-tuning al- +gorithms [13, 14]). We also consider only clusters con- +sisting of at least 20 pixels to ensure better reliability of +the linear fit. +After pixel classification, contiguous clusters of pixels +belonging to a given class of transitions are analyzed in- +dividually, resulting in a cluster-based fit and standard +deviation. Cases where more than one cluster belongs +to a given charge transition result in separate fits, as in +Fig. 5(b) and (e) where the LT lines are split into groups +to either side of the RT lines. This separation serves two +purposes: to ensure that variation along a given transi- +tion isn’t included and to treat each additional line inde- +pendent of the charge on another QD. +Within a class and group of transitions, the magni- +tude of the capacitive coupling is expected to increase +monotonically as the charge is added. Such behavior is +clearly visible in Fig. 5(k) and (l), with the latter having +to separate groups of fits (shown with different shades of +green) for the groups of clusters. On the contrary, a spu- +rious QD manifests itself by a non-monotonic behavior of +the capacitive coupling between transitions, as depicted +graphically by the center point (or group of points) in +Fig. 5(m), (n), and (o). The severity of this divergence +can be quantified using a Z-test [40]. +In practical applications, the automated detection of +spurious QD fits nicely within the auto-tuning paradigm. +As mentioned earlier, many of the proposed approaches +utilize a series of small 2D scans [13, 14, 27, 29] or +1D rays [41, 42] as means to improve the tuning effi- + +6 +2.26 +2.32 +2.33 +2.39 +VP2(V) +(a) +2.26 +2.32 +2.33 +2.39 +(f) +2.33 +2.39 +-0.31 +-0.28 +-0.26 +α12 +(k) +2.38 +2.44 +2.31 +2.37 +VR1 +(b) +0.6 +0.8 +1.0 +1.2 +Current +(arb. units) +2.38 +2.44 +2.31 +2.37 +(g) +NT +LT +CT +RT +PL +2.31 +2.37 +-0.4 +-0.3 +-0.2 (l) +1.89 +1.96 +2.18 +2.24 +(c) +1.89 +1.94 +2.18 +2.24 +(h) +2.18 +2.25 +-0.4 +-0.3 +-0.2 (m) +0.42 +0.46 +0.46 +0.50 +(d) +0.42 +0.46 +0.46 +0.50 +(i) +0.46 +0.50 +-0.5 +-0.4 +-0.3 (n) +0.48 +0.52 +0.45 +0.49 +VR2 +(e) +0.7 +0.8 +0.9 +1.0 +Current +(arb. units) +0.48 +0.52 VP1(V) +0.45 +0.49 +(j) +NT +LT +CT +RT +PL +0.45 +0.49 VP2(V) +-0.6 +-0.4 +-0.2 (o) +FIG. 5. Spurious dot detection. The top row shows two charge stability diagrams capturing properly formed QD [panels (a) +and (b)] and three charge stability capturing spurious QD [panels (c), (d), and (e)]. The black boxes in (b) and (e) in highlight +small 2D scans, denoted as VR1 and VR2, typical of the auto-tuning approaches proposed in Ref. [13, 14]. Panels (f)–(j) in the +middle row show pixel classification results for charge stability diagrams shown in the top row. Plots of fitting results used to +determine whether a spurious QD is present are shown in the bottom row [panels (k)–(o)]. The different groups of transition +are shown with different shades of green. The monotonicity within each group of transitions is clearly visible in panels (k) and +(l). On the contrary, in the three plots shown in panels (m), (n), and (o), there is a clear divergence from the expected trend +for the spurious QD (middle) transitions. Error bars indicate one standard deviation. +ciency. +While these approaches deliver measurement- +cost-effective solutions, they are prone to unexpected and +difficult-to-detect failure even when the data quality is +high. Fig. 5(b) and (e) show examples of such potentially +problematic cases. The small 2D regions in the plunger- +plunger space, highlighted in these scans with the black +boxes, are typical for topology setting algorithms. +In +both cases, they are classified by a state classifier model +as double QD state, with state prediction vectors being +p(VR1) = [0.01, 0.04, 0., 0.04, 0.92] for VR1 region high- +lighted in Fig. 5(b) and p(VR2) = [0., 0., 0.18, .05, 0.76] +for region VR2 highlighted in Fig. 5(b), where p(VR) = +[pND, pSDL, pSDC, pSDR, pDD] with ND denoting no QDs +formed, SDL, SDC, and SDR denoting the left, central, +and right single QD, respectively, and DD denoting the +double-QD state. Moreover, the data quality for these +images is high in both cases, with q(VR1) = [1.0, 0.0, 0.0] +for region VR1 and q(VR1) = [0.99, 0.01, 0.0] for VR2, +where q(VR) = [phigh, pmod, plow] with phigh, pmod, and +plow denoting the probability of region VR being assessed +by the data quality control module as “high,”, “moder- +ate,” and “low” quality, respectively. Thus, from the ML +perspective, both these predictions are confidently cor- +rect. However, when looked at within a slightly larger +voltage range, it is clear that in the latter case the +small scan captures an anti-crossing with a spurious QD, +which for practical tuning purposes is a failure. If not +recognized and corrected for, termination at this point +will result in an incorrect charge setting and virtualiza- +tion [13, 29]. +The spurious QD detection algorithm can be easily +implemented in the auto-tuning algorithm proposed in +Ref. [27] as a safety check before the unloading step is ini- +tiated. An automated identification and characterization +of spurious QDs may also be useful to inform fabrication +procedures and prevent them in future devices [30]. +IV. +CONCLUSIONS +As quantum dot devices grow in size and complexity, +the need for reliable and automated tune-up procedures +becomes more pressing. +Establishing orthogonal con- +trol of the chemical potentials of quantum dots is one +of the first steps in the tune-up of any larger quantum +dot array. Here, we demonstrated a method that com- +bines machine-learning-based pixel-classification and tra- +ditional curve fitting to reliably determine voltage cross- +talk coefficients. The advantage of this method over pre- +vious approaches is highlighted by increased reliability + +7 +and resilience to experimental noise. +Further on, un- +wanted spurious dots that would reduce or inhibit de- +vice performance can be detected and flagged when this +module is used as part of a larger tune-up algorithm [29]. +The capability to automatically and reliably detect spuri- +ous dots is especially important on wafer-scale fabrication +characterization tools that produce more data than can +efficiently be processed by human analysis. In extensions, +our tools could allow for automated navigation of volt- +age space for more targeted measurement of all chemical +potential and tunnel barrier cross-capacitances [17, 32]. +ACKNOWLEDGMENTS +This research was performed while J.Z. held a NRC +Research Associateship award at the National Institute +of Standards and Technology (NIST). The views and con- +clusions contained in this paper are those of the authors +and should not be interpreted as representing the of- +ficial policies, either expressed or implied, of the U.S. +Government. +The U.S. Government is authorized to +reproduce and distribute reprints for Government pur- +poses notwithstanding any copyright noted herein. Any +mention of commercial products is for information only; +it does not imply recommendation or endorsement by +NIST. +[1] R. Pillarisetty, +T. Watson, +B. Mueller, +E. Henry, +H. George, S. Bojarski, L. Lampert, F. Luthi, R. Kotl- +yar, O. Zietz, S. Neyens, F. Borjans, R. Caudillo, +D. Michalak, R. Nahm, J. Park, M. Ramsey, J. Roberts, +S. Schaal, G. Zheng, T. Kr¨ahenmann, M. Lodari, A. Zw- +erver, M. Veldhorst, G. Scappucci, L. Vandersvpen, and +J. Clarke, Si mos and si/sige quantum well spin qubit +platforms for scalable quantum computing, in 2021 IEEE +International Electron Devices Meeting (IEDM) (2021) +pp. 14.1.1–14.1.4. +[2] P. L. 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Appl. 18, 064040 (2022). + diff --git a/AtFAT4oBgHgl3EQfrx4U/content/tmp_files/load_file.txt b/AtFAT4oBgHgl3EQfrx4U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..723b1ed1ed64eddf95811a3d8cc2943437820e05 --- /dev/null +++ b/AtFAT4oBgHgl3EQfrx4U/content/tmp_files/load_file.txt @@ -0,0 +1,981 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf,len=980 +page_content='Automated extraction of capacitive coupling for quantum dot systems Joshua Ziegler,1, ∗ Florian Luthi,2 Mick Ramsey,2 Felix Borjans,2 Guoji Zheng,2 and Justyna P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Zwolak1, † 1National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 2Intel Components Research, Intel Corporation, 2501 NW 229th Avenue, Hillsboro, Oregon 97124, USA (Dated: January 23, 2023) Gate-defined quantum dots (QDs) have appealing attributes as a quantum computing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' However, near-term devices possess a range of possible imperfections that need to be accounted for during the tuning and operation of QD devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' One such problem is the capacitive cross-talk between the metallic gates that define and control QD qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' A way to compensate for the capacitive cross-talk and enable targeted control of specific QDs independent of coupling is by the use of virtual gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Here, we demonstrate a reliable automated capacitive coupling identification method that combines machine learning with traditional fitting to take advantage of the desirable properties of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' We also show how the cross-capacitance measurement may be used for the identification of spurious QDs sometimes formed during tuning experimental devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Our systems can autonomously flag devices with spurious dots near the operating regime which is crucial information for reliable tuning to a regime suitable for qubit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' INTRODUCTION Quantum dot (QD) arrays, in which charge carriers are trapped in localized potential wells and qubits can be made by use of the spin and permutation symmetries of the carriers, are a promising quantum computing plat- form [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In fact, last year has shown the first demon- stration of QD two-qubit gates with fidelities exceeding the thresholds for fault-tolerant computing [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' How- ever, because the individual charge carriers that make up qubits have electrochemical sensitivity to minor im- purities and imperfections, calibration and tuning of QD devices is a nontrivial and time-consuming process, with each QD requiring a careful adjustment of a gate voltage to define charge number, and multiple gate voltages to specify tunnel coupling between QDs for two-qubit gates or to reservoirs for reset and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' While manual calibration is achievable for small, few-QD devices, with increasing size and complexity of QD arrays, the relevant control parameter space grows quickly, necessitating the development of autonomous tuning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' There have been numerous demonstrations of automa- tion of the various phases of the tuning process for sin- gle and double-QD devices [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Some approaches seek to tackle tuning starting from device turn-on to coarse tun- ing [8–11] while others assume that bootstrapping (cal- ibration of measurement devices and identification of a nominal regime for further investigation) and basic tun- ing (confirmation of controllability and device character- istics) have been completed and focus on a more tar- geted automation of the coarse and charge tuning [12–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' While the initial auto-tuning approaches relied mainly on the appealingly intuitive and relatively easy to imple- ment conventional algorithms that typically involved a ∗ Current address: Intel Components Research, Intel Corporation, 2501 NW 229th Avenue, Hillsboro, Oregon 97124, USA † jpzwolak@nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='gov combination of techniques from regression analysis, pat- tern matching, and quantum control theory, the more re- cent algorithms take advantage of the modern computer vision and machine learning [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' A typical accumulation-mode QD device consists of two sets of gates—plungers and barriers—that collec- tively control the overall potential profile, QD-specific single-particle energy detuning of individual QDs, the tunnel couplings between QDs, and tunnel rates between the most outer QDs and reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Ideally, each plunger gate would affect only the electrochemical potential of a single targeted QD and each barrier gate only one in- tended tunnel barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Due to the tight proximity, how- ever, each gate capacitively couples to nearby potential and tunnel barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' This makes careful control of these key parameters challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' One way to compensate for the capacitive cross-talk between gates is to enable orthogonal control of the QDs potential by implementing so-called virtual gates [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Specifically, linear combinations of gate voltage changes can be mapped onto onsite energy differences [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' These approaches have been key for the initialization and control of larger QD arrays [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To autonomously identify capacitive couplings in a device, various approaches have been demonstrated us- ing both conventional fitting and machine learning (ML) techniques [23–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' However, these approaches, typically relying on the Hough transform or conventional least- squares fitting procedures, may be unreliable in the pres- ence of data imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Hough transforms can ex- tract slopes directly but may be sensitive to noise or be excessively complex to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The conventional fitting can be more flexible but is susceptible to local minima and can be time-consuming at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Convolutional neural networks (CNN) are well suited for extracting high-level features from images and can remain effective in the presence of noise or other imper- fections [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' However, ML methods can have difficulties identifying data outside of the training distribution even if it contains similar features [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Fortunately, given a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='08654v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='mes-hall] 20 Jan 2023 2 simplified, high-level representation of the data, conven- tional fitting approaches can be more targeted to extract key information more effectively and quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Here we develop a reliable automated capacitive cou- pling identification method that combines ML with tra- ditional fitting to take advantage of the desirable proper- ties of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' We use an ML module for pixel classification followed by linear regression for extracting targeted in- formation and demonstrate effective performance across noise levels and data variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Testing each of these methods on a set of eight simulated QD devices with large variability and realistic noise variation mimicking experimental conditions shows that the approach com- bining ML and traditional fitting works well, with a root mean square error (RMSE) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='034(14), corresponding to a roughly 7 % error, for predicting virtual gate ma- trix off-diagonal values (normalizing such that diagonal values are one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' We also demonstrate how the cross-capacitance mea- surement may be used for the identification of spurious QDs formed during tuning experimental devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Many of the auto-tuning approaches proposed to date rely on a series of small 2D scans capturing a relatively narrow range of the voltage space [13, 14, 27, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' While such ap- proaches improve the efficiency of tuning, they may result in unexpected and difficult to assess failure modes when the tuning algorithm terminates at an anti-crossing with a spurious QD that may form in small potential wells due to interface defects, surface roughness, or strain within the device [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' They are highly undesirable since they may interfere with the QDs intended for use as qubits and cannot themselves be used as qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To avoid de- vice tuning failure, spurious QDs must be identified when present and avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' We test the utility of our approach for capacitive coupling estimation by identifying spurious QD in experimental measurements of QD devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The manuscript is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' II we introduce the framework of combining traditional fitting techniques with a pixel classifier to process the high-level information extracted from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' III we show the utility of the proposed framework to auto- matically extract virtual gates as well as identify charge transitions resulting from a formation of spurious QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' IV we summarize the results and discuss the outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' METHODS: MACHINE LEARNING AND FIT Capacitive couplings in a QD device can be measured and, in a constant capacitance approximation, described by a matrix that maps the physical gate voltages onto the effect they each have on the QD’s chemical poten- tials or barriers [17, 23, 24, 31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Measurement of the elements of this matrix must be performed distinctly for electrochemical potentials and tunnel barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Cou- plings of the chemical potentials to each QD—which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='29 VP2(V) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='4 Current (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='32 VP1(V) (b) NT LT CT RT PL (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' An example 2D scan and corresponding pixel classifi- cation, class clusters, and linear fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' (a) A simulated voltage scan showing left and right transitions as well as a polariza- tion line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' (b) Pixel classification for the scan shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' (c) Regions of pixels and linear fits from the pixel classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The large dark points indicate the centers of pixel regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' the focus of this work—can be extracted from shifts in charge transition lines when each voltage is varied [17] while the effect of each gate on tunnel barriers can be assessed by measuring changes in the width of inter-dot transitions, assuming the electron temperature is suffi- ciently low [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Measured this way, the couplings are relative, usually scaled with respect to the coupling of the QD to the nearest gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' An absolute energy scale can be obtained by measuring the gate lever arms with photon-assisted tunneling, Coulomb diamonds, or bias triangles [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' However, for establishing the orthogonal control the relative scale is sufficient [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' For a double QD, the virtualization matrix Gvirt re- lating the physical plunger gates to virtual gates can be represented by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Each row is normalized such that the diagonal entries are 1 to reflect the relative nature of our virtual gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Gvirt ≡ �VP ′ 1 VP ′ 2 � = � 1 α12 α21 1 � � VP1 VP2 � (1) The relative cross-capacitances for chemical potentials manifest themselves via the slopes of charge transition lines, with the dominant terms of the cross-capacitance matrix determined from a measurement in the space of neighboring pairs of gates [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' We address the iden- tification of the cross-capacitances as captured in two- dimensional (2D) plunger-plunger gate scans, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To translate the low-level QD data into high- level information useful for automation we use a pixel classifier, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=', a CNN model with a structure similar to a feature pyramid network [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The pixel classifier takes as 3 an input a small 2D plunger voltage scan obtained using a charge sensor, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' It then identi- fies each pixel within the scan as belonging to one of the charge transition classes—left, right, central, or inter-dot (polarization line) transition, denoted as LT, RT, CT, or PL, respectively—or to the no transition (NT) class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In other words, the CNN provides a high-level classification of the raw experimental data while keeping spatial infor- mation about the relative location and orientation of the detected features, which is useful for algorithmic process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Figure 1(b) shows the pixel classification of a scan from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To translate pixel classifications to capacitive cou- plings, we identify contiguous regions within each class of pixels in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' A labeling algorithm from the mul- tidimensional image processing package in SciPy is then used to determine the relevant clusters of connected pix- els [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' This separates fragments of charge transitions into distinct clusters so that each can be processed indi- vidually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Each region of pixels classified as LT, CT, or RT is then independently fitted using linear regression, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' When multiple segments for a given class are present in an image, the capacitive coupling re- turned is the average for all fitted lines weighted by the standard deviations of each fit, yielding the solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1(b) (offset arbitrarily for comparison with the pixel regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Standard deviations σ are computed from the standard error of the fit, S, by σ = S/√n, where n is the size of the pixel region, as in Student’s t-distribution [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In addition, each region is tagged with its center in volt- age space, shown by the large black points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1(c), which allows tracking the changes in charge transitions and their slopes within the larger space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Data The data used for training the ML tools and testing the methods was generated using a simulation of QD de- vices [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The simulation is composed of a calculation of the electron density in the Thomas-Fermi approximation and a capacitance matrix to determine the stable electron configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To improve the robustness of the models, the data is augmented with synthetic white, pink (1/f), and telegraph noise [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The effect of a QD charge sen- sor strongly coupled to the plunger gates is varied during the scan to improve performance on this type of experi- mental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The training dataset consists of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='6 × 105 devices with parameters varied over a uniform distribution with a standard deviation equal to 1 % of each parameter’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To train the ML models we randomly sample 10 small scans per device and use charge state ground truth to label each scan on a pixel level with the presence and type of transition, yielding NT, LT, CT, RT, and PL la- bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Additionally, we extract the slopes of the transition lines directly using the gradients of the device charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The test data is composed of eight simulated devices with large variations in screening length and device pitch 1× 5× 10× 15× 20× 25× 30× 35× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='6 RMSE (a) 1× 15× 30× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='6 (b) 1× 15× 30× Noise Level (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' (a) Root mean square error (RMSE) for all transition classes (left, central, and right [LT, CT, RT]) as a function of the synthetic noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' (b) RMSE as a function of noise level for the LT class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' (c) RMSE as a function of noise level for the RT class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' and with large shifts in the position of one of the plunger gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' These changes lead to large variations in the slopes of the charge transition lines, the capacitive coupling between QDs, spacing between lines, and the relative amount of left and right QD, making them largely dis- tinct from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To facilitate a controlled study and track the performance of the pixel classifier as data quality degrades, each large scan is randomly sampled 50 times and the resulting small scans are aug- mented with increasing levels of synthetic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' This results in a set of 400 simulated test scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Finally, sev- eral large experimental measurements acquired using a double-QD configuration on a three-QD Six/SiGe1−x de- vice, fabricated on an industrial 300 mm process line [1], are used to test the performance of the virtualization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Experimental scans capturing spurious QDs are used to demonstrate the algorithm for spurious QD detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' RESULTS We test the effectiveness of our automated approach to extracting the cross-capacitance by first evaluating the performance of each component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=', the pixel classifier and the slope extractions, on each scan in the simulated test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The error of the pixel classifier in our frame- work is defined as a fraction of pixels belonging to true transitions that are not contained in line segments in the CNN output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' This captures type-I errors without the 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='20 VP1(V) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='40 VP2(V) (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='55 VP ′ 1(V) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='70 VP ′ 2(V) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='35 VP ′′ 1 (V) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='30 VP ′′ 2 (V) (c) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='2 Virt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' gate off diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' (a) Large experimentally measured charge stability diagram with a scatter plot of centers of pixel class regions overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The colors of the points indicate the virtual gate off-diagonal values identified by fits to the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The sizes of the points indicate the weights used when averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Only points with relative error less than 20 % are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' (a,b) Charge stability diagram after applying virtual gates acquired near the (0, 0) − (1, 1) charge transition in (a) and near the (1, 3) − (2, 4) charge transition in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In both (b) and (c) the virtualization is performed off-line, via an affine transform to the original scan shown in (a) and the points are plotted using the same parameters as in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' effect of false type-II errors due to imperfect labels [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Figure 2(a) shows the change in RMSE as a function of the noise level in the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' At the noise level of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=', the noise level estimated from experimental data in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' [29], we observe an RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='17(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The RMSE increases significantly to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='50(11) at the noise level of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' For reference, a pixel classifier that always predicts the NT class would have an RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='62 ( √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' For the LT and RT classes relevant to cross-capacitances, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 2(b) and (c), the pixel classifier for noise level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0 has an RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='20(8) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='11(8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To verify that the slope extraction tool works as in- tended, we test it across the eight large simulated test devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' For these tests, we evaluate the pixel classifier in windows of size roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='5× the charging energy, as estimated by the spacing of the first two charge transi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Outputs from the pixel classifier are cropped by one pixel from the edge of the image before processing due to missing context reducing CNN performance [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The resulting classes of pixels are then grouped into dis- tinct clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' For each cluster consisting of more than five pixels an independent linear fit is performed, return- ing both the slope and the standard error of the fitted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' This information can be used to find the orthog- onal “virtual” control space or to flag transitions that potentially belong to spurious dots, as described in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Deriving virtual gates As stated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' II, in our approach the off-diagonal elements, defining the virtual gates transformation, are determined based on the slopes of the LT and RT cap- tured in a given image, and the diagonal elements of the capacitance matrix are set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' When multiple lines belonging to the same class are detected, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1(a), the capacitive coupling is calculated through a weighted average, with the weight accounting for both the size of the clusters and the standard deviations of respective fits [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The off-diagonal elements of the virtualization matrix computed this way have an RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='034(14) at the noise level of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0 defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' [29], corresponding to a roughly 8 % error compared to the ground truth values derived from simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' We further test them on a range of levels of synthetic noise and find the RMSE rises by a factor of two at a level of noise of roughly 15× the level of noise defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' [29], consistent with the pixel classifier error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To better understand the trends of the virtualization matrix in the plunger-plunger space, we carry out a per- formance analysis using the test set of eight large sim- ulated charge stability diagrams and several experimen- tally measured scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' For each scan, we calculate the fits to the pixel classification clusters based on a series of small scans sampled at each point within the large scan with the exclusion of a margin implemented to ensure that all sampled scans fall within the full scan bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The small scans and the margins are set to have a size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='5× the charging energy of a given simulated device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Figure 3(a) shows the centers of the pixel region identi- fied in each small scan [as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 1(c)] as the sampling window is swept across a large experimentally measured charge stability diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The regions identified by the pixel classification are consistently placed correctly on the charge transition lines regardless of the position of the 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='18 VP1(V) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='50 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' virt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' gate off-diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' (a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='34 VP2(V) (b) 0 1 2 3 4 Density ×10−2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Histograms of the off-diagonal elements of the virtu- alization matrix for an experimentally measured scan shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 3(a) as a function of plunger gates, (a) VP1 and (b) VP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Off-diagonal values are normalized to the mean of the virtual gates in the (1, 1) charge state for ease of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Virtual gate values are extracted from a strip of small scans shifted by 6 mV (four pixels) to better visualize variation at each plunger gate value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' line within a small scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Region centers shift along the charge transition lines as different portions of the line are captured within the small scan and remain fixed when- ever the same fragment of the charge transition is cap- tured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The color of the points indicates the off-diagonal values of the virtual gate matrix, α12 and α21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' As ex- pected, these coupling constants get larger in magnitude as charges are added to each QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Finally, the size of the points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 3(a) indicates the 1/σ2 weight of the slopes used when averaging multiple slopes from the same type of transition within a small scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' As desired, the posi- tions of the points with smaller sizes indicate that lines that are smaller or less captured within a small scan have fits with larger errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Overall, this plot confirms that the pixel classification and the fits are working as intended at capturing charge transition lines and their slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To demonstrate the spatial relevance of the virtual gates derived from a set of fits across a device’s charge landscape, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 3(b) and (c) we plot affine-transformed charge stability diagrams, with points indicating fits overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The points plotted are the centers of pixel re- gions with colors indicating the α12 and α21 values and size indicating the inverse of the fit error squared (the weight of the fit in the average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The affine transfor- mation applied in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 3(b) corresponds to virtual gates derived from an image near the (0, 0) − (1, 1) charge transition with off-diagonal values α12 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='282(4), α21 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='331(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' For Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 3(c), the affine transformation applied has virtual gates from the (1, 3) − (2, 4) charge transition, with off-diagonal values α12 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='363(4), α21 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='480(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' As can be seen in the insets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 3(b) and (c), these virtual gates are very effective at trans- forming the target region to an orthogonal space, but the difference between the extracted virtual gate off-diagonal values are about 50 % higher for the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' This highlights the importance of an efficient local method for determining virtual gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To further understand how capacitive coupling varies across a charge stability diagram, we can calculate varia- tion as each plunger gate is adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Figure 4(a) and (b) show how virtual gates extracted from small scans change as VP1 and VP2 are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' To better show the trend, vir- tual gates from small scans shifted by 3 mV (two pixels) in the opposing direction are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' This shows that the virtual gates extracted from small scans effectively capture variation across charge stability diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Detection of spurious dots Visually, spurious QDs are recognized in large 2D scans as charge transitions with slopes diverging from a mono- tonic trend, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In this framework, they may be identified as transition lines with anomalous capaci- tive couplings relative to the transitions around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' As a demonstration, we use the pixel classification and fit tools to analyze five experimental charge stability di- agrams: two capturing properly formed QD, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5(a) and (b), and three capturing spurious QDs, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5(a), (b), and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' While for extraction of the virtualization matrix small scans are sufficient, de- tection of spurious QD requires somewhat bigger scans to ensure that the neighboring charge transitions are ad- equately captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In our analysis, we rely on 2D scans of a size roughly three times the charging energy (four times the area of scans typically used in auto-tuning al- gorithms [13, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' We also consider only clusters con- sisting of at least 20 pixels to ensure better reliability of the linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' After pixel classification, contiguous clusters of pixels belonging to a given class of transitions are analyzed in- dividually, resulting in a cluster-based fit and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Cases where more than one cluster belongs to a given charge transition result in separate fits, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5(b) and (e) where the LT lines are split into groups to either side of the RT lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' This separation serves two purposes: to ensure that variation along a given transi- tion isn’t included and to treat each additional line inde- pendent of the charge on another QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Within a class and group of transitions, the magni- tude of the capacitive coupling is expected to increase monotonically as the charge is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Such behavior is clearly visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5(k) and (l), with the latter having to separate groups of fits (shown with different shades of green) for the groups of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' On the contrary, a spu- rious QD manifests itself by a non-monotonic behavior of the capacitive coupling between transitions, as depicted graphically by the center point (or group of points) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5(m), (n), and (o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The severity of this divergence can be quantified using a Z-test [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In practical applications, the automated detection of spurious QD fits nicely within the auto-tuning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' As mentioned earlier, many of the proposed approaches utilize a series of small 2D scans [13, 14, 27, 29] or 1D rays [41, 42] as means to improve the tuning effi- 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='39 VP2(V) (a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='39 (f) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='26 α12 (k) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='31 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='37 (g) NT LT CT RT PL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='3 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='52 VP1(V) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='49 (j) NT LT CT RT PL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='49 VP2(V) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='2 (o) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Spurious dot detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The top row shows two charge stability diagrams capturing properly formed QD [panels (a) and (b)] and three charge stability capturing spurious QD [panels (c), (d), and (e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The black boxes in (b) and (e) in highlight small 2D scans, denoted as VR1 and VR2, typical of the auto-tuning approaches proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Panels (f)–(j) in the middle row show pixel classification results for charge stability diagrams shown in the top row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Plots of fitting results used to determine whether a spurious QD is present are shown in the bottom row [panels (k)–(o)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The different groups of transition are shown with different shades of green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The monotonicity within each group of transitions is clearly visible in panels (k) and (l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' On the contrary, in the three plots shown in panels (m), (n), and (o), there is a clear divergence from the expected trend for the spurious QD (middle) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Error bars indicate one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' While these approaches deliver measurement- cost-effective solutions, they are prone to unexpected and difficult-to-detect failure even when the data quality is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5(b) and (e) show examples of such potentially problematic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The small 2D regions in the plunger- plunger space, highlighted in these scans with the black boxes, are typical for topology setting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In both cases, they are classified by a state classifier model as double QD state, with state prediction vectors being p(VR1) = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='92] for VR1 region high- lighted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5(b) and p(VR2) = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='18, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='76] for region VR2 highlighted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 5(b), where p(VR) = [pND, pSDL, pSDC, pSDR, pDD] with ND denoting no QDs formed, SDL, SDC, and SDR denoting the left, central, and right single QD, respectively, and DD denoting the double-QD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Moreover, the data quality for these images is high in both cases, with q(VR1) = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0] for region VR1 and q(VR1) = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='99, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='0] for VR2, where q(VR) = [phigh, pmod, plow] with phigh, pmod, and plow denoting the probability of region VR being assessed by the data quality control module as “high,”, “moder- ate,” and “low” quality, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Thus, from the ML perspective, both these predictions are confidently cor- rect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' However, when looked at within a slightly larger voltage range, it is clear that in the latter case the small scan captures an anti-crossing with a spurious QD, which for practical tuning purposes is a failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' If not recognized and corrected for, termination at this point will result in an incorrect charge setting and virtualiza- tion [13, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The spurious QD detection algorithm can be easily implemented in the auto-tuning algorithm proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' [27] as a safety check before the unloading step is ini- tiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' An automated identification and characterization of spurious QDs may also be useful to inform fabrication procedures and prevent them in future devices [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' CONCLUSIONS As quantum dot devices grow in size and complexity, the need for reliable and automated tune-up procedures becomes more pressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Establishing orthogonal con- trol of the chemical potentials of quantum dots is one of the first steps in the tune-up of any larger quantum dot array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Here, we demonstrated a method that com- bines machine-learning-based pixel-classification and tra- ditional curve fitting to reliably determine voltage cross- talk coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The advantage of this method over pre- vious approaches is highlighted by increased reliability 7 and resilience to experimental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Further on, un- wanted spurious dots that would reduce or inhibit de- vice performance can be detected and flagged when this module is used as part of a larger tune-up algorithm [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The capability to automatically and reliably detect spuri- ous dots is especially important on wafer-scale fabrication characterization tools that produce more data than can efficiently be processed by human analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' In extensions, our tools could allow for automated navigation of volt- age space for more targeted measurement of all chemical potential and tunnel barrier cross-capacitances [17, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' ACKNOWLEDGMENTS This research was performed while J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' held a NRC Research Associateship award at the National Institute of Standards and Technology (NIST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The views and con- clusions contained in this paper are those of the authors and should not be interpreted as representing the of- ficial policies, either expressed or implied, of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Government is authorized to reproduce and distribute reprints for Government pur- poses notwithstanding any copyright noted herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Any mention of commercial products is for information only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' it does not imply recommendation or endorsement by NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Pillarisetty, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Watson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Mueller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Henry, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' George, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Bojarski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Lampert, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Luthi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Kotl- yar, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Zietz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Neyens, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Borjans, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Caudillo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' Michalak, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} +page_content=' 18, 064040 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf'} diff --git a/INAyT4oBgHgl3EQfffiq/content/tmp_files/2301.00342v1.pdf.txt b/INAyT4oBgHgl3EQfffiq/content/tmp_files/2301.00342v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d5542cebf4d62d2dc447c7f5461d741b8906c86f --- /dev/null +++ b/INAyT4oBgHgl3EQfffiq/content/tmp_files/2301.00342v1.pdf.txt @@ -0,0 +1,855 @@ +Many-body collective neutrino oscillations: +recent developments +Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, +A. B. Balantekin +Abstract Neutrino flavor transformations in core-collapse supernovae and binary +neutron star mergers represent a complex and unsolved problem that is integral to +our understanding of the dynamics and nucleosynthesis in these environments. The +high number densities of neutrinos present in these environments can engender var- +ious collective effects in neutrino flavor transformations, driven either by neutrino- +neutrino coherent scattering, or in some cases, through collisional (incoherent) in- +teractions. An ensemble of neutrinos undergoing coherent scattering among them- +selves is an interacting quantum many-body system—as such, there is a tantalising +prospect of quantum entanglement developing between the neutrinos, which can +leave imprints on their flavor evolution histories. Here, we seek to summarize re- +cent progress that has been made towards understanding this phenomenon. +Amol V. Patwardhan +SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025 +e-mail: apatward@slac.stanford.edu +Michael J. Cervia +George Washington University, 725 21st St NW, Washington, DC 20052 +e-mail: cervia@gwu.edu +Ermal Rrapaj +University of California, Berkeley, CA 94720-7300 +e-mail: ermalrrapaj@berkeley.edu +Pooja Siwach +University of Wisconsin, 1150 University Ave, Madison, WI 53706 +e-mail: psiwach@physics.wisc.edu +A.B. Balantekin +University of Wisconsin, 1150 University Ave, Madison, WI 53706 +e-mail: baha@physics.wisc.edu +1 +arXiv:2301.00342v1 [hep-ph] 1 Jan 2023 + +2 +Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin +Motivation: Supernovae, Mergers, and the Early Universe +In extreme astrophysical environments such as core-collapse supernova explosions, +and binary neutron star (or black hole - neutron star) mergers, as well as during +certain epochs in the early universe, neutrinos dominate the transport of energy, +entropy, and lepton number (for example, see [Janka et al., 2007, Burrows and +Vartanyan, 2021, Fuller and Haxton, 2022, Foucart, 2022, Kyutoku et al., 2018, +Grohs et al., 2016], etc.). The key processes governing neutrino transport in these +environments are electron neutrino and antineutrino captures on nucleons, i.e., +νe +n ⇌ p+e− +(1) +¯νe + p ⇌ n+e+ +(2) +A consequence of the typical temperatures and densities of these environments is +that neutrinos decouple with energies of O(1–10)MeV, and therefore, the µ and τ +flavor (anti-)neutrinos are unable to participate in these charged-current processes, +due to there not being enough energy to produce µ and τ leptons in the final state. +Given the importance of these processes in the energy transport, as well as in de- +termining the neutron-to-proton ratio and the resulting nucleosynthesis prospects +(e.g., [Surman and McLaughlin, 2004, Mart´ınez-Pinedo et al., 2017, Kajino et al., +2014, Fr¨ohlich et al., 2015, Langanke et al., 2019, Roberts et al., 2017, Steigman, +2012, Grohs et al., 2016]), the flavor-asymmetric nature of charged-current capture +necessitates a thorough understanding of neutrino flavor evolution in these envi- +ronments. The potential impact of neutrino flavor evolution on nucleosynthesis has +already been studied in various contexts (e.g., [Qian et al., 1993, Yoshida et al., +2006, Duan et al., 2011, Kajino et al., 2012, Wu et al., 2015, Sasaki et al., 2017, Bal- +antekin, 2018, Xiong et al., 2019, Xiong et al., 2020]). +In what follows, we shall summarize recent progress in our understanding of +a particular facet of neutrino oscillations in extreme astrophysical environments— +namely, the quantum many-body nature of collective neutrino oscillations engen- +dered by ν-ν interactions in dense neutrino streams. +Introduction to collective neutrino oscillations +The neutral current weak term of the Standard Model (SM) allows neutrinos to +interact pairwise via virtual Z-boson exchange or, more simply, in the low-energy +effective theory, via the Fermi four-point interaction +Hint ≡ GF +√ +2 ∑ +f,g +νgγµνgν f γµνf , +(3) +where f,g span the flavor state indices. The relevance of these interactions in en- +vironments where the number densities of neutrinos are comparable to (or larger + +Many-body collective neutrino oscillations: recent developments +3 +than) those of charged leptons, e.g., in core-collapse supernovae, binary neutron +star mergers, as well as in the early universe, had been discussed in [Notzold and +Raffelt, 1988, Fuller et al., 1987]. But the extent of their importance in changing +the flavor content of neutrinos, via diagonal and off-diagonal contributions to the +neutrino Hamiltonian, was not fully recognized until later [Pantaleone, 1992a, Pan- +taleone, 1992b, Samuel, 1993]. +Considering pairs of neutrinos with well-defined incoming momenta p and q +(i.e., plane wave states) and the same pair of outgoing momenta (i.e., “forward +scattering” neutrinos, the contributions of which can be added coherently), the off- +diagonal matrix elements of the interaction Hamiltonian Hint may be interpreted as +arising from “flavor swaps” between neutrino pairs (in the flavor basis). Because +the off-diagonal term exchanges flavor between the “test” and the “background” +neutrinos, the flavor evolution of the interacting neutrinos constitutes a many-body +problem, potentially rendering the one-particle propagation formalism [Samuel, +1993, Sigl and Raffelt, 1993, Qian and Fuller, 1995] inadequate for describing +the resulting dynamics. Notably, the interaction Hamiltonian Hint does not com- +mute with the Hamiltonian terms corresponding to flavor oscillations in vacuum +and neutrino interactions with background matter. Consequently, in a regime where +the strength of these terms is comparable in scale to the ν-ν interaction strength, +diagonalizing this Hamiltonian is not straightforward and the many-body problem +acquires a nontrivial nature. Here, the entire Hilbert space of N interacting neutrinos +and antineutrinos in nf flavors has dimension nN +f . +Despite emphasis on the high nonlinearity of this problem, [Samuel, 1993] had +proposed that a statistical mechanical approach, whereby a two-flavor neutrino den- +sity matrix is treated as interacting with a background of neutrinos and antineutri- +nos, could describe the evolution of a dense neutrino gas for certain portions of this +parameter space. This analysis was extended by [Sigl and Raffelt, 1993] to nf ≥ 2 +flavors with proposed evolution of nf ×n f density matrices via quantum Boltzmann +equations, including collision integrals as well as more general, potentially non-SM +coupling between flavors. In these treatments, the collisional contributions can lead +to a nontrivial loss of coherence being reflected in the density matrices of individual +neutrinos. However, the ability to calculate multi-body wave functions that exhibit +ν-ν correlations is relinquished, in exchange for a more favourable scaling of com- +putational complexity with the number of neutrinos in the simulation. Along these +lines, [Qian and Fuller, 1995] proposed a physical ansatz that the wave function of +the ensemble is not a coherent many-body state, but simply composed of single- +neutrino wave functions with random relative phases, to be summed incoherently, +called the Random Phase Approximation (RPA). In this way, each neutrino density +matrix is taken to be pure, and the effective Hilbert space dimension is reduced to +nf N. In kind, the complexity of collective oscillations calculations becomes greatly +simplified. This ansatz amounts to a “mean field approximation” wherein expecta- +tion values of operator products may be replaced by products of the individual op- +erator expectation values. Notably, this physical description of neutrinos expressly +prohibits the quantum entanglement between neutrinos. As such, assessing the va- +lidity of this ansatz involves determining the extent to which quantum effects are + +4 +Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin +needed to correct this approximation. In this chapter, we discuss recent progress +along this front. +Before delving into the chapter, we mention in passing that recent years have seen +a rapid growth of interest in flavor instabilities and resulting fast flavor oscillation, +even within the scope of the mean field approximation. For more information we +refer the reader to the chapter on “Fast Flavor Transformations” by [Richers and Sen, +2022], or the review articles by [Chakraborty et al., 2016, Tamborra and Shalgar, +2021]. +Many-body Hamiltonian for interacting neutrinos +The Hamiltonian describing a system of interacting neutrinos can be written in terms +of generators of SU(nf ), and it possesses a SU(nf ) rotation symmetry in neutrino +flavor. A significant feature of ν-ν interactions is the dependence of the interac- +tion strength on the intersection angle between their trajectories. This dependence +introduces a geometric complexity to the problem, in addition to the complexity +associated with the exponential scaling of the Hilbert space. +For simplicity, if we consider neutrino mixing between only two flavors, νe and +νx, then a Hamiltonian consisting of terms that represent vacuum mixing as well as +ν-ν interactions can be written as +H = ∑ +p +ωp⃗B· ⃗Jp + +√ +2GF +V +∑ +p,q +(1−�p·�q) ⃗Jp · ⃗Jq , +(4) +where ⃗B = (0,0,−1) in the mass-basis representation, and ωp = δm2/(2|p|) are +the vacuum oscillation frequencies for neutrinos with momenta p. Here, �p and �q +are the unit vectors along the momenta of the interacting neutrinos, and V is the +quantization volume. For ease of notation, one can define a ν-ν coupling parameter +µ ≡ +√ +2GFN/V, where N is the total number of interacting neutrinos. The oper- +ators ⃗Jp represent the neutrino “isospin” in the mass basis, where isospin up and +down correspond to the mass basis states |ν1⟩ and |ν2⟩. In this depiction, ⃗B can be +interpreted as a “background field” with which the neutrino isospins interact. Here, +we exclude the term representing neutrino interactions with ordinary matter (e.g., +charged leptons), since it has a structure that is conceptually similar to the vacuum +oscillation term—i.e., consisting of individual neutrinos interacting with a back- +ground. In contrast, the ν-ν interaction term consists of pairs of neutrino isospins +interacting with each other. +In terms of the Fermionic creation and annihilation operators, the neutrino +isospins are described as [Balantekin and Pehlivan, 2006] +J+ +p = a† +1(p)a2(p) , +Jz +p = 1 +2 +� +a† +1 (p)a1(p)−a† +2 (p)a2(p) +� +, +(5) + +Many-body collective neutrino oscillations: recent developments +5 +with J− +p = (J+ +p )†. In the spin-1/2 representation, one can write the isospin operators +in terms of Pauli matrices: i.e., ⃗Jp = ⃗σp/2, where σp is a vector of Pauli matrices +defined in the subspace of the neutrino with momentum p. +Path integral formulation +An assessment of quantum corrections to a mean field picture can in principle be +performed via a coherent state analysis, as formulated by [Balantekin and Pehlivan, +2006]. Schematically, in this procedure, one seeks to calculate the matrix elements +of the time evolution operator U(tf ;ti) for a single neutrino in the basis of SU(nf ) +coherent states |z⟩ for neutrinos (and/or antineutrinos) in n f flavors, equivalent to a +path integral +⟨zf |U(t f ;ti)|zi⟩ = +� +D[z,z∗] exp(iS[z,z∗]) +(6) +of the derived action +S[z,z∗] = +� tf +ti +dt +� +⟨z(t)|(i∂t −H)|z(t)⟩−ilog⟨z f |zi⟩ +� +, +(7) +where H is the Hamiltonian of the many-body system. A saddle-point approxima- +tion of the resulting path integral yields the classical action, which is in complete +agreement with the RPA used to derive the mean field theory for collective neutrino +oscillations. However, in this perspective, analyzing quantum corrections to this ap- +proximation entails a careful analysis of the Hessian matrix of the action integral +derived from this procedure. Such mathematical analysis has not yet been presented +to date. +Beyond the Mean-Field: Entanglement, Correlations, and +Dynamical Phase Transitions +Early literature +The seminal work describing the ν-ν interaction Hamiltonian from Eq. (3) recog- +nized that these interactions give rise to a quantum many-body problem, which may +not in the general case be factorizable in terms of a one-particle effective approxi- +mation [Pantaleone, 1992a, Pantaleone, 1992b]. Subsequently, there were some at- +tempts to ascertain the validity of the one-particle effective approximation [Bell +et al., 2003, Friedland and Lunardini, 2003b, Friedland and Lunardini, 2003a, Fried- +land et al., 2006]. In these works, the flavor evolution of interacting neutrinos was +analyzed with two different approaches: (i) using two intersecting beams of neutri- + +6 +Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin +nos, where the flavor evolution was described in terms of a sequence of elementary +scattering amplitudes, and (ii) using a neutrino ensemble represented as interacting +plane waves in a box. +Following initial disagreement regarding whether substantial quantum entangle- +ment can develop among interacting neutrinos [Bell et al., 2003, Friedland and Lu- +nardini, 2003b], it was subsequently concluded that the build-up of entanglement +and resulting flavor conversion would occur on timescales whose scaling is sugges- +tive of incoherent effects [Friedland and Lunardini, 2003a]. These conclusions were +further generalized in [Friedland et al., 2006]. However, these analyses nevertheless +involved several simplifications, most notably, the omission of the one-body terms +in the Hamiltonian. The interplay between vacuum oscillations and ν-ν interaction +terms has been shown to give rise to interesting collective phenomena such as “spec- +tral splits” [Duan et al., 2006a, Duan et al., 2006b, Duan et al., 2007b, Raffelt and +Smirnov, 2007b, Raffelt and Smirnov, 2007a], even in the mean-field approxima- +tion. Therefore, studying the quantum many-body dynamics of collective neutrino +oscillations, with both one- and two-body terms fully incorporated, remains an in- +teresting problem. +With these seemingly conflicting results in the past predicting either a vanish- +ingly small contribution in the large system size limit [Friedland and Lunardini, +2003a, Friedland et al., 2006] or substantial flavor evolution over time scales τF ∼ +µ−1 log(N) that can remain relevant for large systems [Bell et al., 2003, Sawyer, +2004], the role of entanglement and quantum effects in the out-of-equilibrium dy- +namics [Eisert et al., 2015] of neutrinos has received renewed interest recently (e.g., +[Cervia et al., 2019, Rrapaj, 2020] and subsequent works mentioned later in this +chapter). Note that flavor oscillations on the time scale τF can be considered to be +“fast”, different from “slow” oscillations occurring over τL ∼ µ−1√ +N. In the lit- +erature on collective flavor effects in the mean field approximation, one can more +commonly find “fast” and “slow” oscillations associated with time scales ∼ µ−1 +and ∼ √µω (or ω), respectively. +Single-angle approximation, invariants, and integrability +To circumvent the geometric complexity of the problem, the frequently-employed +single-angle approximation replaces the angle-dependent (i.e., �p,�q-dependent) in- +teraction strengths among pairs of neutrinos with a single, appropriately chosen +classical average over the various neutrino trajectories. In this limit, one can de- +fine a trajectory-averaged interaction parameter µ ≡ ( +√ +2GFN/V)⟨1 − �p · �q⟩, and +approximate the Hamiltonian as +H = ∑ +ωp +ωp⃗B· ⃗Jωp + µ +N +⃗J · ⃗J , +(8) + +Many-body collective neutrino oscillations: recent developments +7 +where ⃗J = ∑ωp ⃗Jωp is the total neutrino isospin. Note that, in this limit, the neutrino +flavor state becomes trajectory-independent, introducing a considerable simplifica- +tion in the problem. As a result, the neutrinos may be indexed simply by the mag- +nitudes of their momenta (or equivalently, by their vacuum oscillation frequencies +ωp), rather than by the momenta themselves (magnitude and direction). The ν-ν +coupling in general will depend on time. In the context of supernovae, a commonly +employed expression for µ is derived from the spherically symmetric single-angle +neutrino bulb model, first described in [Duan et al., 2006c]: +µ(r) = µ0 +� +�1− +� +1− +�Rν +r +�2 +� +� +2 +, +(9) +where r is the distance from the center of a “neutrino-sphere” of radius Rν, which +represents a sharp surface where neutrinos decouple from nuclear matter and be- +gin free streaming outwards from the proto-neutron star. We also define µ0 ≡ +(GF/ +√ +2)(N/V) = µ(Rν) to be the interaction strength at the neutrino-sphere. Here, +the neutrino emission is assumed to be time-invariant over the short time scales as- +sociated with neutrino propagation through the supernova envelope, so the interac- +tion strength depends explicitly only on position, rather than time. In the neutrino- +driven wind phase of core-collapse supernovae, which occurs over a time window +of O(1–10) s after core bounce, one may expect Rν ≃ 20km and µ0 ∼ 105ω0, where +ω0 ∼ 10−16 MeV is the scale of the vacuum oscillations. During the shock breakout +or “neutronization burst” phase that occurs earlier, around 10 ms after core bounce, +the proto-neutron star can be more extended, with Rν ≳ 50–60 km, but the neutrino +luminosity is also much higher, resulting in µ0 ∼ 106ω0. +It has been shown that a single-angle Hamiltonian describing neutrino mixing +in vacuum and ν-ν interactions possesses a number of conserved charges which +commute with the Hamiltonian [Pehlivan et al., 2011]. These are analogous to +the “Gaudin magnets” [Gaudin, M., 1976] that had been previously identified as +the conserved charges of the pairing-force Hamiltonian in nuclear and condensed- +matter physics [Richardson, 1963, Richardson and Sherman, 1964, Richardson, +1965]. These conserved charges are related to the integrability of the Hamiltonian— +meaning that it is possible to obtain, in principle, exact eigenvalues and eigenstates +of this Hamiltonian in terms of closed-form solutions to a set of algebraic “Bethe- +Ansatz” equations [Bethe, 1931]. Based on these ideas, specific procedures for the +eigen-decomposition of a single-angle neutrino Hamiltonian have been outlined in +the literature [Pehlivan et al., 2011, Birol et al., 2018, Patwardhan et al., 2019]. +Besides descriptions in terms of instantaneously conserved charges, analogies +with other many-body problems have been fruitful to yield an explanation of the +neutrino flavor spectral split in terms of a Bardeen-Cooper-Schrieffer (BCS)-Bose- +Einstein Condensate (BEC) crossover-like phenomenon [Pehlivan et al., 2017], as +well as to help provide many-body predictions of a spectral split [Birol et al., 2018] +specifically in the case of an initial many-body wave function with all neutrinos in +the electron flavor state. + +8 +Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin +Instabilities and dynamical phase transitions +Collective neutrino oscillations are generally assumed to be caused by unstable +modes in the mean field dynamics generated by the Hamiltonian described in +Eq. (4) (for two flavors). These instabilities are able to amplify initially small +flavor perturbations exponentially fast (e.g., [Sawyer, 2004, Sawyer, 2005, Duan +et al., 2010, Chakraborty et al., 2016, Izaguirre et al., 2017, Tamborra and Shalgar, +2021, Richers and Sen, 2022] and references therein). The presence of the forward- +scattering interaction can allow collective effects to develop when µ ≳ ωp, giving +rise to interesting phenomena like synchronization [Pastor et al., 2002, Fuller and +Qian, 2006, Raffelt and Tamborra, 2010, Akhmedov and Mirizzi, 2016], bipolar os- +cillations [Kosteleck´y and Samuel, 1995, Duan et al., 2006c, Duan et al., 2007a] and +spectral splits/swaps [Duan et al., 2006b, Duan et al., 2007b, Raffelt and Smirnov, +2007b, Dasgupta et al., 2009, Martin et al., 2020]. +On the other hand, in descriptions of interacting neutrino systems that per- +mit many-body quantum dynamics, oscillations that develop on “fast” timescales +are generally associated with rapid dynamical development of the neutrino en- +tanglement entropy [Cervia et al., 2019, Rrapaj, 2020, Roggero, 2021a, Roggero, +2021b, Patwardhan et al., 2021]. The dynamically generated entanglement between +neutrinos is seen to be correlated with deviations from the mean-field dynamics of +the system [Cervia et al., 2019, Rrapaj, 2020] and with the presence of spectral splits +in the neutrino energy distributions [Patwardhan et al., 2021]. An example of such +a calculation is depicted in Fig. 1. In [Roggero et al., 2022], rapid entanglement and +mean field instabilities were also found to be linked for certain angular setups. +As shown in [Roggero, 2021a, Roggero, 2021b] in the single angle approxima- +tion, when the frequency difference between two neutrino beams (δω) is positive +and comparable to the ν-ν interaction coupling (µ), 0 < δω ≲ µ, rapid and strong +flavor oscillations develop. This rather particular finding can be understood in terms +of the presence of a Dynamic Phase Transition (DPT) [Heyl et al., 2013, Heyl, +2018], which can be characterized by the introduction of the Loschmidt echo, +L (t) = |⟨Φ|exp(−itH)|Φ⟩|2 , +(10) +with |Φ⟩ the initial state at t = 0. The quantity L (t) is a fidelity measure [Gorin +et al., 2006] that quantifies the probability for the system to return to its initial state. +A DPT is then characterized by non-analyticities in the rate function +λ(t) = − 1 +N log[L (t)] , +(11) +where N is the total number of particles in the system and λ(t) an intensive “free +energy” [Heyl et al., 2013, Gambassi and Silva, 2012]. Here, the rate λ(t) plays the +role of a non-equilibrium equivalent of the thermodynamic free-energy. Notably, +other definitions of DPT are possible, for instance, time-averaged order parame- +ters [Sciolla and Biroli, 2011, Sciolla and Biroli, 2013, ˇZunkoviˇc et al., 2018]. + +Many-body collective neutrino oscillations: recent developments +9 +−1 +−0.8 +−0.6 +−0.4 +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +200 +500 +1000 +2000 +P MB +z +(ωp) +r (in units of ω−1 +0 ) +Pz(ω1) +Pz(ω2) +Pz(ω3) +Pz(ω4) +Pz(ω5) +Pz(ω6) +Pz(ω7) +Pz(ω8) +−1 +−0.8 +−0.6 +−0.4 +−0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +200 +500 +1000 +2000 +P MF +z +(ωp) +r (in units of ω−1 +0 ) +Pz(ω1) +Pz(ω2) +Pz(ω3) +Pz(ω4) +Pz(ω5) +Pz(ω6) +Pz(ω7) +Pz(ω8) +0 +0.2 +0.4 +0.6 +0.8 +1 +200 +500 +1000 +2000 +S(ωp) +r (in units of ω−1 +0 ) +S(ω1) +S(ω2) +S(ω3) +S(ω4) +S(ω5) +S(ω6) +S(ω7) +S(ω8) +−1 +−0.5 +0 +0.5 +1 +1 +2 +3 +4 +5 +6 +7 +8 +0 +0.2 +0.4 +0.6 +Pz(ωp) +S(ωp) +ω (in units of ω0) +Pz (initial) +P MB +z +(nal) +P MF +z +(nal) +S(ωp) (nal) +Fig. 1 Evolution of an initial state |νe⟩⊗4 |νx⟩⊗4 from a starting radius r0 such that µ(r0) = 5ω0, +with a small mixing angle (θ = 0.161) and discrete, equally spaced oscillation frequencies +ωk = kω0, and a time-varying neutrino interaction strength µ(r) motivated by the neutrino bulb +model [Duan et al., 2006b], in the single-angle approximation according to Eqs. (8) and (9). Details +of this calculation can be found in [Cervia et al., 2019]. Top left: Evolution of the z-components +of the neutrino isospin expectation values (also known as “Polarization vectors”) in the mass basis, +i.e., Pz ≡ 2⟨Jz⟩, for the full many-body quantum system. Top right: Same as top left, but in the +mean-field approximation. Bottom left: Evolution of the entanglement entropy of each neutrino, +with respect to the rest of the ensemble. Bottom right: Asymptotic values of Pz vs ωk, in the full +many-body calculation (purple), and in the mean-field approximation (green), together with the +initial Pz values (red), and the asymptotic entanglement entropies (dark orange). Neutrinos located +closest to the spectral splits in the energy distributions (in this case, at ω2 and ω7) develop the +largest amount of entanglement and thereby experience the most significant deviations compared +to their mean-field evolution. +Phase-space analysis +In a recent work [Lacroix et al., 2022], this problem was further explored by ana- +lyzing the evolution of neutrino flavor and entanglement in phase space. The setup +consisted of two sets (beams) of neutrinos interacting with each other. In this anal- +ysis, the Husimi quasi-probability or “Q” representation [Husimi, 1940] was con- +structed for the reduced density operator of neutrinos in one of the beams, using an +over-complete basis of coherent states. In the limit of infinite neutrino number, the +Q representation acquires the interpretation of a classical phase-space probability +distribution. + +10 +Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. Balantekin +For this two-beam interacting neutrino system, it was demonstrated that, while +at early times the quasi-probability distribution remains relatively localized, at late +times it develops a multi-modal structure with several localized peaks. This delocal- +ization is indicative of non-Gaussian entanglement, which suggests that any approx- +imate method beyond the mean-field relying on only the first and second moments +of neutrino observables may not be sufficient to describe the long-term evolution +of this system. Based on the phase space analysis, a new method for approximat- +ing the exact evolution of the interacting neutrino system was proposed, wherein +the quantum mechanical many-body evolution is replaced by a statistical average of +‘mean-field’ solutions, with a Gaussian distribution of initial conditions around the +exact starting point of the system [Lacroix and Ayik, 2014]. +Compact Representations for studying many body effects +Still allowing for possibilities of mixed one-neutrino density matrices, one pro- +posal [Volpe et al., 2013] to determine quantum corrections is to systematically +incorporate n-body density matrices ρ1...n for n ≥ 1, given by +ρ1...n = +N! +(N −n)!Trn+1...Nρ1...N, +(12) +into the coupled equations of motion for N neutrinos, as follows: +i∂tρ1...n = [H1...n,ρ1...n]+ +n +∑ +s=1 +Trn+1[V(s,n+1),ρ1...n+1], +(13) +where H1...n is the Hamiltonian truncated for the first n neutrinos in a given ordering +and V(i, j) is the two-body interaction potential for a pair of neutrinos (i, j). This +procedure is based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hi- +erarchy for density matrices. Here, the mean field theory interaction of neutrinos +and antineutrinos with the background gas is reproduced by restricting to n = 2 +and estimating ρ12 ≈ ρ1ρ2 (i.e., requiring the two-body correlation function to be +zero) in this picture, in a sense as a loop Feynman diagram for neutrino propagation. +In principle, investigating the importance of quantum corrections would practically +entail checking for convergence of results for physical observables as the n-body +correlation functions are incorporated for progressively increasing values of n in the +BBGKY hierarchy. +Owing to the exponential growth in the Hilbert space, classical (conventional) +computers are unable to exactly simulate systems of more than ≃ 20 neutrinos. +To overcome this difficulty, one can resort to compact representations of the wave- +function through tensor network methods [Roggero, 2021a, Roggero, 2021b, Cervia +et al., 2022], and more specifically matrix product states [Vidal, 2003, Schollw¨ock, +2011, Paeckel et al., 2019]. In simplified setups, these methods allow for the com- +putation of systems of hundreds of neutrinos. Alternatively, when considering very + +Many-body collective neutrino oscillations: recent developments +11 +dense neutrino gases (vacuum oscillations can be ignored), methods based on gen- +eralized angular momentum representations, by analogy between two flavor oscilla- +tions and spin systems, can reach up to thousands of neutrinos and predict the ther- +modynamic limit [Friedland and Lunardini, 2003a, Friedland et al., 2006, Xiong, +2022, Roggero et al., 2022]. +In the case of time-dependent interaction strength and all-to-all ν-ν interactions, +the more sophisticated tensor network method, namely, the time-dependent varia- +tional principle (TDVP) method has been utilized in [Cervia et al., 2022]. These +techniques provided considerable computational benefit for an initial state with all +neutrinos in the same flavor, allowing for evolution of a system with ≈ 50 oscil- +lation modes. This was a consequence of the entanglement among neutrinos being +more localized in certain regions of the neutrino energy distribution. For systems +with initial states being a mixture of νe and νx flavors, the entanglement is more de- +localized, and therefore, the comparative advantage gained through TDVP methods +is less dramatic, although work remains in progress on this front. +For a general setup, quantum computers are a promising tool to solve the quan- +tum many-body problem. Initial steps [Hall et al., 2021, Yeter-Aydeniz et al., +2022, Illa and Savage, 2022, Amitrano et al., 2022] to simulate the collective neu- +trino oscillations on a quantum computer are already taken in this direction. In [Hall +et al., 2021] a sytem of four neutrinos was simulated on IBM’s quantum devices +using the real-time evolution. The unitary evolution operator U(t) = exp(−iHt) +was decomposed using the first order Trotter-Suzuki decomposition, where error +scales as O(t2). Since the interaction is long-range, a device with all-to-all con- +nectivity among qubits is preferred. As an alternative, SWAP operations have been +used to implement this interaction on a quantum device having connectivity among +neighboring qubits [Hall et al., 2021]. In [Yeter-Aydeniz et al., 2022], the hybrid +quantum-classical algorithm QLanczos (quantum Lanczos) was used to calculate +the eigenvalues of neutrino many-body interaction Hamiltonian [Patwardhan et al., +2019] on a quantum computer. Furthermore, the transition probabilities of collec- +tive neutrino oscillations were obtained by performing the real-time evolution using +trotterization. However, all these earlier quantum computing studies were limited to +a small system of four neutrinos due to constraints in the form of currently avail- +able quantum devices, which can perform only a limited number of operations with +low accuracy. More recently in [Amitrano et al., 2022], a trapped-ion quantum de- +vice was utilized to perform the simulations for up to eight neutrinos, thanks to the +all-to-all qubit connectivity in trapped-ion based architecture. +Concluding remarks +Studying the many-body quantum dynamics of dense neutrino systems remains an +active area of research, with various groups attempting to investigate the problem +using different types of classical and quantum computational tools, as well as ana- +lytic or semi-analytic descriptions. In environments where neutrinos are present in + +12 +Amol V. Patwardhan, Michael J. Cervia, Ermal Rrapaj, Pooja Siwach, A. B. 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Lett., 96:091101, astro-ph/0602195. + diff --git a/INAyT4oBgHgl3EQfffiq/content/tmp_files/load_file.txt b/INAyT4oBgHgl3EQfffiq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ae2c2c3caf866ea0e644cae91f9ada5944b9153 --- /dev/null +++ b/INAyT4oBgHgl3EQfffiq/content/tmp_files/load_file.txt @@ -0,0 +1,1302 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf,len=1301 +page_content='Many-body collective neutrino oscillations: recent developments Amol V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Patwardhan, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Cervia, Ermal Rrapaj, Pooja Siwach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Balantekin Abstract Neutrino flavor transformations in core-collapse supernovae and binary neutron star mergers represent a complex and unsolved problem that is integral to our understanding of the dynamics and nucleosynthesis in these environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The high number densities of neutrinos present in these environments can engender var- ious collective effects in neutrino flavor transformations, driven either by neutrino- neutrino coherent scattering, or in some cases, through collisional (incoherent) in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' An ensemble of neutrinos undergoing coherent scattering among them- selves is an interacting quantum many-body system—as such, there is a tantalising prospect of quantum entanglement developing between the neutrinos, which can leave imprints on their flavor evolution histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Here, we seek to summarize re- cent progress that has been made towards understanding this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Amol V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Patwardhan SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025 e-mail: apatward@slac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='edu Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Cervia George Washington University, 725 21st St NW, Washington, DC 20052 e-mail: cervia@gwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='edu Ermal Rrapaj University of California, Berkeley, CA 94720-7300 e-mail: ermalrrapaj@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='edu Pooja Siwach University of Wisconsin, 1150 University Ave, Madison, WI 53706 e-mail: psiwach@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='edu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Balantekin University of Wisconsin, 1150 University Ave, Madison, WI 53706 e-mail: baha@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='00342v1 [hep-ph] 1 Jan 2023 2 Amol V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Patwardhan, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Cervia, Ermal Rrapaj, Pooja Siwach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Balantekin Motivation: Supernovae, Mergers, and the Early Universe In extreme astrophysical environments such as core-collapse supernova explosions, and binary neutron star (or black hole - neutron star) mergers, as well as during certain epochs in the early universe, neutrinos dominate the transport of energy, entropy, and lepton number (for example, see [Janka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2007, Burrows and Vartanyan, 2021, Fuller and Haxton, 2022, Foucart, 2022, Kyutoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2018, Grohs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2016], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The key processes governing neutrino transport in these environments are electron neutrino and antineutrino captures on nucleons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', νe +n ⇌ p+e− (1) ¯νe + p ⇌ n+e+ (2) A consequence of the typical temperatures and densities of these environments is that neutrinos decouple with energies of O(1–10)MeV, and therefore, the µ and τ flavor (anti-)neutrinos are unable to participate in these charged-current processes, due to there not being enough energy to produce µ and τ leptons in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Given the importance of these processes in the energy transport, as well as in de- termining the neutron-to-proton ratio and the resulting nucleosynthesis prospects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', [Surman and McLaughlin, 2004, Mart´ınez-Pinedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2017, Kajino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2014, Fr¨ohlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2015, Langanke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2019, Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2017, Steigman, 2012, Grohs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2016]), the flavor-asymmetric nature of charged-current capture necessitates a thorough understanding of neutrino flavor evolution in these envi- ronments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The potential impact of neutrino flavor evolution on nucleosynthesis has already been studied in various contexts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', [Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 1993, Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2011, Kajino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2012, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2015, Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2017, Bal- antekin, 2018, Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2019, Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2020]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In what follows, we shall summarize recent progress in our understanding of a particular facet of neutrino oscillations in extreme astrophysical environments— namely, the quantum many-body nature of collective neutrino oscillations engen- dered by ν-ν interactions in dense neutrino streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Introduction to collective neutrino oscillations The neutral current weak term of the Standard Model (SM) allows neutrinos to interact pairwise via virtual Z-boson exchange or, more simply, in the low-energy effective theory, via the Fermi four-point interaction Hint ≡ GF √ 2 ∑ f,g νgγµνgν f γµνf , (3) where f,g span the flavor state indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The relevance of these interactions in en- vironments where the number densities of neutrinos are comparable to (or larger Many-body collective neutrino oscillations: recent developments 3 than) those of charged leptons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', in core-collapse supernovae, binary neutron star mergers, as well as in the early universe, had been discussed in [Notzold and Raffelt, 1988, Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 1987].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' But the extent of their importance in changing the flavor content of neutrinos, via diagonal and off-diagonal contributions to the neutrino Hamiltonian, was not fully recognized until later [Pantaleone, 1992a, Pan- taleone, 1992b, Samuel, 1993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Considering pairs of neutrinos with well-defined incoming momenta p and q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', plane wave states) and the same pair of outgoing momenta (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', “forward scattering” neutrinos, the contributions of which can be added coherently), the off- diagonal matrix elements of the interaction Hamiltonian Hint may be interpreted as arising from “flavor swaps” between neutrino pairs (in the flavor basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Because the off-diagonal term exchanges flavor between the “test” and the “background” neutrinos, the flavor evolution of the interacting neutrinos constitutes a many-body problem, potentially rendering the one-particle propagation formalism [Samuel, 1993, Sigl and Raffelt, 1993, Qian and Fuller, 1995] inadequate for describing the resulting dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Notably, the interaction Hamiltonian Hint does not com- mute with the Hamiltonian terms corresponding to flavor oscillations in vacuum and neutrino interactions with background matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Consequently, in a regime where the strength of these terms is comparable in scale to the ν-ν interaction strength, diagonalizing this Hamiltonian is not straightforward and the many-body problem acquires a nontrivial nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Here, the entire Hilbert space of N interacting neutrinos and antineutrinos in nf flavors has dimension nN f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Despite emphasis on the high nonlinearity of this problem, [Samuel, 1993] had proposed that a statistical mechanical approach, whereby a two-flavor neutrino den- sity matrix is treated as interacting with a background of neutrinos and antineutri- nos, could describe the evolution of a dense neutrino gas for certain portions of this parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' This analysis was extended by [Sigl and Raffelt, 1993] to nf ≥ 2 flavors with proposed evolution of nf ×n f density matrices via quantum Boltzmann equations, including collision integrals as well as more general, potentially non-SM coupling between flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In these treatments, the collisional contributions can lead to a nontrivial loss of coherence being reflected in the density matrices of individual neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' However, the ability to calculate multi-body wave functions that exhibit ν-ν correlations is relinquished, in exchange for a more favourable scaling of com- putational complexity with the number of neutrinos in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Along these lines, [Qian and Fuller, 1995] proposed a physical ansatz that the wave function of the ensemble is not a coherent many-body state, but simply composed of single- neutrino wave functions with random relative phases, to be summed incoherently, called the Random Phase Approximation (RPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In this way, each neutrino density matrix is taken to be pure, and the effective Hilbert space dimension is reduced to nf N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In kind, the complexity of collective oscillations calculations becomes greatly simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' This ansatz amounts to a “mean field approximation” wherein expecta- tion values of operator products may be replaced by products of the individual op- erator expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Notably, this physical description of neutrinos expressly prohibits the quantum entanglement between neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' As such, assessing the va- lidity of this ansatz involves determining the extent to which quantum effects are 4 Amol V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Patwardhan, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Cervia, Ermal Rrapaj, Pooja Siwach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Balantekin needed to correct this approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In this chapter, we discuss recent progress along this front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Before delving into the chapter, we mention in passing that recent years have seen a rapid growth of interest in flavor instabilities and resulting fast flavor oscillation, even within the scope of the mean field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' For more information we refer the reader to the chapter on “Fast Flavor Transformations” by [Richers and Sen, 2022], or the review articles by [Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2016, Tamborra and Shalgar, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Many-body Hamiltonian for interacting neutrinos The Hamiltonian describing a system of interacting neutrinos can be written in terms of generators of SU(nf ), and it possesses a SU(nf ) rotation symmetry in neutrino flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' A significant feature of ν-ν interactions is the dependence of the interac- tion strength on the intersection angle between their trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' This dependence introduces a geometric complexity to the problem, in addition to the complexity associated with the exponential scaling of the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' For simplicity, if we consider neutrino mixing between only two flavors, νe and νx, then a Hamiltonian consisting of terms that represent vacuum mixing as well as ν-ν interactions can be written as H = ∑ p ωp⃗B· ⃗Jp + √ 2GF V ∑ p,q (1−�p·�q) ⃗Jp · ⃗Jq , (4) where ⃗B = (0,0,−1) in the mass-basis representation, and ωp = δm2/(2|p|) are the vacuum oscillation frequencies for neutrinos with momenta p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Here, �p and �q are the unit vectors along the momenta of the interacting neutrinos, and V is the quantization volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' For ease of notation, one can define a ν-ν coupling parameter µ ≡ √ 2GFN/V, where N is the total number of interacting neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The oper- ators ⃗Jp represent the neutrino “isospin” in the mass basis, where isospin up and down correspond to the mass basis states |ν1⟩ and |ν2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In this depiction, ⃗B can be interpreted as a “background field” with which the neutrino isospins interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Here, we exclude the term representing neutrino interactions with ordinary matter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', charged leptons), since it has a structure that is conceptually similar to the vacuum oscillation term—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', consisting of individual neutrinos interacting with a back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In contrast, the ν-ν interaction term consists of pairs of neutrino isospins interacting with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In terms of the Fermionic creation and annihilation operators, the neutrino isospins are described as [Balantekin and Pehlivan, 2006] J+ p = a† 1(p)a2(p) , Jz p = 1 2 � a† 1 (p)a1(p)−a† 2 (p)a2(p) � , (5) Many-body collective neutrino oscillations: recent developments 5 with J− p = (J+ p )†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In the spin-1/2 representation, one can write the isospin operators in terms of Pauli matrices: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', ⃗Jp = ⃗σp/2, where σp is a vector of Pauli matrices defined in the subspace of the neutrino with momentum p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Path integral formulation An assessment of quantum corrections to a mean field picture can in principle be performed via a coherent state analysis, as formulated by [Balantekin and Pehlivan, 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Schematically, in this procedure, one seeks to calculate the matrix elements of the time evolution operator U(tf ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='ti) for a single neutrino in the basis of SU(nf ) coherent states |z⟩ for neutrinos (and/or antineutrinos) in n f flavors, equivalent to a path integral ⟨zf |U(t f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='ti)|zi⟩ = � D[z,z∗] exp(iS[z,z∗]) (6) of the derived action S[z,z∗] = � tf ti dt � ⟨z(t)|(i∂t −H)|z(t)⟩−ilog⟨z f |zi⟩ � , (7) where H is the Hamiltonian of the many-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' A saddle-point approxima- tion of the resulting path integral yields the classical action, which is in complete agreement with the RPA used to derive the mean field theory for collective neutrino oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' However, in this perspective, analyzing quantum corrections to this ap- proximation entails a careful analysis of the Hessian matrix of the action integral derived from this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Such mathematical analysis has not yet been presented to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Beyond the Mean-Field: Entanglement, Correlations, and Dynamical Phase Transitions Early literature The seminal work describing the ν-ν interaction Hamiltonian from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' (3) recog- nized that these interactions give rise to a quantum many-body problem, which may not in the general case be factorizable in terms of a one-particle effective approxi- mation [Pantaleone, 1992a, Pantaleone, 1992b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Subsequently, there were some at- tempts to ascertain the validity of the one-particle effective approximation [Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2003, Friedland and Lunardini, 2003b, Friedland and Lunardini, 2003a, Fried- land et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In these works, the flavor evolution of interacting neutrinos was analyzed with two different approaches: (i) using two intersecting beams of neutri- 6 Amol V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Patwardhan, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Cervia, Ermal Rrapaj, Pooja Siwach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Balantekin nos, where the flavor evolution was described in terms of a sequence of elementary scattering amplitudes, and (ii) using a neutrino ensemble represented as interacting plane waves in a box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Following initial disagreement regarding whether substantial quantum entangle- ment can develop among interacting neutrinos [Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2003, Friedland and Lu- nardini, 2003b], it was subsequently concluded that the build-up of entanglement and resulting flavor conversion would occur on timescales whose scaling is sugges- tive of incoherent effects [Friedland and Lunardini, 2003a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' These conclusions were further generalized in [Friedland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' However, these analyses nevertheless involved several simplifications, most notably, the omission of the one-body terms in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The interplay between vacuum oscillations and ν-ν interaction terms has been shown to give rise to interesting collective phenomena such as “spec- tral splits” [Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006a, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006b, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2007b, Raffelt and Smirnov, 2007b, Raffelt and Smirnov, 2007a], even in the mean-field approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Therefore, studying the quantum many-body dynamics of collective neutrino oscillations, with both one- and two-body terms fully incorporated, remains an in- teresting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' With these seemingly conflicting results in the past predicting either a vanish- ingly small contribution in the large system size limit [Friedland and Lunardini, 2003a, Friedland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006] or substantial flavor evolution over time scales τF ∼ µ−1 log(N) that can remain relevant for large systems [Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2003, Sawyer, 2004], the role of entanglement and quantum effects in the out-of-equilibrium dy- namics [Eisert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2015] of neutrinos has received renewed interest recently (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', [Cervia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2019, Rrapaj, 2020] and subsequent works mentioned later in this chapter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Note that flavor oscillations on the time scale τF can be considered to be “fast”, different from “slow” oscillations occurring over τL ∼ µ−1√ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In the lit- erature on collective flavor effects in the mean field approximation, one can more commonly find “fast” and “slow” oscillations associated with time scales ∼ µ−1 and ∼ √µω (or ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Single-angle approximation, invariants, and integrability To circumvent the geometric complexity of the problem, the frequently-employed single-angle approximation replaces the angle-dependent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', �p,�q-dependent) in- teraction strengths among pairs of neutrinos with a single, appropriately chosen classical average over the various neutrino trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In this limit, one can de- fine a trajectory-averaged interaction parameter µ ≡ ( √ 2GFN/V)⟨1 − �p · �q⟩, and approximate the Hamiltonian as H = ∑ ωp ωp⃗B· ⃗Jωp + µ N ⃗J · ⃗J , (8) Many-body collective neutrino oscillations: recent developments 7 where ⃗J = ∑ωp ⃗Jωp is the total neutrino isospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Note that, in this limit, the neutrino flavor state becomes trajectory-independent, introducing a considerable simplifica- tion in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' As a result, the neutrinos may be indexed simply by the mag- nitudes of their momenta (or equivalently, by their vacuum oscillation frequencies ωp), rather than by the momenta themselves (magnitude and direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The ν-ν coupling in general will depend on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In the context of supernovae, a commonly employed expression for µ is derived from the spherically symmetric single-angle neutrino bulb model, first described in [Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006c]: µ(r) = µ0 � �1− � 1− �Rν r �2 � � 2 , (9) where r is the distance from the center of a “neutrino-sphere” of radius Rν, which represents a sharp surface where neutrinos decouple from nuclear matter and be- gin free streaming outwards from the proto-neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' We also define µ0 ≡ (GF/ √ 2)(N/V) = µ(Rν) to be the interaction strength at the neutrino-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Here, the neutrino emission is assumed to be time-invariant over the short time scales as- sociated with neutrino propagation through the supernova envelope, so the interac- tion strength depends explicitly only on position, rather than time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In the neutrino- driven wind phase of core-collapse supernovae, which occurs over a time window of O(1–10) s after core bounce, one may expect Rν ≃ 20km and µ0 ∼ 105ω0, where ω0 ∼ 10−16 MeV is the scale of the vacuum oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' During the shock breakout or “neutronization burst” phase that occurs earlier, around 10 ms after core bounce, the proto-neutron star can be more extended, with Rν ≳ 50–60 km, but the neutrino luminosity is also much higher, resulting in µ0 ∼ 106ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' It has been shown that a single-angle Hamiltonian describing neutrino mixing in vacuum and ν-ν interactions possesses a number of conserved charges which commute with the Hamiltonian [Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' These are analogous to the “Gaudin magnets” [Gaudin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 1976] that had been previously identified as the conserved charges of the pairing-force Hamiltonian in nuclear and condensed- matter physics [Richardson, 1963, Richardson and Sherman, 1964, Richardson, 1965].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' These conserved charges are related to the integrability of the Hamiltonian— meaning that it is possible to obtain, in principle, exact eigenvalues and eigenstates of this Hamiltonian in terms of closed-form solutions to a set of algebraic “Bethe- Ansatz” equations [Bethe, 1931].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Based on these ideas, specific procedures for the eigen-decomposition of a single-angle neutrino Hamiltonian have been outlined in the literature [Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2011, Birol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2018, Patwardhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Besides descriptions in terms of instantaneously conserved charges, analogies with other many-body problems have been fruitful to yield an explanation of the neutrino flavor spectral split in terms of a Bardeen-Cooper-Schrieffer (BCS)-Bose- Einstein Condensate (BEC) crossover-like phenomenon [Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2017], as well as to help provide many-body predictions of a spectral split [Birol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2018] specifically in the case of an initial many-body wave function with all neutrinos in the electron flavor state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' 8 Amol V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Patwardhan, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Cervia, Ermal Rrapaj, Pooja Siwach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Balantekin Instabilities and dynamical phase transitions Collective neutrino oscillations are generally assumed to be caused by unstable modes in the mean field dynamics generated by the Hamiltonian described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' (4) (for two flavors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' These instabilities are able to amplify initially small flavor perturbations exponentially fast (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', [Sawyer, 2004, Sawyer, 2005, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2010, Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2016, Izaguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2017, Tamborra and Shalgar, 2021, Richers and Sen, 2022] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The presence of the forward- scattering interaction can allow collective effects to develop when µ ≳ ωp, giving rise to interesting phenomena like synchronization [Pastor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2002, Fuller and Qian, 2006, Raffelt and Tamborra, 2010, Akhmedov and Mirizzi, 2016], bipolar os- cillations [Kosteleck´y and Samuel, 1995, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006c, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2007a] and spectral splits/swaps [Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006b, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2007b, Raffelt and Smirnov, 2007b, Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2009, Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' On the other hand, in descriptions of interacting neutrino systems that per- mit many-body quantum dynamics, oscillations that develop on “fast” timescales are generally associated with rapid dynamical development of the neutrino en- tanglement entropy [Cervia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2019, Rrapaj, 2020, Roggero, 2021a, Roggero, 2021b, Patwardhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The dynamically generated entanglement between neutrinos is seen to be correlated with deviations from the mean-field dynamics of the system [Cervia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2019, Rrapaj, 2020] and with the presence of spectral splits in the neutrino energy distributions [Patwardhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' An example of such a calculation is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In [Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2022], rapid entanglement and mean field instabilities were also found to be linked for certain angular setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' As shown in [Roggero, 2021a, Roggero, 2021b] in the single angle approxima- tion, when the frequency difference between two neutrino beams (δω) is positive and comparable to the ν-ν interaction coupling (µ), 0 < δω ≲ µ, rapid and strong flavor oscillations develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' This rather particular finding can be understood in terms of the presence of a Dynamic Phase Transition (DPT) [Heyl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2013, Heyl, 2018], which can be characterized by the introduction of the Loschmidt echo, L (t) = |⟨Φ|exp(−itH)|Φ⟩|2 , (10) with |Φ⟩ the initial state at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The quantity L (t) is a fidelity measure [Gorin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006] that quantifies the probability for the system to return to its initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' A DPT is then characterized by non-analyticities in the rate function λ(t) = − 1 N log[L (t)] , (11) where N is the total number of particles in the system and λ(t) an intensive “free energy” [Heyl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2013, Gambassi and Silva, 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Here, the rate λ(t) plays the role of a non-equilibrium equivalent of the thermodynamic free-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Notably, other definitions of DPT are possible, for instance, time-averaged order parame- ters [Sciolla and Biroli, 2011, Sciolla and Biroli, 2013, ˇZunkoviˇc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Many-body collective neutrino oscillations: recent developments 9 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='8 1 200 500 1000 2000 P MB z (ωp) r (in units of ω−1 0 ) Pz(ω1) Pz(ω2) Pz(ω3) Pz(ω4) Pz(ω5) Pz(ω6) Pz(ω7) Pz(ω8) −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='8 1 200 500 1000 2000 P MF z (ωp) r (in units of ω−1 0 ) Pz(ω1) Pz(ω2) Pz(ω3) Pz(ω4) Pz(ω5) Pz(ω6) Pz(ω7) Pz(ω8) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='8 1 200 500 1000 2000 S(ωp) r (in units of ω−1 0 ) S(ω1) S(ω2) S(ω3) S(ω4) S(ω5) S(ω6) S(ω7) S(ω8) −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='5 1 1 2 3 4 5 6 7 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='6 Pz(ωp) S(ωp) ω (in units of ω0) Pz (initial) P MB z ( nal) P MF z ( nal) S(ωp) ( nal) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' 1 Evolution of an initial state |νe⟩⊗4 |νx⟩⊗4 from a starting radius r0 such that µ(r0) = 5ω0, with a small mixing angle (θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='161) and discrete, equally spaced oscillation frequencies ωk = kω0, and a time-varying neutrino interaction strength µ(r) motivated by the neutrino bulb model [Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006b], in the single-angle approximation according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Details of this calculation can be found in [Cervia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Top left: Evolution of the z-components of the neutrino isospin expectation values (also known as “Polarization vectors”) in the mass basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', Pz ≡ 2⟨Jz⟩, for the full many-body quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Top right: Same as top left, but in the mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Bottom left: Evolution of the entanglement entropy of each neutrino, with respect to the rest of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Bottom right: Asymptotic values of Pz vs ωk, in the full many-body calculation (purple), and in the mean-field approximation (green), together with the initial Pz values (red), and the asymptotic entanglement entropies (dark orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Neutrinos located closest to the spectral splits in the energy distributions (in this case, at ω2 and ω7) develop the largest amount of entanglement and thereby experience the most significant deviations compared to their mean-field evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Phase-space analysis In a recent work [Lacroix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2022], this problem was further explored by ana- lyzing the evolution of neutrino flavor and entanglement in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The setup consisted of two sets (beams) of neutrinos interacting with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In this anal- ysis, the Husimi quasi-probability or “Q” representation [Husimi, 1940] was con- structed for the reduced density operator of neutrinos in one of the beams, using an over-complete basis of coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In the limit of infinite neutrino number, the Q representation acquires the interpretation of a classical phase-space probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' 10 Amol V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Patwardhan, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Cervia, Ermal Rrapaj, Pooja Siwach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Balantekin For this two-beam interacting neutrino system, it was demonstrated that, while at early times the quasi-probability distribution remains relatively localized, at late times it develops a multi-modal structure with several localized peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' This delocal- ization is indicative of non-Gaussian entanglement, which suggests that any approx- imate method beyond the mean-field relying on only the first and second moments of neutrino observables may not be sufficient to describe the long-term evolution of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Based on the phase space analysis, a new method for approximat- ing the exact evolution of the interacting neutrino system was proposed, wherein the quantum mechanical many-body evolution is replaced by a statistical average of ‘mean-field’ solutions, with a Gaussian distribution of initial conditions around the exact starting point of the system [Lacroix and Ayik, 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Compact Representations for studying many body effects Still allowing for possibilities of mixed one-neutrino density matrices, one pro- posal [Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2013] to determine quantum corrections is to systematically incorporate n-body density matrices ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='n for n ≥ 1, given by ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='n = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' (N −n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='Trn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='Nρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='N, (12) into the coupled equations of motion for N neutrinos, as follows: i∂tρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='n = [H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='n,ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='n]+ n ∑ s=1 Trn+1[V(s,n+1),ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='n+1], (13) where H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='n is the Hamiltonian truncated for the first n neutrinos in a given ordering and V(i, j) is the two-body interaction potential for a pair of neutrinos (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' This procedure is based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hi- erarchy for density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Here, the mean field theory interaction of neutrinos and antineutrinos with the background gas is reproduced by restricting to n = 2 and estimating ρ12 ≈ ρ1ρ2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', requiring the two-body correlation function to be zero) in this picture, in a sense as a loop Feynman diagram for neutrino propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In principle, investigating the importance of quantum corrections would practically entail checking for convergence of results for physical observables as the n-body correlation functions are incorporated for progressively increasing values of n in the BBGKY hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Owing to the exponential growth in the Hilbert space, classical (conventional) computers are unable to exactly simulate systems of more than ≃ 20 neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' To overcome this difficulty, one can resort to compact representations of the wave- function through tensor network methods [Roggero, 2021a, Roggero, 2021b, Cervia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2022], and more specifically matrix product states [Vidal, 2003, Schollw¨ock, 2011, Paeckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In simplified setups, these methods allow for the com- putation of systems of hundreds of neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Alternatively, when considering very Many-body collective neutrino oscillations: recent developments 11 dense neutrino gases (vacuum oscillations can be ignored), methods based on gen- eralized angular momentum representations, by analogy between two flavor oscilla- tions and spin systems, can reach up to thousands of neutrinos and predict the ther- modynamic limit [Friedland and Lunardini, 2003a, Friedland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2006, Xiong, 2022, Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In the case of time-dependent interaction strength and all-to-all ν-ν interactions, the more sophisticated tensor network method, namely, the time-dependent varia- tional principle (TDVP) method has been utilized in [Cervia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' These techniques provided considerable computational benefit for an initial state with all neutrinos in the same flavor, allowing for evolution of a system with ≈ 50 oscil- lation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' This was a consequence of the entanglement among neutrinos being more localized in certain regions of the neutrino energy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' For systems with initial states being a mixture of νe and νx flavors, the entanglement is more de- localized, and therefore, the comparative advantage gained through TDVP methods is less dramatic, although work remains in progress on this front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' For a general setup, quantum computers are a promising tool to solve the quan- tum many-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Initial steps [Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2021, Yeter-Aydeniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2022, Illa and Savage, 2022, Amitrano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2022] to simulate the collective neu- trino oscillations on a quantum computer are already taken in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In [Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2021] a sytem of four neutrinos was simulated on IBM’s quantum devices using the real-time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' The unitary evolution operator U(t) = exp(−iHt) was decomposed using the first order Trotter-Suzuki decomposition, where error scales as O(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Since the interaction is long-range, a device with all-to-all con- nectivity among qubits is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' As an alternative, SWAP operations have been used to implement this interaction on a quantum device having connectivity among neighboring qubits [Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In [Yeter-Aydeniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2022], the hybrid quantum-classical algorithm QLanczos (quantum Lanczos) was used to calculate the eigenvalues of neutrino many-body interaction Hamiltonian [Patwardhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2019] on a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Furthermore, the transition probabilities of collec- tive neutrino oscillations were obtained by performing the real-time evolution using trotterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' However, all these earlier quantum computing studies were limited to a small system of four neutrinos due to constraints in the form of currently avail- able quantum devices, which can perform only a limited number of operations with low accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' More recently in [Amitrano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 2022], a trapped-ion quantum de- vice was utilized to perform the simulations for up to eight neutrinos, thanks to the all-to-all qubit connectivity in trapped-ion based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Concluding remarks Studying the many-body quantum dynamics of dense neutrino systems remains an active area of research, with various groups attempting to investigate the problem using different types of classical and quantum computational tools, as well as ana- lytic or semi-analytic descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' In environments where neutrinos are present in 12 Amol V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Patwardhan, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Cervia, Ermal Rrapaj, Pooja Siwach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Balantekin high number densities, they almost inevitably become the main carriers of energy and lepton number, and as a result, the physics of neutrino flavor transformation in these environments becomes particularly relevant for the dynamics and nucle- osynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Moreover, the close parallels between this problem and other quantum many-body systems in nuclear and condensed-matter physics suggests that the re- sults and insights obtained through these studies could have a much broader scope, beyond just the field of neutrino physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' References Akhmedov and Mirizzi, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Akhmedov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' and Mirizzi, A.' 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+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Supernova neutrino nucleosynthesis of light elements with neutrino oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} +page_content=', 96:091101, astro-ph/0602195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAyT4oBgHgl3EQfffiq/content/2301.00342v1.pdf'} diff --git a/LNE4T4oBgHgl3EQf8A7c/content/tmp_files/2301.05345v1.pdf.txt b/LNE4T4oBgHgl3EQf8A7c/content/tmp_files/2301.05345v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9cac8b68f4287bcfa6f3d69aa677435531e4316 --- /dev/null +++ b/LNE4T4oBgHgl3EQf8A7c/content/tmp_files/2301.05345v1.pdf.txt @@ -0,0 +1,1246 @@ +GOHSP: A Unified Framework of Graph and Optimization-based Heterogeneous +Structured Pruning for Vision Transformer +Miao Yin1* Burak Uzkent2, Yilin Shen2, Hongxia Jin2, Bo Yuan1 +1 Rutgers University, 2 Samsung Research America +Abstract +The recently proposed Vision transformers (ViTs) have +shown very impressive empirical performance in various +computer vision tasks, and they are viewed as an impor- +tant type of foundation model. However, ViTs are typically +constructed with large-scale sizes, which then severely hin- +der their potential deployment in many practical resources- +constrained applications. To mitigate this challenging prob- +lem, structured pruning is a promising solution to compress +model size and enable practical efficiency. However, unlike +its current popularity for CNNs and RNNs, structured prun- +ing for ViT models is little explored. +In this paper, we propose GOHSP, a unified framework of +Graph and Optimization-based Structured Pruning for ViT +models. We first develop a graph-based ranking for measur- +ing the importance of attention heads, and the extracted im- +portance information is further integrated to an optimization- +based procedure to impose the heterogeneous structured spar- +sity patterns on the ViT models. Experimental results show +that our proposed GOHSP demonstrates excellent compres- +sion performance. On CIFAR-10 dataset, our approach can +bring 40% parameters reduction with no accuracy loss for +ViT-Small model. On ImageNet dataset, with 30% and 35% +sparsity ratio for DeiT-Tiny and DeiT-Small models, our ap- +proach achieves 1.65% and 0.76% accuracy increase over the +existing structured pruning methods, respectively. +Introduction +Recently applying transformer architecture to computer vi- +sion has emerged as an important forefront of foundation +model design (Dosovitskiy et al. 2020). Thanks to the del- +icate vision-specific self-attention, inherent minimal induc- +tive biases and high scalability and parallelism, vision trans- +formers (ViTs) (Dosovitskiy et al. 2020; Touvron et al. +2021; Zhou et al. 2021) have shown very outstanding and +even state-of-the-art performance in many fundamental and +downstream image and video processing tasks, such as im- +age classification, object detection, super-resolution, video +classification etc. +Motivated by the scaling success of the giant natural lan- +guage processing (NLP) transformers (e.g., BERT (Devlin +*This work was done during Miao Yin’s internship at Samsung +Research America. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +et al. 2018) and GPT-3 (Brown et al. 2020)), the existing +ViTs are also constructed with large model sizes to adapt for +massive data training (Zhai et al. 2021). Consequently, they +are suffering from huge memory footprints and extensive +computational costs. These limitations, if not being properly +addressed, could severely hinder the widespread adoption of +ViTs in many practical scenarios, especially on the resource- +constrained mobile platforms and Internet-of-things (IoT) +devices. +To mitigate this challenging problem, one attractive solu- +tion is to perform model compression (Yu et al. 2017; Kim +et al. 2016; Pan et al. 2019) to reduce the network costs with- +out affecting task performance. However, unlike the current +popularity of compressing convolutional and recurrent neu- +ral networks (CNNs and RNNs), ViT-oriented model com- +pression has not been systematically studied yet. In partic- +ular, structured pruning, as an important hardware-friendly +compression strategy that can bring practical efficiency on +the off-the-shelf hardware, is little explored for ViT models. +To date, a rich set of structured pruning approaches have +been proposed and investigated in the existing literatures, +and most of them focus on sparsifying the CNNs at the chan- +nel level (He, Zhang, and Sun 2017; Ye et al. 2018). On the +other hand, as will be analyzed and elaborated in Section , +because of the difference of the underlying architecture, the +structured sparse ViT models can exhibit multi-granularity +sparsity (i.e., head-level and column-level) in the different +component modules (i.e., attention head and multi-layer per- +ception (MLP)). The co-existence of such heterogeneous +sparse patterns raises a series of new research challenges and +questions when we consider the efficient structured pruning +strategy for ViT models. For instance, for each component +module what is the corresponding suitable pruning criterion +to obtain its specific sparse pattern? Also, how should we +perform the entire pruning process across different modules +with different levels of granularity sparsity to optimize the +overall compression and task performance? +Technical Preview & Contributions. To answer these +questions, in this paper we propose GOHSP, a unified frame- +work of Graph and Optimization-based Structure Prun- +ing for vision transformer. To be specific, we first de- +velop a graph-based ranking approach to measure the im- +portance of attention heads. As a soft-pruning guideline, +such importance information is then integrated to the overall +arXiv:2301.05345v1 [cs.AI] 13 Jan 2023 + +optimization-based procedure to impose the different types +of structured sparsity in a joint and global way. Overall, the +contributions of this paper are summarized as follows: +• We propose a graph-based ranking algorithm to mea- +sure and determine the importance of attention heads. +By modeling the inter-head correlation as a converged +Markov chain, the head importance can be interpreted +and calculated as the stationary distribution, which is fur- +ther used as a soft guideline for the overall pruning pro- +cedure. +• We propose a unified framework to jointly optimize dif- +ferent types of structured sparsity in the different mod- +ules. The complicated coordination for different sparse +patterns are automatically learned and optimized in a sys- +tematic way. +• We evaluate the performance of our structured pruning +approach of different ViT models on different datasets. +On CIFAR-10 dataset, our approach can bring 40% pa- +rameters reduction with no accuracy loss for ViT-Small +model. On ImageNet dataset, with 30% and 40% spar- +sity ratio for DeiT-Tiny and DeiT-Small models, our +approach achieves 1.65% and 0.76% accuracy increase +than the existing structured pruning methods, respec- +tively. +Related Work +Vision Transformer. Inspired by the grand success of trans- +former architecture in NLP domains, deep learning re- +searchers have actively explored the efficient transformer- +based neural networks for computer vision. Most recently, +several vision transformers (ViTs) and their variants have +already shown very impressive performance in several im- +age and video processing tasks (Dosovitskiy et al. 2020; +Touvron et al. 2021; Zhou et al. 2021). However, in order +to achieve competitive performance with the state-of-the-art +CNNs, ViTs typically have to scale up their model sizes and +therefore they suffer from costly computation and storage. +Dynamic Inference with ViTs. To reduce the deploy- +ment costs of ViTs, several works (Wang et al. 2021; +Bakhtiarnia, Zhang, and Iosifidis 2021; Rao et al. 2021; +Meng et al. 2022; Xu et al. 2022; Uzkent, Yeh, and Er- +mon 2020; Uzkent and Ermon 2020) have been proposed to +improve the processing speed via dynamically pruning the +tokens/patches or skipping transformer components adap- +tively. Essentially as dynamic inference approaches, this set +of works do not pursue to reduce the model sizes but focus +on input-aware inference to obtain practical speedup. Our +structured pruning-based solution is orthogonal to them, and +these two different strategies can be potentially combined +together to achieve higher speed and smaller memory foot- +print. +Structured Pruning. Model compression is a promising +strategy to reduce the deployment costs of neural networks. +Among various model compression techniques, structured +pruning is a very popular choice because its hardware- +friendly nature can bring practical efficiency on the real- +world devices. Based on different pruning criterias, various +structured pruning approaches have been extensively studied +Head-based Sparsity +(Multi-Head Attention) +(a) Conventional Unstructured Sparsity +(b) Unified Structured Sparsity (Ours) +Column-based Sparsity +(Multi-Head Attention) +Column-based Sparsity +(MLP) +Unstructured Sparsity +(MLP) +Unstructured Sparsity +(Multi-Head Attention) +Figure 1: (a) Sparsity pattern of ViT models after un- +structured pruning. Only part of the Multi-Head Attention +and MLP columns are pruned which are not hardware- +friendly.(b) Heterogeneous sparsity patterns of ViT models +after structured pruning. Certain MLP and Multi-Head At- +tention columns are removed which is hardware-friendly. +in the existing literature (Yu et al. 2018; Zhuang et al. 2018; +Liu et al. 2019; He et al. 2019; Lin et al. 2020; Tiwari et al. +2021; Lou et al. 2022), and most of them focus on pruning +CNN models; while the efficient structured pruning of ViTs +is little explored. One of these studies, (Chen et al. 2021), +prunes the vision transformers using structured pruning. (Yu +et al. 2022), on the other hand, focuses on FLOPs reduction +with the vision transformers using pruning, layer skipping, +and knowledge distillation whereas in our study we focus on +structured pruning to mainly reduce the number of parame- +ters for building hardware-friendly compressed models. For +this reason, we compare our method to (Chen et al. 2021). +Structured Pruning of ViTs: Analysis +Notation. Considering an L-block vision transformer, +W (l) +attn = {W (l) +qkv, W (l) +proj} and W (l) +mlp = {W (l) +fc1, W (l) +fc2} rep- +resent the weights of the attention layer and the MLP layer +at l-th block, respectively. For each attention layer, there are +H self-attention heads, namely W (l) +qkv = {W (l,h) +qkv }H +h=1 and +W (l) +proj = {W (l,h) +proj }H +h=1. To simplify the notation, in the fol- +lowing content we take one block as the example and omit +the superscript (layer index). +Heterogeneity of structured sparsity. Because of the +difference of the network architecture, the meaning of +‘structured sparsity’ varies with different model types. As +described and performed in (Wen et al. 2016; Anwar, +Hwang, and Sung 2017; Liu et al. 2018, 2020), the struc- +tured pruning of CNN and RNN typically indicates the re- +moval of the entire channels of the weight tensors and the +entire columns of the weight matrices, respectively. Notice +that here for either of these two cases, only one type of the +structured sparse pattern exist because of the architectural +homogeneity of the CNN and RNN. +On the other hand, a ViT model exhibits inherent archi- +tectural heterogeneity. Within the same block, the front-end +multi-head attention module and the back-end MLP mod- +ule represent two types of design philosophy for information +processing, and thereby leading to huge difference on both +computing procedures and the available structured sparse +patterns. +To be specific, when we consider performing structured + +pruning of ViT model, three types of structured sparse pat- +terns can co-exist with different levels of granularity across +different modules. For the multi-head attention module, be- +cause each attention head is processing the information in- +dividually in a parallel way, the pruning can be performed +at the head-level to sparsify this component. In addition, +consider the weights in the heads are represented in the ma- +trix format; the column-level sparsity can also be introduced +towards structured pruning. Meanwhile, because the MLP +consists of multiple weight matrices as well, the column- +level of granularity sparsity can be imposed on this back-end +module at the same time. Consequently, a structured pruned +ViT model can exhibit heterogeneous structured sparsity +(see Fig. 1). +Problem Definition. Based on the above analysis, the +structured pruning of a vision transformer model with loss +function ℓ(·) can be formulated as the following general op- +timization problem: +min +Wattn,Wmlpℓ(Wattn, Wmlp), +s.t. +∥Wattn∥h +0 ≤ κh +attn, +∥Wattn∥c +0 ≤ κc +attn, +∥Wmlp∥c +0 ≤ κc +mlp, +(1) +where κc and κh are the desired number of columns and the +desired number of heads after pruning, respectively. ∥ · ∥c +0 +and ∥ · ∥h +0 are the column-based and head-based group L0- +norm, which denote the number of non-zero columns and +the number of non-zero heads, respectively. +Questions to be Answered. Solving the above opti- +mization problem is non-trivial since it contains the con- +straints involved with multi-granularity sparsity for different +model components. More specifically, two important ques- +tions need to be answered. Question #1: What is the suitable +pruning criterion to obtain head-level sparsity? +Analysis: From the perspective of information process- +ing, multi-head attention shares some interesting similarity +with convolutional layer. Both of them use multiple indi- +vidual computing units, i.e., attention heads and convolu- +tional filters, to perform parallel computations. Therefore, a +naive way to perform head-level pruning is to leverage the +existing criteria developed in the channel pruning of CNNs. +However, such straightforward solution, in principle, may +not be the best choice because of two reasons. First, the re- +ceptive fields and the focused locality of the attention head +and filters are different, and hence simply using the crite- +rion for pruning channels is not a suitable strategy. Second +and more importantly, most of the existing channel pruning +criterias are built on the information of each individual chan- +nel (the corresponding filter weight and/or its feature map). +When adopting this philosophy in the head pruning, the in- +sufficient utilization of inter-head information will probably +cause non-negligible performance loss. Overall, the unique +characteristics of multi-head attention mechanism calls for +attention-specific pruning criterion. +Question #2: How should we coordinate the pruning +across different modules with different levels of granularity? +Analysis: As indicated before, three types of structured +sparse pattern can co-exist in the different modules of the +pruned ViT models. A key component of the to-be-explored +structured pruning strategy is to develop a good coordination +scheme that can properly impose these different structured +sparse patterns in a joint and global way. Consider the com- +plicated interaction among different types of structured spar- +sity, the expected pruning strategy should be able to solve +this problem in a systematic and global way. +Structured Pruning of ViTs: Method +Graph-based Head Ranking +To answer Question #1, we propose a graph-based approach +to measure and determine the importance of different at- +tention heads, which can be further used for the follow-up +pruning. Our key idea is to model the inter-head correla- +tion as a graph, and then leverage the graph-based ranking, +a methodology that has been successfully used in many web +search and NLP algorithms, such as PageRank (Page et al. +1999), TextRank (Mihalcea and Tarau 2004) and LexRank +(Erkan and Radev 2004), to select important attention heads. +Graph Construction of Markov Chain. To be specific, +we first construct a graph G = (A, E) to represent the atten- +tion heads and their similarities in the block of a ViT model. +The set of nodes A denote all the attention heads {Ah}H +h=1, +and E is the set of connected edges. For edge E(Ai, Aj), +its weight is defined as the expected cosine similarity be- +tween Ai and Aj. According to (Mihalcea and Tarau 2004), +the graph defined with such cosine similarity can be inter- +preted as a Markov chain, where each node is a state, and +the transition probability P(i, j) between two states is the +edge weight. In such scenario, P(i, j) can be calculated as: +P(i, j) = EX∼D [CosineSim(Ai(X), Aj(X))] , +(2) +where Ai(X) is the output of i-th attention head with sam- +pled input X and D is the data set. Built upon this calcula- +tion, the entire transition matrix P of a Markov chain. No- +tice that as indicated in (Erkan and Radev 2004), each col- +umn of P should be further normalized. +Batch estimation. Calculating the transition probability +can be very costly since it needs to be performed across the +entire training dataset D (see Eq. 2). To solve this problem, +we adopt a batch-based estimation strategy to improve com- +putation efficiency without sacrificing ranking performance. +To be specific, as described in Eq. 3, only a batch of training +data is sampled and used to to estimate the transition prob- +ability. As our ablation study in Section will show, using +different batch sizes (B) bring very stable ranking results +for the attention heads, thereby empirically verifying the ef- +fectiveness of this estimation strategy. +P(i, j) = CosineSim +� B +� +b=1 +Ai(Xb), +B +� +b=1 +Aj(Xb) +� +. +(3) +Importance Ranking. Mathematically, an irreducible +and aperiodic Markov chain is guaranteed to converge to a +stationary distribution (Seneta 2006). As indicated in (Erkan +and Radev 2004), once converged, the probability of a ran- +dom walker stays in one state can reflect the state impor- +tance. Motivated by this observation, we propose to quantify + +Multi-Head Attention +Embedded +Patches +MLP +Block +0.1 +Graph-based Heads Ranking +Multi-Head Attention +Embedded +Patches +MLP +Block +Data +Optimization-based Soft Pruning +Multi-Head Attention +Embedded +Patches +MLP +Block +Fine-Tuning +X +Score +Mask +0.7 +0.5 +0.3 +Normalize +Figure 2: Procedure of the proposed multi-stage structured pruning approach. +the importance of each attention head via calculating the sta- +tionary distribution in our constructed Markov chain. To that +end, the iterative power method (Erkan and Radev 2004) can +be used via setting a uniform distribution for the states as the +initialization. Overall, the entire graph-based head ranking +procedure is described in Algorithm 1. +Soft-Pruning Mask. Once the importance score for each +state is obtained via calculating the stationary distribution, +the corresponding attention heads can be ranked. Here we +use a binary mark matrix Mattn = {Mqkv, Mproj} to in- +dicate the weight entries associated with the least important +heads that should be removed. Notice that at this stage the +head pruning is not performed yet. Instead such binary mask +serves as the guideline for the next-stage optimization, and +it is essentially integrated into Eq. 1 as follows: +min +Wattn,Wmlpℓ(Wattn, Wmlp) +s.t. +∥(1 − Mattn) ⊙ Wattn∥0 = 0, +∥Wmlp∥0 ≤ κc +mlp, +∥Mattn ⊙ Wattn∥c +0 ≤ κc +attn, +(4) +where ⊙ is element-wise product. In general, because the +overall optimization phase coordinates and adjusts the dif- +ferent types of structured sparse pattern from a global per- +Algorithm 1: Graph-based Attention Head Ranking +Input: Sampled batch {Xb}B +b=1, attention heads {Ah}H +h=1; +Output: Importance score s = [s1, · · · , sH]. +1: Initialize transition matrix: P := zeros(H, H); +2: for i = 1 to H do +3: +for j = 1 to H do +4: +Calculate P(i, j) via Eq. 3; +5: Normalize each column of P ; +6: Initialize s := ones(H)/H; +7: repeat +8: +s′ := s; +9: +s := P s; +10: +δ := ∥s − s′∥; +11: until δ ≤ ϵ +spective, this ranking-only ”soft” pruning strategy, instead +of directly pruning the least important heads, can provide +more flexibility and possibility for the next-stage optimiza- +tion procedure to identify better structured sparse models. +Optimization-based Structured Pruning +As pointed out by Question #2, the co-existence of multi- +granularity and multi-location of the sparsity of ViT models +make the entire structured pruning procedure become very +challenging. To solve this, we propose to use advanced op- +timization technique to perform systematic structured prun- +ing. To be specific, considering the complicated interactions +among different types of structured sparsity, we do not prune +the heads or columns immediately, since any direct hard +pruning at the early stage may cause severe accuracy loss. +Instead, we adopt ”soft-pruning” strategy via optimizing the +entire ViT models towards the desired structured sparse for- +mats. In other words, the three types of sparsity pattern are +gradually imposed onto the attention heads and MLPs. +To that end, we first relax the constraints of Eq. 4 and +rewrite it as follows: +min +Wattn,Wmlpℓ(Wattn, Wmlp) + λ +2 ∥(1 − Mattn) ⊙ Wattn∥2 +F , +s.t. +∥Wmlp∥c +0 ≤ κc +mlp, +∥Mattn ⊙ Wattn∥c +0 ≤ κc +attn, +(5) +where λ is the coefficient that controls the influence of +quadratic term. +Optimization-based Soft Pruning. As indicated in +(Boyd, Parikh, and Chu 2011), when the constraints of con- +tinuous non-convex problem are sparsity related (as Eq. +5 shows), Douglas—Rachford splitting method (Eckstein +and Bertsekas 1992) can be a suitable optimization solution +for such types of problem. Following this philosophy, we +first introduce auxiliary variables Zattn, Zmlp and indicator +functions as: +g(Zattn) = +�0 +∥Mattn ⊙ Zattn∥c +0 ≤ κc +attn, ++∞ +otherwise, +(6) +h(Zmlp) = +�0 +∥Zmlp∥c +0 ≤ κc +mlp, ++∞ +otherwise. +(7) + +Algorithm 2: Overall Procedure of GOHSP Framework +Input: Dense weight {Wattn, Wmlp}, desired model size +{κattn, κmlp}, training data D, number of epochs E; +Output: Structured sparse weight { ˜ +Wattn, ˜ +Wmlp}; +1: Sample a batch of data {Xb}B +b=1 from D; +2: Calculate importance score s via Alg. 1; +3: Obtain structured mask Mattn according to s; +4: Zattn := Wattn, Zmlp := Wmlp; // Initialize auxiliary +variables +5: Uattn := 0, Umlp := 0; // Initialize Lagrangian multi- +pliers +6: for e = 1 to E do +7: +Update Wattn, Wattn via Eq. 10 and Eq. 11; +8: +Update Zattn, Zmlp via Eq. 12 and Eq. 13; +9: +Update Uattn, Umlp via Eq. 14 and Eq. 15; +10: Fine-tune pruned weight { ˜ +Wattn, ˜ +Wmlp}. +Then, we can rewrite Eq. 5 as the following equivalent form: +min +W ,Z +ℓ(Wattn, Wmlp) + g(Zattn) + h(Zmlp)+ +λ +2 ∥(1 − Mattn) ⊙ Wattn∥2 +F , +s.t. +Wmlp = Zmlp, +Wattn = Zattn. +(8) +In such scenario, the corresponding augmented Lagrangian +function of the above optimization objective is: +Lρ(Wattn, Wmlp, Zmlp) = ℓ(Wattn, Wmlp) + g(Zattn)+ +h(Zmlp) + λ +2 ∥(1 − Mattn) ⊙ Wattn∥2 +F + +ρ +2∥Wattn − Zattn + Uattn∥2 +F + +ρ +2∥Uattn∥2 +F + ρ +2∥Wmlp − Zmlp + Umlp∥2 +F + ρ +2∥Umlp∥2 +F , +(9) +where ρ > 0 is the penalty parameter, and Uattn, Umlp are +the Lagrangian multipliers. Then the variables at step t can +be iteratively updated as: +W t +attn = W t−1 +attn−η +ℓ(Wattn, W t−1 +mlp ) +Wattn +− +λ +� +(1 − Mattn) ⊙ W t−1 +attn +� +−ρ(W t−1 +attn − Zt−1 +attn + U t−1 +attn), +(10) +W t +mlp = W t−1 +mlp − η ℓ(W t +attn, Wmlp) +Wmlp +− +ρ(W t−1 +mlp − Zt−1 +mlp + U t−1 +mlp ), +(11) +Zt +attn = P(W t +attn + U t−1 +attn), +(12) +Zt +mlp = P(W t +mlp + U t−1 +mlp ), +(13) +U t +attn = U t−1 +attn + W t +attn − Zt +attn, +(14) +U t +mlp = U t−1 +mlp + W t +mlp − Zt +mlp. +(15) +Here η is the optimizer learning rate for training the ViT, and +P is the Euclidean projection for the sparse constraint. +Final Hard-Pruning and Fine-Tuning. After the above +described optimization procedure, the structured sparse pat- +terns have been gradually imposed onto the ViT model. +In other words, the weight values of the masked attention +heads, as well as some columns of MLPs and attention +heads, become extremely small. At this stage, we can now +prune those small weights and then perform a few rounds of +fine-tuning to achieve higher performance. +Overall, +by +using +graph-based +head +ranking +and +optimization-based structured pruning, the previously raised +Question #1 and #2 can be properly addressed. The overall +GOHSP framework is summarized in Fig. 2. +Experiments +Experimental Settings +Dataset and Baseline. We evaluate the performance of +our proposed GOHSP approach on CIFAR-10 and Ima- +geNet datasets (Deng et al. 2009). For experiments on +the CIFAR-10 dataset, the original dense model is ViT- +Small1(Dosovitskiy et al. 2020) with 48M parameters. For +experiments on the ImageNet dataset, the original dense +models are DeiT-Tiny and DeiT-Small (Touvron et al. 2021) +with 5.7M and 22.1M parameters, respectively. +Hyper-parameters and Sparsity Ratio. For our experi- +ments on the CIFAR-10 dataset, the batch size, learning rate +and ρ are set as 256, 0.1 and 0.001, respectively. For Ima- +geNet dataset, the batch size, learning rate and ρ are set as +256, 0.01 and 0.001, respectively. For both of these two ex- +periments, SGD is selected as the training optimizer with- +out using weight decay, and we apply Erd˝os-R´enyi (Mo- +canu et al. 2018) to determine the sparsity distribution of +each layer given an overall sparsity ratio. In particular, soft- +pruning maintains high accuracy at the high sparsity ratios. +Performance Evaluation +CIFAR-10 Dataset. Table 1 shows performance compari- +son on CIFAR-10 dataset between our proposed GOHSP +and other structured pruning method (structured one-shot +magnitude pruning (SOMP) (Han, Mao, and Dally 2015) +and structured gradually magnitude pruning (SGMP) (Zhu +and Gupta 2017)) for ViT-Small model. It is seen that +with the same sparsity ratio, our approach brings significant +performance improvement. Compared to SGMP approach, +1We take this model from open source library timm. +Table 1: Performance comparison between our GOHSP +and structured one-shot/gradually magnitude-based pruning +(SOMP/SGMP) of ViT-Small model on CIFAR-10 dataset. +Method +Sparsity +# Paramters +Top-1 (%) +Baseline +- +48.0M +97.85 +SOMP +40% +28.8M +96.07 +SGMP +40% +28.8M +96.93 +GOHSP (Ours) +40% +28.8M +97.89 +GOHSP (Ours) +80% +9.6M +97.40 + +Table 2: Comparison results of our method, GOHSP, with other structured and unstructured pruning methods on ImageNet. +Model +Method +Sparsity +# Parameters +FLOPs ↓ +Run-time ↓ +Top-1 (%) +DeiT-Tiny +Baseline +- +5.7M +- +- +72.20 +OMP (Unstructured) +30% +4.02M +25.56% +- +68.35 +GMP (Unstructured) +30% +4.02M +25.56% +- +69.56 +TP (Unstructured) +30% +4.02M +25.56% +- +68.38 +SSP (Structured) +30% +4.2M +23.69% +- +68.59 +S2ViTE (Structured) +30% +4.2M +23.69% +10.57 % +70.12 +GOHSP (Structured) +30% +4.0M +30% +13.41% +70.24 +DeiT-Small +Baseline +- +22.1M +- +- +79.90 +SSP (Structured) +40% +14.6M +31.63% +- +77.74 +S2ViTE (Structured) +40% +14.6M +31.63% +22.65% +79.22 +GOHSP (Structured) +40% +14.4M +35% +24.61% +79.98 +GOHSP (Structured) +50% +11.1M +39% +26.57% +79.86 +GOHSP achieves 0.96% accuracy increase with the same +pruned model size. Even compared with the baseline, the +structured sparse model pruned by GOHSP can outperform +the uncompressed model with 40% fewer parameters while +80% compressed model achieves only 0.45% worse than the +full ViT-Small model. +ImageNet Dataset. Table 2 summarizes the performance +on ImageNet dataset between GOHSP and other structured +pruning approaches (SOMP, SGMP, Talyer pruning (TP), +Salience-based Structured Pruning (SSP) and S2ViTE(Chen +et al. 2021)) for DeiT-Tiny and DeiT-Small models. It is seen +that due to the limited redundancy in such small-size model, +the existing pruning approaches suffer from more than 2.5% +accuracy loss when compressing DeiT-Tiny. Instead, with +the even fewer parameters and more FLOPs reduction, our +GOHSP approach can achieve at least 0.68% accuracy in- +crease over the unstructured pruning approaches. Compared +to the structured pruning approach (SSP), our method enjoys +1.65% accuracy improvement with lower storage cost and +computational cost. In addition, when compressing DeiT- +Small model, with fewer parameters and more FLOPs re- +duction, our GOHSP approach can achieve 0.76% accuracy +increase as compared to the state-of-the-art structured prun- +ing method S2ViTE (Chen et al. 2021) and can even outper- +form the original DeiT-Small. With 50% pruned DeiT-Small +we achieve similar accuracy to the full DeiT-Small. Finally, +we report 26.57% improvement in run-time efficiency with +our 50% pruned DeiT-Small. +0.2 +0.3 +0.4 +0.5 +0.6 +Sparsity +90.0 +92.0 +94.0 +96.0 +98.0 +Top-1 Accuracy (%) +Ours +Hard Pruning +Figure 3: Results on the effect of soft-pruning (ours) and +hard-pruning for ViT-Small model on CIFAR-10 dataset. +Ablation Study, Visualization and Discussion +To obtain the deep understanding of the effect of our pro- +posed approach, we perform several ablation studies and a +detailed analysis. Here the experiments conducted in the ab- +lation study focus on compressing ViT-Small on CIFAR-10. +Soft Pruning vs Hard Pruning. As described in Opti- +mization section, after ranking the attention heads, we use +the ranking information as a soft-pruning mask to guide +the next-phase optimization. The optimization itself is also +a soft-pruning procedure that does not directly zero the +weights but gradually impose the structured sparsity. To ana- +lyze the effect of this strategy, we conduct an ablation exper- +iment via performing the direct hard pruning. In this ablation +study, the least important attention heads are removed ac- +cording to their ranks, and the columns of MLPs with least +group L1 norm are also pruned. Such hard pruned models +are still trained with the same hyper-parameters settings that +are used for soft pruning method. Fig. 3 shows the curves +of top-1 test accuracy with different target sparsity settings. +1 +2 +3 +4 +5 +6 +7 +8 +Head Index +1 +2 +3 +4 +5 +6 +7 +8 +Block Index +Batch Size=256 +0 +2 +4 +6 +1 +2 +3 +4 +5 +6 +7 +8 +Head Index +1 +2 +3 +4 +5 +6 +7 +8 +Block Index +Batch Size=512 +0 +2 +4 +6 +1 +2 +3 +4 +5 +6 +7 +8 +Head Index +1 +2 +3 +4 +5 +6 +7 +8 +Block Index +Batch Size=1024 +0 +2 +4 +6 +1 +2 +3 +4 +5 +6 +7 +8 +Head Index +1 +2 +3 +4 +5 +6 +7 +8 +Block Index +Batch Size=1536 +0 +2 +4 +6 +Figure 4: The effect of batch sizes for ranking results. Dif- +ferent colors represent different ranking scores. We can see +that our head ranking algorithm is not sensitive to batch size. + +The soft-pruning strategy brings very significant accuracy +improvement over the direct hard pruning with the same +sparsity ratio. +Effect of Batch Size on Head Ranking. As shown in Eq. +3, the importance scores of attention head is calculated on +a batch of data. To investigate the potential impact of batch +sizes for the ranking results, we observe the change of rank- +ing with different batch sizes. As shown in Fig. 4, the rank- +ing results are very stable (almost the same) when the batch +size varies. Therefore we can conclude that using batches of +data can already achieve very good estimation of head rank- +ing. In other words, our ranking approach has low sensitivity +to the distribution of input data. +Sensitivity of Penalty Parameter ρ. We also explore the +effect of hyperparameter ρ on the structured pruning proce- +dure. Fig. 5 (a) shows the convergence of training process +with respect to different ρ. It is seen that the convergence +speed is always fast, and hence it demonstrates the promis- +ing convergence property of our approach in practice. Fig. 5 +(b) illustrates the L2-norm of the masked entries. It is seen +that the larger ρ makes the model exhibit more sparsity at the +earlier stage, thereby indicating that larger ρ can bring fewer +epochs in the final fine-tuning stage. However, as shown in +Fig. 5 (c), too large ρ brings accuracy degradation, so ρ can +be considered as a parameter that controls the trade-off be- +tween the speed of imposing sparsity and task performance. +Visualization. Fig. 6 illustrates the sparsity patterns in +the pruned ViT models after performing our GOHSP ap- +proach. It is seen that three types of structured sparsity pat- +terns (head-level sparsity, column-level sparsity in the head +and column-level sparsity in the MLP) are imposed on the +0 +10 +20 +30 +40 +50 +60 +Epoch +0 +250 +500 +Loss +(a) Curves of training loss +=0.001 +=0.002 +=0.0005 +0 +10 +20 +30 +40 +50 +60 +Epoch +0 +50 +100 +L2-Norm +(b) Curves of sparsity strength +=0.001 +=0.002 +=0.0005 +0 +10 +20 +30 +40 +50 +60 +Epoch +50 +75 +100 +Top-1 (%) +(c) Curves of test accuracy +=0.001 +=0.002 +=0.0005 +40 +50 +60 +0 +3 +6 +40 +50 +60 +96 +97 +98 +Figure 5: Effect of ρ on the structured pruning procedure. ρ +controls the trade-off between the speed of imposing sparsity +and task performance. +pruned models. Such pruning can be more effective on hard- +ware than the unstructured pruning methods. +Block9 +Block10 +Block11 +Multi-Head Attention Layer +MLP Layer +Block9 +Block10 +Block11 +Multi-Head Attention Layer +MLP Layer +Block9 +Block10 +Block11 +Multi-Head Attention Layer +MLP Layer +Figure 6: Visualization of the imposed structured sparsity on +the DeiT-Small model. The columns and heads with lighter +color are pruned. Our method can prune columns (Block9, +Block10, and Block11), and heads (Block10, Block11) of +the Multi-Head Attention layer. On the other hand, we can +prune columns of MLP layers in all the blocks. +Why +Douglas—Rachford +splitting +method? +As +shown in our Optimization section, the iterative Dou- +glas—Rachford splitting technique is adopted to solve Eq. +5. Such choice is built on two reasons. 1) Convergence: +Douglas—Rachford splitting method is a primal-dual +optimization method that enjoys fast convergence speed. +According to (Boyd, Parikh, and Chu 2011), within a few +iterations it can provide satisfied solution for large-scale +problems – particularly attractive for DNN applications. +More specifically for this work, the fast convergence of +Douglas—Rachford splitting method can avoid gradient +explosion problem introduced by the additional sparsity +loss in Eq. 9. 2) Flexibility: Douglas—Rachford splitting +method, by its nature, divides the original difficult optimiza- +tion problem into several less complicated sub-problems, +each of which can be then addressed independently. This +divide-and-conquer property is very suitable for optimizing +the heterogeneous structured pruning of ViT, which explores +the different types of structured sparsity across different +attention heads and MLPs (Eq. 10 and 11). +Conclusion +In this paper we propose GOHSP, a unified framework to +perform graph and optimization-based heterogeneous struc- +tured pruning for vision transformers. By using graph-based +ranking and leveraging the advanced optimization tech- +nique, our approach can efficiently impose different types +of structured sparse patterns on the vision transformers with +high compression rate and task performance. Our experi- +ments show that, on ImageNet, with 30 − 50% sparsity, +GOHSP compresses the DeiT-Tiny and DeiT-Small mod- +els with minor or no loss in accuracy and with ∼ 25 im- +provement in rum-time efficiency. Finally, we compress ViT- +Small up to 80% on CIFAR10 with minor loss in accuracy. + +References +Anwar, S.; Hwang, K.; and Sung, W. 2017. 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In Advances in Neural +Information Processing Systems, 875–886. + diff --git a/LNE4T4oBgHgl3EQf8A7c/content/tmp_files/load_file.txt b/LNE4T4oBgHgl3EQf8A7c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3b7b7c6395119425707e153eaf85f895e35c97f --- /dev/null +++ b/LNE4T4oBgHgl3EQf8A7c/content/tmp_files/load_file.txt @@ -0,0 +1,974 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf,len=973 +page_content='GOHSP: A Unified Framework of Graph and Optimization-based Heterogeneous Structured Pruning for Vision Transformer Miao Yin1* Burak Uzkent2, Yilin Shen2, Hongxia Jin2, Bo Yuan1 1 Rutgers University, 2 Samsung Research America Abstract The recently proposed Vision transformers (ViTs) have shown very impressive empirical performance in various computer vision tasks, and they are viewed as an impor- tant type of foundation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' However, ViTs are typically constructed with large-scale sizes, which then severely hin- der their potential deployment in many practical resources- constrained applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To mitigate this challenging prob- lem, structured pruning is a promising solution to compress model size and enable practical efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' However, unlike its current popularity for CNNs and RNNs, structured prun- ing for ViT models is little explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In this paper, we propose GOHSP, a unified framework of Graph and Optimization-based Structured Pruning for ViT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' We first develop a graph-based ranking for measur- ing the importance of attention heads, and the extracted im- portance information is further integrated to an optimization- based procedure to impose the heterogeneous structured spar- sity patterns on the ViT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Experimental results show that our proposed GOHSP demonstrates excellent compres- sion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' On CIFAR-10 dataset, our approach can bring 40% parameters reduction with no accuracy loss for ViT-Small model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' On ImageNet dataset, with 30% and 35% sparsity ratio for DeiT-Tiny and DeiT-Small models, our ap- proach achieves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='65% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='76% accuracy increase over the existing structured pruning methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Introduction Recently applying transformer architecture to computer vi- sion has emerged as an important forefront of foundation model design (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Thanks to the del- icate vision-specific self-attention, inherent minimal induc- tive biases and high scalability and parallelism, vision trans- formers (ViTs) (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Touvron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021) have shown very outstanding and even state-of-the-art performance in many fundamental and downstream image and video processing tasks, such as im- age classification, object detection, super-resolution, video classification etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Motivated by the scaling success of the giant natural lan- guage processing (NLP) transformers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=', BERT (Devlin This work was done during Miao Yin’s internship at Samsung Research America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2018) and GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2020)), the existing ViTs are also constructed with large model sizes to adapt for massive data training (Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Consequently, they are suffering from huge memory footprints and extensive computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' These limitations, if not being properly addressed, could severely hinder the widespread adoption of ViTs in many practical scenarios, especially on the resource- constrained mobile platforms and Internet-of-things (IoT) devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To mitigate this challenging problem, one attractive solu- tion is to perform model compression (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2019) to reduce the network costs with- out affecting task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' However, unlike the current popularity of compressing convolutional and recurrent neu- ral networks (CNNs and RNNs), ViT-oriented model com- pression has not been systematically studied yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In partic- ular, structured pruning, as an important hardware-friendly compression strategy that can bring practical efficiency on the off-the-shelf hardware, is little explored for ViT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To date, a rich set of structured pruning approaches have been proposed and investigated in the existing literatures, and most of them focus on sparsifying the CNNs at the chan- nel level (He, Zhang, and Sun 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' On the other hand, as will be analyzed and elaborated in Section , because of the difference of the underlying architecture, the structured sparse ViT models can exhibit multi-granularity sparsity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=', head-level and column-level) in the different component modules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=', attention head and multi-layer per- ception (MLP)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' The co-existence of such heterogeneous sparse patterns raises a series of new research challenges and questions when we consider the efficient structured pruning strategy for ViT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For instance, for each component module what is the corresponding suitable pruning criterion to obtain its specific sparse pattern?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Also, how should we perform the entire pruning process across different modules with different levels of granularity sparsity to optimize the overall compression and task performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Technical Preview & Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To answer these questions, in this paper we propose GOHSP, a unified frame- work of Graph and Optimization-based Structure Prun- ing for vision transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To be specific, we first de- velop a graph-based ranking approach to measure the im- portance of attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' As a soft-pruning guideline, such importance information is then integrated to the overall arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='05345v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='AI] 13 Jan 2023 optimization-based procedure to impose the different types of structured sparsity in a joint and global way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Overall, the contributions of this paper are summarized as follows: We propose a graph-based ranking algorithm to mea- sure and determine the importance of attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' By modeling the inter-head correlation as a converged Markov chain, the head importance can be interpreted and calculated as the stationary distribution, which is fur- ther used as a soft guideline for the overall pruning pro- cedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' We propose a unified framework to jointly optimize dif- ferent types of structured sparsity in the different mod- ules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' The complicated coordination for different sparse patterns are automatically learned and optimized in a sys- tematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' We evaluate the performance of our structured pruning approach of different ViT models on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' On CIFAR-10 dataset, our approach can bring 40% pa- rameters reduction with no accuracy loss for ViT-Small model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' On ImageNet dataset, with 30% and 40% spar- sity ratio for DeiT-Tiny and DeiT-Small models, our approach achieves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='65% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='76% accuracy increase than the existing structured pruning methods, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Related Work Vision Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Inspired by the grand success of trans- former architecture in NLP domains, deep learning re- searchers have actively explored the efficient transformer- based neural networks for computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Most recently, several vision transformers (ViTs) and their variants have already shown very impressive performance in several im- age and video processing tasks (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Touvron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' However, in order to achieve competitive performance with the state-of-the-art CNNs, ViTs typically have to scale up their model sizes and therefore they suffer from costly computation and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Dynamic Inference with ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To reduce the deploy- ment costs of ViTs, several works (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Bakhtiarnia, Zhang, and Iosifidis 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Uzkent, Yeh, and Er- mon 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Uzkent and Ermon 2020) have been proposed to improve the processing speed via dynamically pruning the tokens/patches or skipping transformer components adap- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Essentially as dynamic inference approaches, this set of works do not pursue to reduce the model sizes but focus on input-aware inference to obtain practical speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Our structured pruning-based solution is orthogonal to them, and these two different strategies can be potentially combined together to achieve higher speed and smaller memory foot- print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Structured Pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Model compression is a promising strategy to reduce the deployment costs of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Among various model compression techniques, structured pruning is a very popular choice because its hardware- friendly nature can bring practical efficiency on the real- world devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Based on different pruning criterias, various structured pruning approaches have been extensively studied Head-based Sparsity (Multi-Head Attention) (a) Conventional Unstructured Sparsity (b) Unified Structured Sparsity (Ours) Column-based Sparsity (Multi-Head Attention) Column-based Sparsity (MLP) Unstructured Sparsity (MLP) Unstructured Sparsity (Multi-Head Attention) Figure 1: (a) Sparsity pattern of ViT models after un- structured pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Only part of the Multi-Head Attention and MLP columns are pruned which are not hardware- friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (b) Heterogeneous sparsity patterns of ViT models after structured pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Certain MLP and Multi-Head At- tention columns are removed which is hardware-friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' in the existing literature (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Lou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2022), and most of them focus on pruning CNN models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' while the efficient structured pruning of ViTs is little explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' One of these studies, (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021), prunes the vision transformers using structured pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2022), on the other hand, focuses on FLOPs reduction with the vision transformers using pruning, layer skipping, and knowledge distillation whereas in our study we focus on structured pruning to mainly reduce the number of parame- ters for building hardware-friendly compressed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For this reason, we compare our method to (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Structured Pruning of ViTs: Analysis Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Considering an L-block vision transformer, W (l) attn = {W (l) qkv, W (l) proj} and W (l) mlp = {W (l) fc1, W (l) fc2} rep- resent the weights of the attention layer and the MLP layer at l-th block, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For each attention layer, there are H self-attention heads, namely W (l) qkv = {W (l,h) qkv }H h=1 and W (l) proj = {W (l,h) proj }H h=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To simplify the notation, in the fol- lowing content we take one block as the example and omit the superscript (layer index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Heterogeneity of structured sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Because of the difference of the network architecture, the meaning of ‘structured sparsity’ varies with different model types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' As described and performed in (Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Anwar, Hwang, and Sung 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2018, 2020), the struc- tured pruning of CNN and RNN typically indicates the re- moval of the entire channels of the weight tensors and the entire columns of the weight matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Notice that here for either of these two cases, only one type of the structured sparse pattern exist because of the architectural homogeneity of the CNN and RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' On the other hand, a ViT model exhibits inherent archi- tectural heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Within the same block, the front-end multi-head attention module and the back-end MLP mod- ule represent two types of design philosophy for information processing, and thereby leading to huge difference on both computing procedures and the available structured sparse patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To be specific, when we consider performing structured pruning of ViT model, three types of structured sparse pat- terns can co-exist with different levels of granularity across different modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For the multi-head attention module, be- cause each attention head is processing the information in- dividually in a parallel way, the pruning can be performed at the head-level to sparsify this component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In addition, consider the weights in the heads are represented in the ma- trix format;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' the column-level sparsity can also be introduced towards structured pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Meanwhile, because the MLP consists of multiple weight matrices as well, the column- level of granularity sparsity can be imposed on this back-end module at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Consequently, a structured pruned ViT model can exhibit heterogeneous structured sparsity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Problem Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Based on the above analysis, the structured pruning of a vision transformer model with loss function ℓ(·) can be formulated as the following general op- timization problem: min Wattn,Wmlpℓ(Wattn, Wmlp), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' ∥Wattn∥h 0 ≤ κh attn, ∥Wattn∥c 0 ≤ κc attn, ∥Wmlp∥c 0 ≤ κc mlp, (1) where κc and κh are the desired number of columns and the desired number of heads after pruning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' ∥ · ∥c 0 and ∥ · ∥h 0 are the column-based and head-based group L0- norm, which denote the number of non-zero columns and the number of non-zero heads, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Questions to be Answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Solving the above opti- mization problem is non-trivial since it contains the con- straints involved with multi-granularity sparsity for different model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' More specifically, two important ques- tions need to be answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Question #1: What is the suitable pruning criterion to obtain head-level sparsity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Analysis: From the perspective of information process- ing, multi-head attention shares some interesting similarity with convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Both of them use multiple indi- vidual computing units, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=', attention heads and convolu- tional filters, to perform parallel computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Therefore, a naive way to perform head-level pruning is to leverage the existing criteria developed in the channel pruning of CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' However, such straightforward solution, in principle, may not be the best choice because of two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' First, the re- ceptive fields and the focused locality of the attention head and filters are different, and hence simply using the crite- rion for pruning channels is not a suitable strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Second and more importantly, most of the existing channel pruning criterias are built on the information of each individual chan- nel (the corresponding filter weight and/or its feature map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' When adopting this philosophy in the head pruning, the in- sufficient utilization of inter-head information will probably cause non-negligible performance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Overall, the unique characteristics of multi-head attention mechanism calls for attention-specific pruning criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Question #2: How should we coordinate the pruning across different modules with different levels of granularity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Analysis: As indicated before, three types of structured sparse pattern can co-exist in the different modules of the pruned ViT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' A key component of the to-be-explored structured pruning strategy is to develop a good coordination scheme that can properly impose these different structured sparse patterns in a joint and global way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Consider the com- plicated interaction among different types of structured spar- sity, the expected pruning strategy should be able to solve this problem in a systematic and global way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Structured Pruning of ViTs: Method Graph-based Head Ranking To answer Question #1, we propose a graph-based approach to measure and determine the importance of different at- tention heads, which can be further used for the follow-up pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Our key idea is to model the inter-head correla- tion as a graph, and then leverage the graph-based ranking, a methodology that has been successfully used in many web search and NLP algorithms, such as PageRank (Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 1999), TextRank (Mihalcea and Tarau 2004) and LexRank (Erkan and Radev 2004), to select important attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Graph Construction of Markov Chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To be specific, we first construct a graph G = (A, E) to represent the atten- tion heads and their similarities in the block of a ViT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' The set of nodes A denote all the attention heads {Ah}H h=1, and E is the set of connected edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For edge E(Ai, Aj), its weight is defined as the expected cosine similarity be- tween Ai and Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' According to (Mihalcea and Tarau 2004), the graph defined with such cosine similarity can be inter- preted as a Markov chain, where each node is a state, and the transition probability P(i, j) between two states is the edge weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In such scenario, P(i, j) can be calculated as: P(i, j) = EX∼D [CosineSim(Ai(X), Aj(X))] , (2) where Ai(X) is the output of i-th attention head with sam- pled input X and D is the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Built upon this calcula- tion, the entire transition matrix P of a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' No- tice that as indicated in (Erkan and Radev 2004), each col- umn of P should be further normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Batch estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Calculating the transition probability can be very costly since it needs to be performed across the entire training dataset D (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To solve this problem, we adopt a batch-based estimation strategy to improve com- putation efficiency without sacrificing ranking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To be specific, as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 3, only a batch of training data is sampled and used to to estimate the transition prob- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' As our ablation study in Section will show, using different batch sizes (B) bring very stable ranking results for the attention heads, thereby empirically verifying the ef- fectiveness of this estimation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' P(i, j) = CosineSim � B � b=1 Ai(Xb), B � b=1 Aj(Xb) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (3) Importance Ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Mathematically, an irreducible and aperiodic Markov chain is guaranteed to converge to a stationary distribution (Seneta 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' As indicated in (Erkan and Radev 2004), once converged, the probability of a ran- dom walker stays in one state can reflect the state impor- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Motivated by this observation, we propose to quantify Multi-Head Attention Embedded Patches MLP Block 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='1 Graph-based Heads Ranking Multi-Head Attention Embedded Patches MLP Block Data Optimization-based Soft Pruning Multi-Head Attention Embedded Patches MLP Block Fine-Tuning X Score Mask 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='3 Normalize Figure 2: Procedure of the proposed multi-stage structured pruning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' the importance of each attention head via calculating the sta- tionary distribution in our constructed Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To that end, the iterative power method (Erkan and Radev 2004) can be used via setting a uniform distribution for the states as the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Overall, the entire graph-based head ranking procedure is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Soft-Pruning Mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Once the importance score for each state is obtained via calculating the stationary distribution, the corresponding attention heads can be ranked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Here we use a binary mark matrix Mattn = {Mqkv, Mproj} to in- dicate the weight entries associated with the least important heads that should be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Notice that at this stage the head pruning is not performed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Instead such binary mask serves as the guideline for the next-stage optimization, and it is essentially integrated into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 1 as follows: min Wattn,Wmlpℓ(Wattn, Wmlp) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' ∥(1 − Mattn) ⊙ Wattn∥0 = 0, ∥Wmlp∥0 ≤ κc mlp, ∥Mattn ⊙ Wattn∥c 0 ≤ κc attn, (4) where ⊙ is element-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In general, because the overall optimization phase coordinates and adjusts the dif- ferent types of structured sparse pattern from a global per- Algorithm 1: Graph-based Attention Head Ranking Input: Sampled batch {Xb}B b=1, attention heads {Ah}H h=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Output: Importance score s = [s1, · · · , sH].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 1: Initialize transition matrix: P := zeros(H, H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2: for i = 1 to H do 3: for j = 1 to H do 4: Calculate P(i, j) via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 5: Normalize each column of P ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 6: Initialize s := ones(H)/H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 7: repeat 8: s′ := s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 9: s := P s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 10: δ := ∥s − s′∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 11: until δ ≤ ϵ spective, this ranking-only ”soft” pruning strategy, instead of directly pruning the least important heads, can provide more flexibility and possibility for the next-stage optimiza- tion procedure to identify better structured sparse models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Optimization-based Structured Pruning As pointed out by Question #2, the co-existence of multi- granularity and multi-location of the sparsity of ViT models make the entire structured pruning procedure become very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To solve this, we propose to use advanced op- timization technique to perform systematic structured prun- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To be specific, considering the complicated interactions among different types of structured sparsity, we do not prune the heads or columns immediately, since any direct hard pruning at the early stage may cause severe accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Instead, we adopt ”soft-pruning” strategy via optimizing the entire ViT models towards the desired structured sparse for- mats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In other words, the three types of sparsity pattern are gradually imposed onto the attention heads and MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To that end, we first relax the constraints of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 4 and rewrite it as follows: min Wattn,Wmlpℓ(Wattn, Wmlp) + λ 2 ∥(1 − Mattn) ⊙ Wattn∥2 F , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' ∥Wmlp∥c 0 ≤ κc mlp, ∥Mattn ⊙ Wattn∥c 0 ≤ κc attn, (5) where λ is the coefficient that controls the influence of quadratic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Optimization-based Soft Pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' As indicated in (Boyd, Parikh, and Chu 2011), when the constraints of con- tinuous non-convex problem are sparsity related (as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 5 shows), Douglas—Rachford splitting method (Eckstein and Bertsekas 1992) can be a suitable optimization solution for such types of problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Following this philosophy, we first introduce auxiliary variables Zattn, Zmlp and indicator functions as: g(Zattn) = �0 ∥Mattn ⊙ Zattn∥c 0 ≤ κc attn, +∞ otherwise, (6) h(Zmlp) = �0 ∥Zmlp∥c 0 ≤ κc mlp, +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (7) Algorithm 2: Overall Procedure of GOHSP Framework Input: Dense weight {Wattn, Wmlp}, desired model size {κattn, κmlp}, training data D, number of epochs E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Output: Structured sparse weight { ˜ Wattn, ˜ Wmlp};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 1: Sample a batch of data {Xb}B b=1 from D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2: Calculate importance score s via Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 3: Obtain structured mask Mattn according to s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 4: Zattn := Wattn, Zmlp := Wmlp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' // Initialize auxiliary variables 5: Uattn := 0, Umlp := 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' // Initialize Lagrangian multi- pliers 6: for e = 1 to E do 7: Update Wattn, Wattn via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 10 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 8: Update Zattn, Zmlp via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 12 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 9: Update Uattn, Umlp via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 14 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 10: Fine-tune pruned weight { ˜ Wattn, ˜ Wmlp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Then, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 5 as the following equivalent form: min W ,Z ℓ(Wattn, Wmlp) + g(Zattn) + h(Zmlp)+ λ 2 ∥(1 − Mattn) ⊙ Wattn∥2 F , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Wmlp = Zmlp, Wattn = Zattn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (8) In such scenario, the corresponding augmented Lagrangian function of the above optimization objective is: Lρ(Wattn, Wmlp, Zmlp) = ℓ(Wattn, Wmlp) + g(Zattn)+ h(Zmlp) + λ 2 ∥(1 − Mattn) ⊙ Wattn∥2 F + ρ 2∥Wattn − Zattn + Uattn∥2 F + ρ 2∥Uattn∥2 F + ρ 2∥Wmlp − Zmlp + Umlp∥2 F + ρ 2∥Umlp∥2 F , (9) where ρ > 0 is the penalty parameter, and Uattn, Umlp are the Lagrangian multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Then the variables at step t can be iteratively updated as: W t attn = W t−1 attn−η ℓ(Wattn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' W t−1 mlp ) Wattn − λ � (1 − Mattn) ⊙ W t−1 attn � −ρ(W t−1 attn − Zt−1 attn + U t−1 attn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (10) W t mlp = W t−1 mlp − η ℓ(W t attn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Wmlp) Wmlp − ρ(W t−1 mlp − Zt−1 mlp + U t−1 mlp ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (11) Zt attn = P(W t attn + U t−1 attn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (12) Zt mlp = P(W t mlp + U t−1 mlp ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (13) U t attn = U t−1 attn + W t attn − Zt attn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (14) U t mlp = U t−1 mlp + W t mlp − Zt mlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' (15) Here η is the optimizer learning rate for training the ViT, and P is the Euclidean projection for the sparse constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Final Hard-Pruning and Fine-Tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' After the above described optimization procedure, the structured sparse pat- terns have been gradually imposed onto the ViT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In other words, the weight values of the masked attention heads, as well as some columns of MLPs and attention heads, become extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' At this stage, we can now prune those small weights and then perform a few rounds of fine-tuning to achieve higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Overall, by using graph-based head ranking and optimization-based structured pruning, the previously raised Question #1 and #2 can be properly addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' The overall GOHSP framework is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Experiments Experimental Settings Dataset and Baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' We evaluate the performance of our proposed GOHSP approach on CIFAR-10 and Ima- geNet datasets (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For experiments on the CIFAR-10 dataset, the original dense model is ViT- Small1(Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2020) with 48M parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For experiments on the ImageNet dataset, the original dense models are DeiT-Tiny and DeiT-Small (Touvron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021) with 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='7M and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='1M parameters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Hyper-parameters and Sparsity Ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For our experi- ments on the CIFAR-10 dataset, the batch size, learning rate and ρ are set as 256, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='001, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For Ima- geNet dataset, the batch size, learning rate and ρ are set as 256, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='001, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' For both of these two ex- periments, SGD is selected as the training optimizer with- out using weight decay, and we apply Erd˝os-R´enyi (Mo- canu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2018) to determine the sparsity distribution of each layer given an overall sparsity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In particular, soft- pruning maintains high accuracy at the high sparsity ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Performance Evaluation CIFAR-10 Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Table 1 shows performance compari- son on CIFAR-10 dataset between our proposed GOHSP and other structured pruning method (structured one-shot magnitude pruning (SOMP) (Han, Mao, and Dally 2015) and structured gradually magnitude pruning (SGMP) (Zhu and Gupta 2017)) for ViT-Small model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' It is seen that with the same sparsity ratio, our approach brings significant performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Compared to SGMP approach, 1We take this model from open source library timm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Table 1: Performance comparison between our GOHSP and structured one-shot/gradually magnitude-based pruning (SOMP/SGMP) of ViT-Small model on CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Method Sparsity # Paramters Top-1 (%) Baseline 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0M 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='85 SOMP 40% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='8M 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='07 SGMP 40% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='8M 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='93 GOHSP (Ours) 40% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='8M 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='89 GOHSP (Ours) 80% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='6M 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='40 Table 2: Comparison results of our method, GOHSP, with other structured and unstructured pruning methods on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Model Method Sparsity # Parameters FLOPs ↓ Run-time ↓ Top-1 (%) DeiT-Tiny Baseline 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='7M 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='20 OMP (Unstructured) 30% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='02M 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='56% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='35 GMP (Unstructured) 30% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='02M 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='56% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='56 TP (Unstructured) 30% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='02M 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='56% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='38 SSP (Structured) 30% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='2M 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='69% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='59 S2ViTE (Structured) 30% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='2M 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='69% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='57 % 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='12 GOHSP (Structured) 30% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0M 30% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='41% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='24 DeiT-Small Baseline 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='1M 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='90 SSP (Structured) 40% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='6M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='63% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='74 S2ViTE (Structured) 40% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='6M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='63% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='65% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='22 GOHSP (Structured) 40% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='4M 35% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='61% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='98 GOHSP (Structured) 50% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='1M 39% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='57% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='86 GOHSP achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='96% accuracy increase with the same pruned model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Even compared with the baseline, the structured sparse model pruned by GOHSP can outperform the uncompressed model with 40% fewer parameters while 80% compressed model achieves only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='45% worse than the full ViT-Small model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' ImageNet Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Table 2 summarizes the performance on ImageNet dataset between GOHSP and other structured pruning approaches (SOMP, SGMP, Talyer pruning (TP), Salience-based Structured Pruning (SSP) and S2ViTE(Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021)) for DeiT-Tiny and DeiT-Small models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' It is seen that due to the limited redundancy in such small-size model, the existing pruning approaches suffer from more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='5% accuracy loss when compressing DeiT-Tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Instead, with the even fewer parameters and more FLOPs reduction, our GOHSP approach can achieve at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='68% accuracy in- crease over the unstructured pruning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Compared to the structured pruning approach (SSP), our method enjoys 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='65% accuracy improvement with lower storage cost and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In addition, when compressing DeiT- Small model, with fewer parameters and more FLOPs re- duction, our GOHSP approach can achieve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='76% accuracy increase as compared to the state-of-the-art structured prun- ing method S2ViTE (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2021) and can even outper- form the original DeiT-Small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' With 50% pruned DeiT-Small we achieve similar accuracy to the full DeiT-Small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Finally, we report 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='57% improvement in run-time efficiency with our 50% pruned DeiT-Small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='6 Sparsity 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0 Top-1 Accuracy (%) Ours Hard Pruning Figure 3: Results on the effect of soft-pruning (ours) and hard-pruning for ViT-Small model on CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Ablation Study, Visualization and Discussion To obtain the deep understanding of the effect of our pro- posed approach, we perform several ablation studies and a detailed analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Here the experiments conducted in the ab- lation study focus on compressing ViT-Small on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Soft Pruning vs Hard Pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' As described in Opti- mization section, after ranking the attention heads, we use the ranking information as a soft-pruning mask to guide the next-phase optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' The optimization itself is also a soft-pruning procedure that does not directly zero the weights but gradually impose the structured sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To ana- lyze the effect of this strategy, we conduct an ablation exper- iment via performing the direct hard pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In this ablation study, the least important attention heads are removed ac- cording to their ranks, and the columns of MLPs with least group L1 norm are also pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Such hard pruned models are still trained with the same hyper-parameters settings that are used for soft pruning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 3 shows the curves of top-1 test accuracy with different target sparsity settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 Head Index 1 2 3 4 5 6 7 8 Block Index Batch Size=256 0 2 4 6 1 2 3 4 5 6 7 8 Head Index 1 2 3 4 5 6 7 8 Block Index Batch Size=512 0 2 4 6 1 2 3 4 5 6 7 8 Head Index 1 2 3 4 5 6 7 8 Block Index Batch Size=1024 0 2 4 6 1 2 3 4 5 6 7 8 Head Index 1 2 3 4 5 6 7 8 Block Index Batch Size=1536 0 2 4 6 Figure 4: The effect of batch sizes for ranking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Dif- ferent colors represent different ranking scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' We can see that our head ranking algorithm is not sensitive to batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' The soft-pruning strategy brings very significant accuracy improvement over the direct hard pruning with the same sparsity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Effect of Batch Size on Head Ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 3, the importance scores of attention head is calculated on a batch of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' To investigate the potential impact of batch sizes for the ranking results, we observe the change of rank- ing with different batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 4, the rank- ing results are very stable (almost the same) when the batch size varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Therefore we can conclude that using batches of data can already achieve very good estimation of head rank- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' In other words, our ranking approach has low sensitivity to the distribution of input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Sensitivity of Penalty Parameter ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' We also explore the effect of hyperparameter ρ on the structured pruning proce- dure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 5 (a) shows the convergence of training process with respect to different ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' It is seen that the convergence speed is always fast, and hence it demonstrates the promis- ing convergence property of our approach in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 5 (b) illustrates the L2-norm of the masked entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' It is seen that the larger ρ makes the model exhibit more sparsity at the earlier stage, thereby indicating that larger ρ can bring fewer epochs in the final fine-tuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' However, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 5 (c), too large ρ brings accuracy degradation, so ρ can be considered as a parameter that controls the trade-off be- tween the speed of imposing sparsity and task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 6 illustrates the sparsity patterns in the pruned ViT models after performing our GOHSP ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' It is seen that three types of structured sparsity pat- terns (head-level sparsity, column-level sparsity in the head and column-level sparsity in the MLP) are imposed on the 0 10 20 30 40 50 60 Epoch 0 250 500 Loss (a) Curves of training loss =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='001 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='002 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0005 0 10 20 30 40 50 60 Epoch 0 50 100 L2-Norm (b) Curves of sparsity strength =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='001 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='002 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0005 0 10 20 30 40 50 60 Epoch 50 75 100 Top-1 (%) (c) Curves of test accuracy =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='001 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='002 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content='0005 40 50 60 0 3 6 40 50 60 96 97 98 Figure 5: Effect of ρ on the structured pruning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' ρ controls the trade-off between the speed of imposing sparsity and task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' pruned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Such pruning can be more effective on hard- ware than the unstructured pruning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Block9 Block10 Block11 Multi-Head Attention Layer MLP Layer Block9 Block10 Block11 Multi-Head Attention Layer MLP Layer Block9 Block10 Block11 Multi-Head Attention Layer MLP Layer Figure 6: Visualization of the imposed structured sparsity on the DeiT-Small model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' The columns and heads with lighter color are pruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Our method can prune columns (Block9, Block10, and Block11), and heads (Block10, Block11) of the Multi-Head Attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' On the other hand, we can prune columns of MLP layers in all the blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Why Douglas—Rachford splitting method?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' As shown in our Optimization section, the iterative Dou- glas—Rachford splitting technique is adopted to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Such choice is built on two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 1) Convergence: Douglas—Rachford splitting method is a primal-dual optimization method that enjoys fast convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' According to (Boyd, Parikh, and Chu 2011), within a few iterations it can provide satisfied solution for large-scale problems – particularly attractive for DNN applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' More specifically for this work, the fast convergence of Douglas—Rachford splitting method can avoid gradient explosion problem introduced by the additional sparsity loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 2) Flexibility: Douglas—Rachford splitting method, by its nature, divides the original difficult optimiza- tion problem into several less complicated sub-problems, each of which can be then addressed independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' This divide-and-conquer property is very suitable for optimizing the heterogeneous structured pruning of ViT, which explores the different types of structured sparsity across different attention heads and MLPs (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' 10 and 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Conclusion In this paper we propose GOHSP, a unified framework to perform graph and optimization-based heterogeneous struc- tured pruning for vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' By using graph-based ranking and leveraging the advanced optimization tech- nique, our approach can efficiently impose different types of structured sparse patterns on the vision transformers with high compression rate and task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Our experi- ments show that, on ImageNet, with 30 − 50% sparsity, GOHSP compresses the DeiT-Tiny and DeiT-Small mod- els with minor or no loss in accuracy and with ∼ 25 im- provement in rum-time efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Finally, we compress ViT- Small up to 80% on CIFAR10 with minor loss in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' References Anwar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=' Hwang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf'} +page_content=';' metadata={'source': 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0000000000000000000000000000000000000000..550ae8f417fb48880c65773b7addf676f923406a --- /dev/null +++ b/LtFRT4oBgHgl3EQf2Di0/content/tmp_files/2301.13659v1.pdf.txt @@ -0,0 +1,1263 @@ +1 +Spyker: High-performance Library for Spiking +Deep Neural Networks +Shahriar Rezghi Shirsavar†‡, Mohammad-Reza A. Dehaqani†‡, +†School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran +{shahriar.rezghi, dehaqani}@ut.ac.ir +‡School of Cognitive Sciences, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran +∗Corresponding author: Mohammad-Reza A. Dehaqani, dehaqani@ut.ac.ir +Abstract—Spiking neural networks (SNNs) have been recently +brought to light due to their promising capabilities. SNNs +simulate the brain with higher biological plausibility compared +to previous generations of neural networks. Learning with fewer +samples and consuming less power are among the key features +of these networks. However, the theoretical advantages of SNNs +have not been seen in practice due to the slowness of simulation +tools and the impracticality of the proposed network structures. +In this work, we implement a high-performance library named +Spyker using C++/CUDA from scratch that outperforms its +predecessor. Several SNNs are implemented in this work with +different learning rules (spike-timing-dependent plasticity and +reinforcement learning) using Spyker that achieve significantly +better runtimes, to prove the practicality of the library in the +simulation of large-scale networks. To our knowledge, no such +tools have been developed to simulate large-scale spiking neural +networks with high performance using a modular structure. +Furthermore, a comparison of the represented stimuli extracted +from Spyker to recorded electrophysiology data is performed +to demonstrate the applicability of SNNs in describing the +underlying neural mechanisms of the brain functions. The aim +of this library is to take a significant step toward uncovering the +true potential of the brain computations using SNNs. +Index +Terms—Spiking +Neural +Network, +Learning +Rules, +C++/CUDA, Modular Structure, Biological Plausibility +I. INTRODUCTION +The human brain can operate with amazing robustness and +energy efficiency. Artificial neural networks (ANNs) aim at +modeling the brain, and three generations of these networks +have been developed. Each generation of ANNs improves the +quality of the modeling of the brain compared to the last. The +first generation of ANNs makes use of the McCulloch-Pitts +neurons [1]. Although these neurons are inspired by biological +neurons, time dynamics are not considered in this model, and +the learning rules proposed for them lack power and biological +plausibility. These neurons were used in multi-layer perceptron +(MLPs) [2] and Hopfield [3] networks. +The second generation of ANNs uses a continuous activa- +tion function (ReLU [4] and sigmoid [5], for example) instead +of thresholding, which makes them suitable for processing +analog signals. They have attracted the attention of researchers +in recent years and were able to reach high accuracies [6], [7] +(even surpassing humans) and win different challenges [8]. +Despite the success of DNNs, there are structural differences +between these networks and the human brain. Lack of temporal +dynamics, using analog signals for network propagation and +activation functions, learning rules without biological roots, +and the need for large amounts of data [9] and energy [10] to +achieve acceptable results are among these differences. +The third generation of neural networks is spiking neural +networks (SNNs). The neural models used in these networks +simulate biological neurons more accurately, and the coding +mechanisms used in these networks are found in neural +communications. Furthermore, the learning rules used in these +networks have been discovered in the brain [11]–[13]. Having +lower energy consumption, learning with fewer samples, and +solving more complicated tasks due to time dynamics (several +electrophysiological studies emphasize the role of temporal +dynamics in neural coding [14], [15]) are some of the advan- +tages of SNNs compared to the second generation of ANNs. +SNNs can be used to solve machine learning tasks, study and +explore brain functionality, and run on specialized hardware +with low power consumption. The research being done on +these networks aims to address the disadvantages of DNNs +with more realistic modeling of the brain functionality. +Several high-performance well-established frameworks like +PyTorch [16], TensorFlow [17], and MXNet [18] have been +developed for DNNs in recent years. These libraries have en- +abled DNNs to achieve new highs in solving machine learning +tasks. SNNs are not yet comparable to DNNs due to the lack +of fast simulation tools. There have been some attempts, like +SpykeTorch [19] and BindsNet [20]. SpykeTorch, written on +top of the PyTorch framework, is a simulator for large-scale +spiking neural networks (SDNNs). However, it has a slow +runtime, and training even simple networks can take up to days +to complete. To our knowledge, Spyker is the first toolbox to +simulate large-scale networks with high performance, is easy +to use, has the flexibility to be used in multiple languages, and +has the compatibility to integrate with other commonly used +tools. In order to fill this need, we have developed Spyker. +Spyker is a C++/CUDA library written from scratch with both +C++ and Python interfaces and support for dense and sparse +structures. Although Spyker is a stand-alone library, it has a +highly flexible API and can work with PyTorch tensors and +Numpy arrays. Figure 1 shows an overview of the library. In +order to increase performance, small-sized integers are used +alongside floating-point numbers. It also uses highly-optimized +low-level back-end libraries such as OneDNN and cuDNN to +speed up heavy computations such as convolutions and matrix +multiplications. Spyker can be compiled on various CPUs to be +arXiv:2301.13659v1 [cs.CV] 31 Jan 2023 + +2 +optimized locally and take advantage of native CPU-specific +instructions. +Spiking neural networks are made of different building +blocks (see [21] for more details). The first block is the +modeling of the biological neurons. Some examples of this +are leaky integrate-and-fire [22], spike-response model [23], +and Izhikevich model [24]. Another building block is neural +coding, which can be rate coding [25], temporal coding, phase +coding and synchrony coding [26], or other coding schemes. +The final building block is the learning mechanism. Examples +of these mechanisms are STDP [27], [28], R-STDP [29], +backpropagation [30], and conversion from ANNs to SNNs +[31]. Spyker has a modular implementation of these three +blocks that enables its users to build SNNs. +Spyker provides SNN functionality with a high-performance +and easy-to-use interface with an open-source and permissive +license. It can run on CPU and CUDA devices and has +both dense and sparse interfaces. The library introduces new +features and fixes most of the shortcomings of its prede- +cessor. The improvements include adding batch processing, +strided convolutions, internal padding for convolutions, fully +connected layers, and the rate coding mechanism. Compared +to its predecessor, the interface of the library is simpler, +closer to the current API of deep learning libraries, and more +straightforward to use. In this work, several successful network +structures are implemented using this library to prove its +operability, its runtime speed is compared to SpykeTorch, and +the results indicate Spyker can run up to eight times faster. +The proposed work is able to reduce the gap between SNNs +and DNNs and bring us a step closer to uncovering the true +potential of spiking neural networks. +We start with a description of dimensionality of the input +arrays and how the spike trains are implemented in the library. +Afterward, we provide an explanation of different building +blocks of SNNs and how they are implemented in Spyker and +modeled in the interface. Then, we implement network struc- +tures that have been succesful to prove its operatibility, and we +compare the performance of the library to its predecessor on +these networks. Furthermore, comparison of the represented +stimuli extracted from Spyker to recorded electrophisiology +data is performed to demonstrate the applicability of SNNs +in describing the underlying neural mechanisms of the brain +functions. Finally, we demonstrate an example usage of the +library and discuss the impacts of this work and how it can +be further improved. +II. METHODS +The interface of the Spyker can be better explained when the +classes and methods of the interface are grouped by building +blocks of SNNs. The categories are feature enhancement, +neural coding, neural model, and learning. In this section, the +structure of the input to the network is explained. Afterward, +the sparse and the dense interfaces are compared. Finally, the +building blocks of the library are discussed in detail. +A. Network Input +Arrays passed through convolutional neural networks that +process images are often four-dimensional arrays composed +of batch size (B or N), number of channels (C), image height +(H), and image width (W). The order can either be BCHW +or BHWC (or NCHW or NHWC). SNNs have temporal +dynamics, and it is implemented as a dimension that represents +time steps in Spyker. The library implements five-dimensional +arrays with BTCHW order (T being the time steps). Since +DNNs process analog signals, data types used in these net- +works are (usually four-byte) floating-point numbers. This data +type can be computationally expensive compared to a small- +sized integer type and take up more space in the memory. +Since SNNs process binary signals, Spyker can optionally use +eight-bit (or wider) integers alongside floating-point numbers +to improve performance further. +B. Dense vs Sparse interface +The dense interface of Spyker uses the fully allocated +memory buffers that are used in neural network computations. +However, the sparse interface only needs to hold the indices +of the spikes. Conversion between dense and sparse interfaces +are provided in the library. The sparse interface has some +advantages compared to the dense interface. In the dense +interface, the time consumed by each operation is a function +of the size of each of the 5 dimensions. However, in the sparse +interface, it depends on the number of spikes. This means both +memory and time consumed will be greatly reduced when +processing sparser signals. Furthermore, since neurons fire at +most once when using rank order coding, the increment of the +number of time steps will have a smaller effect in the sparse +interface compared to the dense interface. +C. Feature Enhancement +A transformation can be used to enhance features of the in- +put signal (image) before the neural coding process [32]–[34]. +This results in highlighted features having higher intensities +and appearing in earlier time steps, meaning more excitation. +Feature enhancement is done through filtering the input here. +Various filters are supported in Spyker, and they are introduced +in the following subsections. +1) Difference of Gaussian Filter: The first filter is the Dif- +ference of Gaussian (DoG). This filter increases the intensities +of edges and other details in the image (see Figure 2 for an +example) [35]. It approximates the center-surround properties +of the ganglion cells of the retina [36] (see also [37], [38]). +This operation is implemented as spyker.DoG(size, filters, pad +, device) where size is the size of the width and the height +of the filter, filters is a list of DoG filter descriptions (each +description takes in two standard deviations), pad is the size +of the padding of the image, and device is the device the filter +will run on (CPU, GPU or others). +2) Gabor Filter: The following filter is the Gabor filter +that determines the presence of specific frequency in content +in a specific direction in the image. Research Indicates [39] +that the Gabor filter is used in the human visual cortex. The +Gabor filter is implemented as spyker.Gabor(size, filters, pad +, device). The parameters of this class are the same as the +DoG class, but the filters are Gabor filter descriptions, and +each description takes in sigma, theta, gamma, lambda, and +psi. + +3 +Numpy Array +PyTorch Tensor +Numpy Array +PyTorch Tensor +Feature Enhancement +Neural Coding +Neural Model +Learning +T=0 +T=1 +T=2 +T=3 +A+ +A- +Fig. 1: Overview of the Spyker library. Spyker API supports PyTorch tensors and Numpy arrays as well as a built-in data +wrapper. The output of Spyker operations have the same container type as the input. The functionality of Spyker can be grouped +into subcategories shown in the figure. +3) Laplacian of Gaussian Filter: The Laplacian of Gaus- +sian (LoG) layer is also implemented in Spyker, and it is ap- +proximated using two DoG filters. An LoG filter with standard +deviation σ can be approximated using two DoG filters with +(σ +√ +2, σ/ +√ +2) and (σ/ +√ +2, σ +√ +2) standard deviations. This +filter exists in Spyker as spyker.LoG(size, stds, pad, device) +where stds are a list of standard deviations needed to describe +multiple LoG filters. +4) Shape of the Filters: The previously explained filters +have kernel size Kc × Kh × Kw, which are square kernels +(Kh = Kw). The input can have B × Ci × Hi × Wi shape +which corresponds to batch, channels, height, and width of the +input, respectively. The output will have B × Co × Ho × Wo +shape where: +Co = Ci × Kc +Ho = Hi + 2 × Ph − Kh + 1 +Wo = Wi + 2 × Ph − Kw + 1 +(1) +and Ph and Pw are height and width padding of the filter. The +Kc filters are applied to each channel separately. +5) Zero-phase Component Analysis: +Final implemented +layer is zero-phase component analysis (ZCA) Whitening. +It has been suggested [34] that this transformation can im- +prove the accuracy of SNNs on real-world images. Spyker +implements an efficient version of ZCA whitening by taking +advantage of routines from highly optimized linear algebra +libraries (BLAS and LAPACK) that operate on symmetric +matrices. This layer is implemented as spyker.ZCA class +which has a fit(array, epsilon) and a call function. +D. Neural Coding +SNNs process spike trains, but the input consists of analog +values (for example, images are made of pixel values). In order +to make these inputs suitable for the network, a conversion +scheme is needed. The mapping from stimuli to neural re- +sponses is called neural coding [40]. Coding schemes imple- +mented in Spyker are explained in the following subsections. +1) Rate Coding: Out of several coding schemes suggested, +rate coding is widely used where the rate of firing of the +neurons represents information. In this scheme, the rate of +firing is dependent on the intensity of the input value (higher +intensity corresponds to faster firing) [25]. The exact time +of firing in each neuron is stochastic in nature and may be +modeled with a Poisson distribution. A lengthy window of +time is required to transmit the information in this coding, +and the spikes are not quite sparse. +2) Temporal Coding: Another popular coding scheme is +temporal coding [41]. Recordings in the primary visual cortex +show [42] that the response latency decreases with the stimulus +contrast. This coding scheme can convey information through +the timings of the spikes. Multiple forms of this scheme have +been proposed, including rank order coding [43]. Instead of +computing the exact timing of each spike, the timings are +computed relative to one another in rank order coding. This +relative (instead of exact) timing can increase invariance to +changes in the input intensity and contrast [43]. It has been +suggested [44] that temporal coding might be more efficient +in some situations. +3) Coding in Spyker: Spyker supports rank order and rate +coding. The concept of time is implemented with spikes +occuring in time steps in this library. Rank order coding maps +higher intensities to earlier time steps of a neuron firing. In +order to calculate the time step the neuron will fire in, Spyker +sorts the intensity values by default. This calculates rank order +between spikes, and the spikes will be distributed among +time steps evenly. The sorting operation is computationally + +S +P +Y +K +E +R4 +T=0 T=1 T=2 T=3 +B&W Image +B&W Image +DoG Filtered +Gabor Filtered +T=0 +T=0 +T=1 +T=1 +T=2 +T=2 +T=3 +T=3 +Input Image +(Gray or HSV) +Feature +Enhancement +Encoded input data ready to be processed by the network +Neural Coding +Fig. 2: The figure shows a black and white image being filtered by DoG and Gabor filters. The theta parameter of the Gabor +filter is set to -15 degrees. Then the images are coded using rank order coding into four time steps. Spikes are shown with +white color on a black background through time steps. Spikes carry on from the previous to the current time step (cumulative +structure). +expensive (specially on GPUs), and optionally, it can be +disabled to have runtime improvements (however, accuracy +might be affected). Since processing time steps sequantially is +inefficient and time-consuming, Spyker processes all the time +steps at once. To this end, when a neuron fires in time step ti, +it will also fire at time steps ti+1, ti+2, ..., tn where n is the +number of time steps. An example of this cumulative structure +can be seen in Figure 2. +E. Neural Model +Once the input is filtered and coded, it gets processed +by the network. The network is built using fully connected, +convolution, integrate-and-fire (IF) activation, pooling, and +padding layers. These operations are explained in the follow- +ing subsections. +1) Convolution: The integrate-and-fire mechanism is im- +plemented by combining convolution and the IF activation +layer. The internal potentials of the neurons are computed +using convolution operation, and the IF activation operation +produces spikes where neurons have a potential higher than +a specified threshold. Multiple layers can be assembled and +stacked on top of one another to create deeper structures. +The convolution layer has a kernel with Co×Ci×Kh×Kw +shape. the synaptic weights are initialized randomly with +a normal distribution. It performs two-dimensional convo- +lution with support for padding and stride. The input has +B ×T ×Ci ×Hi ×Wi shape which corresponds to batch, time +steps, channels, height, and width of the input, respectively. +The output has B × T × Co × Ho × Wo shape where: +Ho = ⌊Hi + 2 × Ph − Kh +Sh +⌋ + 1 +Wo = ⌊Wi + 2 × Pw − Kw +Sw +⌋ + 1 +(2) +And Ph, Pw, Sh, Sw are the height and width of convolution +padding and stride. Padding increases the size of the two- +dimensional input before convolution operation by expanding +the edges of the input and filling in the new space with a +constant value (usually zero). Stride is the number of steps +the convolution window takes when it moves on the image. +The output of the convolution layers are internal potentials +of neurons that need to be passed through an IF activation +layer to become output spike trains. This layer is imple- +mented with spyker.Conv(insize, outsize, kernel, stride, pad, +mean, std, device) class in Spyker. +2) Fully Connected: The fully connected layer is combined +with the IF activation to model the IF neurons, much similar +to what happens in the convolution layers. This layer has a +kernel with I × O shape. The synaptic weights are initialized + +5 +randomly with a normal distribution. The input has B ×T ×I +which corresponds to batch, time steps, and input size, respec- +tively. The output has B × T × O shape. The fully connected +layer is represeneted by spyker.FC(insize, outsize, mean, std, +device) in the library. +3) Pooling: The pooling layer performs two-dimensional +max pooling operation with a window size ofLh×Lw, a stride +of Sh ×Sw, and a padding of Ph, Pw. The input has B ×T × +Ci×Hi×Wi shape and the output has B×T ×Co×Ho×Wo +shape where: +Ho = ⌊Hi + 2 × Ph − Lh +Sh +⌋ + 1 +Wo = ⌊Wi + 2 × Pw − Lw +Sw +⌋ + 1 +(3) +The interface of Spyker has the spyker.pool(array, kernel, +stride, pad, rates) function to run the pooling operation on the +input given the kernel, stride, and padding size. rates argument +is the rate of firing of the neurons when rate coding is used. +The pooling operation selects neurons that fire earlier when +rank order coding is used, and selects neurons that have a +higher firing rate when rate coding is used. +F. Learning +Learning in the brain happens when the strength of connec- +tions change between its neurons, and this change in strength +is named synaptic plasticity [45]. Learning methods that utilize +synaptic plasticity have been developed for SNNs [27]–[29]. +1) Spike-timing-dependent Plasticity: +One widely rec- +ognized synaptic plasticity learning rule is spike-timing- +dependent plasticity (STDP) [27], [28]. STDP learning rule op- +erates by adjusting synaptic weights and utilizing the timing of +the spikes. A pre-synaptic neuron firing before (after) the post- +synaptic neuron results in a strengthed (weakened) connection. +STDP allows the neurons to extract and learn frequent features +in the input [46]. STDP layer changes synaptic weights with +stabilization: +∆Wi,j = +� +A+ +k (Wi,j − Lk)(Uk − Wi,j), +tj ≤ ti +A− +k (Wi,j − Lk)(Uk − Wi,j), +tj ≥ ti +(4) +where A+ +k , A− +k , Lk, Uk are the positive learning rate, negative +learning rate, lower bound, and upper bound of the kth +configuration, respectively. If stabilization is not set, then the +formula becomes: +∆Wi,j = +� +A+ +k , +tj ≤ ti +A− +k , +tj ≥ ti +(5) +then the weights are computed: +W + +i,j = max(Lk, min(Uk, ∆Wi,j)) +(6) +Input, neurons selected by the winner-take-all mechanism +(WTA), and the output are passed to a function belonging to +fully connected or convolution layers, and the STDP learn- +ing rule is applied. Convolution or fully connected layers +in Spyker can have multiple STDP configurations (differ- +ent learning rules, weight clipping, enabling/disabling stabi- +lizer) implemented as spyker.STDPConfig(positive, negative, +stabilize, lower, upper). Each winner neuron can be mapped +to an STDP configuration, and that neuron will be updated +using the learning rates and such that belongs to the selected +configuration. SpykeTorch creates an STDP object for each +configuration, and mapping winner neurons to different con- +figurations is done by the user. Compared to SpykeTorch, +Spyker provides a more flexible and easy to use API for +weight updating and enables batch updating, which improves +performance. Samples are processed in mini-batches which +increases performance drastically (see the results section), +and the batch update rule does not differ from single-sample +processing. +2) Reward-modulated STDP: Another approach is using +the reinforcement (RL) learning rule. One method based on RL +is reward-modulated STDP [29]. R-STDP adjusts the STDP +such that neurons that respond correctly are rewarded, and +punished otherwise. It has been suggested [33] that when +the input has non-diagnostic frequent features that are less +effective in decision-making, R-STDP is able to discard these +features and improve the decision-making process. Since con- +volution and fully connected layers accept STDP configura- +tions as input, R-STDP can be implemented by passing two +configurations to a layer (one for rewarding and one for pun- +ishing), and mapping each winner neuron to a configuration +based on data labels. If one formulates this, ∆Wi,j will be: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +A+ +r (Wi,j − Lr)(Ur − Wi,j), +tpre < tpost +A− +r (Wi,j − Lr)(Ur − Wi,j), +tpre ≥ tpost +, +if reward +� +A− +p (Wi,j − Lp)(Up − Wi,j), +tpre < tpost +A+ +p (Wi,j − Lp)(Up − Wi,j), +tpre ≥ tpost +, +if punish +(7) +3) Winner-take-all and Lateral Inhibition: When a neu- +ron fires at a specific location, lateral inhibition [47], [48] +operation inhibits other neurons belonging to other neural +maps from firing in that location. Lateral inhibition for the +convolution operation can be used with spyker.inhibit(array +, thershold, inplace) functions. Winner neurons that STDP +weight updating will be performed on are selected by the +winner-take-all [49], [50] operation. WTA selects neurons that +fire earlier, and if the firing time of neurons is the same, then +the one that has a higher internal potential will be selected. +This operation is implemented with spyker.fcwta(array, radius +, count, threshold) for fully connected and spyker.convwta( +array, radius, count, threshold) for convolution operations. +III. RESULTS +In this section, we will test the performance of the library +against the SpykeTorch library. Afterward, a comparison of the +represented stimuli extracted from Spyker to recorded electro- +physiology data is conducted to demonstrate the applicability +of SNNs in describing the underlying neural mechanisms of +brain functions. +A. Library Performance +In this section, we compare the performance of the library to +SpykeTorch on two networks that classify the MNIST dataset. + +6 +SpykeTorch +Spyker Python +Spyker Python Alt +Spyker C++ +Spyker C++ Alt +95 +96 +97 +98 +99 +100 +Accuracy (%) +96.72 +97.55 +97.632 +97.502 +97.606 +Accuracy results for the MNIST dataset +SpykeTorch +Spyker Python +Spyker Python Alt +Spyker C++ +Spyker C++ Alt +0h +3h +6h +9h +12h +15h +18h +21h +Time (s) +21h 17m +4h 21m +3h 21m +4h 7m +3h 8m +Runtime results for the MNIST dataset +Fig. 3: Comparison plots of the runtime and accuracy of +Spyker aganist SpykeTorch on the Mozafari et al. network. +The plot on the left shows the runtime comparison of Spyker +and SpykeTorch implementations. The plot on the right also +compares accuracy of the two implementations. Comparisons +are between SpykeTorch (ST), implementation using Spyker +in Python (SP Py), alternative version using Spyker in Python +(SPA Py), and their C++ counterparts (SP C++, SPA C++). The +error bars are minimum and maximum values of the samples. +1) R-STDP Network: The first netwrok is the Mozfari et al. +network [33] which has three convolutional layers. The first +layer is trained two times with STDP, the second layer four +times with STDP, and the third layer 680 times with R-STDP +on the training set while compuing the test accuracy at each +iteration while training the third layer. We made a small change +to the structure of the network (named Alt for alternative). +We removed the input padding from the last convolution layer +and changed its window size to 4 and the output channels +to 400. Results can be seen in Figure 3 and Table I. All the +tests are performed on Inte Core i7-9700k with 64G memory +and Nvidia Geforce GTX 1080 Ti with 12G memory (Ubuntu +18.04). +In order to compare the results, we test whether the two- +sample mean difference confidence interval (99.9%) contains +zero. The null hypothesis is having the same means, and the +alternative is having different means. The test results indicate +that the Spyker Python implementation is faster compared +to the SpykeTorch implementation (Confidence intervals are +TABLE I: Comparisons of the the runtime and accuracy of +Spyker aganist SpykeTorch on the Mozafari et al. network. +Implementation +Time +Time +(S±SD) +Accuracy +(%±SD) +Runs +SpykeTorch +21h17m +76,672±916 +96.720±0.163 +12 +Spyker Python +04h49m +15668±52 +97.550±0.169 +30 +Spyker Python Alt +03h31m +12,114±14 +97.632±0.112 +30 +Spyker C++ +03h52m +14,869±50 +97.502±0.157 +30 +[15477, 15859] and [72607, 80737] for Spyker and Spyke- +Torch respectively, showing no intersection). Furthermore, +the alternative implementation is faster both in the Python +implementation with [-3738, -3370] interval and the C++ +implementation with [-3828, -3339] interval. As expected, the +C++ interface is faster compared to the Python interface with [- +1078, -520] interval. The results for the accuracy comparisons +show that there are no significant differences ([96.932, 98.169] +and [95.996, 97.444] for Python vs SpykeTorch implementa- +tions respectively, showing intersection, [-0.89, 0.793] for C++ +vs Python, [-0.649, 0.813] for Python alternative vs Python, +and [-0.763, 0.971] for C++ alternative vs C++). +2) STDP Network: Subsequently, the Kheradpisheh et al. +network [32] is used for comparisons. This network is made of +two convolutional layers. The first layer is trained 2 times with +STDP, and the second layer is trained 20 times with STDP on +the training set. The output of the network is classified uing +the SVM classifier. The elapsed time measured consists of +the time needed to train the network on the training set and +make predictions for the testing set. The time to utilize SVM +is not taken into account because the libraries that simulate +the neural network portion are compared here. The results can +be seen in in Figure 4 and Table II. +TABLE II: Comparisons of the the runtime and accuracy +of Spyker aganist SpykeTorch on the Kheradpisheh et al. +network. +Implementation +Time +Time +(S±SD) +Accuracy +(%±SD) +Runs +SpykeTorch GPU +47m30s +2,850±64 +98.392±0.093 +30 +Spyker GPU Single +21m23s +1,283±6 +98.465±0.095 +30 +Spyker GPU +05m53s +353±9 +98.461±0.079 +30 +Spyker Sparse +08m16s +496±1 +98.464±0.091 +30 +The test results indicate that the Spyker GPU implemen- +tation is faster compared to the SpykeTorch implementation +(confidence interval [-2728, -2265]). Since the SpykeTorch +implementation processes one sample at a time, we also +implemented a single sample version on the GPU, and this +implementation runs faster compared to the SpykeTorch im- +plementation (confidence interval [-1795, -1338]). There is +also an implementation using the sparse interface of the +Spyker (that runs on CPU) that is faster than the SpykeTorch +implementation on the GPU (confidence interval [-2586, - +2120]). These results show that the Spyker implementation is +faster while the accuracy is not significantly different ([-0.373, +0.511] for Spyker GPU, [-0.458, 0.603] for single-sample, +and [-0.405, 0.549] for sparse implementation, all against the + +7 +SpykeTorch GPU +Spyker GPU Single +Spyker GPU +Spyker CPU Sparse +95 +96 +97 +98 +99 +100 +Accuracy (%) +98.393 +98.465 +98.462 +98.465 +Accuracy results for the MNIST dataset +SpykeTorch GPU +Spyker GPU Single +Spyker GPU +Spyker CPU Sparse +0m +10m +20m +30m +40m +50m +Time (s) +47m 30s +21m 23s +5m 53s +8m 16s +Runtime results for the MNIST dataset +Fig. 4: Comparison plots of the runtime and accuracy of +Spyker against SpykeTorch on the Kheradpisheh et al. net- +work. The plot on the left shows shows the runtime com- +parison of Spyker and SpykeTorch implementations. The plot +on the right also compares accuracy of the two implementa- +tions. Comparisons are between GPU implementation using +SpykeTorch (SP GPU), GPU implementation using Spyker +with single-sample instead of batch processing (SP Single), +GPU implementation using Spyker (SP GPU), and Sparse CPU +implementation using Spyker (SP Sparse). The error bars are +minimum and maximum values of the samples. +SpykeTorch implementation). +B. Analyzing the Underlying Structures of the Brain +In order to demonstrate the use case and the importance +of the library in neuroscience research, a similarity analysis is +done in this section to compare the biological plausibility of an +SNN and a deep CNN model. The neural data needed for the +analysis is recorded as spiking activity and LFP signals from +Inferior Temporal (IT) cortex using a single electrode (169 +sessions from two macaque monkeys, the neural data for the +monkeys are pooled together) [51]. The task implemented here +is a Rapid Serial Visual Presentation (RSVP). The intervals +are 50ms for stimulus and 450ms interstimulus. Eighy-one +greyscale images of real-world objects and Gaussian low-pass +filtered and high-pass filtered variations of some are shown +during the task (total 155 images). The categories of the stimuli +are animal face (AF), human face (HF), animal body part(AB), +human body part (HB), natual objects (N), and man-made +objects (MM). +The SNN used here is structurally similar to the one intro- +duced by Shirsavar et al. [52]. The input of the SNN is resized +to 32 and passed through 3 LoG fitlers with stds of 0.471, +1.099, 2.042. The window sizes of the filters are 7. Then, the +output is thresholded and coded into 15 time steps. The first +convolution layer has 16 output channels with awindow size +of 5 and a padding of 2, and the second convolution layer has +32 output channels with a window size of 3 and a padding of +1. The pooling layers have 2 and 3 window sizes, respectively. +The layers are trained 20 times on the images, and the learning +rates are doubled after each image until they reach 0.15. Firing +times (divided by number of time steps) of the final layer is +used as the network output. +The CNN network used here is a ResNet-50 with the +classifier layer replaced. The network is not pretrained. The +input image is resized to 256 and cropped to 224. The network +is trained 15 times on the dataset with Adam optimizer and +0.0001 learning rate. using a linear SVM classifier to classify +the 6 categories. the accuracies for the 6 classes are 51.569 ± +2.240 (SD), 48.623 ± 2.538, and 51.247 ± 2.257 for ResNet- +50, SNN, and an SVM classifier that is used on the average +firing rates of the neural recordings of the monkeys between +150ms and 200m from the onset, respectively. Figure 5 Shows +the results of the analysis. The average Kendall’s Tau value +for the interval between 125ms and 175ms shown in the figure +is tested between the SNN and the ResNet. Using a Mann- +Whitney U test with the alpha value of 0.001 results in a p- +value of 2.028-07, which shows significant difference between +the two. This indicates that the SNN has a closer structure to +monkey brain. +C. Rate Coding Output +In this section, we look at the output of an SNN that uses +rate coding. The SNN network used here is the Shirsavar et +al. [52]. The number of output channels in the convolutional +layers are set to 25 and 50. The training is not changed in +that 15 time steps are used with rank order coding. However, +the inference is done with 300 time steps and rate coding. +Afterward, the spike output of 40 neurons are plotted for one +testing sample for each class shown in Figure 6. The figure also +cointains a plot of T-SNE transformed firing rates as output +fetures and the recall score for each class for the average of +30 runs. The accuracy of the 30 runs is 95.635±0.171 on the +testing set. +IV. LIBRARY DEMONSTRATION +In this section, a sample usage of the library is illustrated. +The network used here is introduced by Shirsavar et al. +[52] to classify the MNIST dataset. The network has two +convolutional layers trained with the STDP learning rule. +The code shown in this section is only a part of the actual +implementation, with the aim of providing a simple example. +For the complete implementation, please visit the GitHub +repository of Spyker1. +1https://github.com/ShahriarRezghi/Spyker + +8 +Inferior Temporal Cortex +450ms +Blank +50ms +Stimulus +450ms +Blank +Time +SNN +ResNet ++ ++ +Fig. 5: Similarity comparison of SNN and ResNet-50 to monkey neural data. The similarity measurement used here is the +cosine similarity. The RDM for the monkey is computed for the 50ms interval after the onset. The RDMs are adjusted with +histogram equalization. The RSA is calculated with 50ms window size and 5ms stride and 95% confidence interval. Kendall’s +Tau measurement is used for the RSA analysis. The RSA is averaged in the interval between 125ms and 175ms and compared +in the plot in the top right with 95% confidence interval. +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Fig. 6: Raster plot of an SNN network for the MNIST test images. In this figure, 40 neurons are plotted in 300 time steps for +10 samples of the MNIST testing set, each image belonging to one class. + +9 +A. Transformation +The transformation from the input image to the network +input consists of feature enhancement and spike coding, shown +in Listing 1. Here, a module named Transform is defined that +performs the transformation when called. This module applies +3 LoG filters with different standard deviations to the input +image with padding to keep the original width and height of +the input. The output is stored in 6 channels. Each channel +of this output is then coded into fifteen time steps using rank +order coding. +Listing 1: Implementation of the Transform module +class Transform : +def +init +( self , device ) : +std = [0.471 , 1.099, 2.042] +self . f i l t = spyker .LoG(3 , std , +pad=3, device=device ) +def +call +( self , data ) : +data = self . f i l t ( data ) +spyker . threshold ( data , 0.01) +return spyker . code( data , 15) +B. Network +The network has two convolutional layers. Here, a module +named Network is defined (shown in Listing 2) to train the +neurons and make predictions. Here, the convolution layers +are initialized, STDP configurations are set, and the winner +selection function is wrapped with a lambda function to keep +the hyperparameters in the initialization of the function of the +network. +Listing 2: Implementation of the Network module +class Network: +def +init +( self , device ) : +self . thresh1 , self . thresh2 = 16, 5 +self .conv1 = spyker .Conv(6 , 100, 5, +pad=2, mean=.5 , std =.02, device=device ) +self .conv2 = spyker .Conv(100, 200, 3, +pad=1, mean=.5 , std =.02, device=device ) +config1 = spyker .STDPConfig(.0004 , −.0003) +config2 = spyker .STDPConfig(.0004 , −.0003) +self .conv1. stdpconfig = [ config1 ] +self .conv2. stdpconfig = [ config2 ] +self .wta1 = lambda x: spyker . convwta(x, 3, 5) +self .wta2 = lambda x: spyker . convwta(x, 1, 8) +C. Learning +Training each layer is done in a separate function shown in +Listing 3. The training of the layers is done in a sequantial +order (one layer after another). Training of the first layer is +done in the train layer1 function with the STDP learning rule. +Here, the output of the first convolution is computed, and +lateral inhibition is performed on it. Then, winner neurons are +selected, and STDP weight updating is performed on them. +The STDP learning rates in the first layer are multiplied by +1.5 every 2000 samples, and the multiplying process stops +once the positive learning rate reaches 0.15. The second layer +is trained in a similar way in the train layer2 function with +the STDP learning rule. +Listing 3: The code for training of the network layers +def train layer1 ( self , data ) : +output = self .conv1( data ) +spyker . threshold ( output , self . thresh1 ) +spyker . inhibit ( output ) +winners = self .wta1( output ) +spikes = spyker . fire ( output ) +self .conv1. stdp ( data , winners , spikes ) +def train layer2 ( self , data ) : +data = self .conv1( data ) +data = spyker . fire (data , self . thresh1 ) +data = spyker . pool(data , 2) +output = self .conv2( data ) +spyker . threshold ( output , self . thresh2 ) +spyker . inhibit ( output ) +winners = self .wta2( output ) +spikes = spyker . fire ( output ) +self .conv2. stdp ( data , winners , spikes ) +After defining the network module, the process of training +and classification is implemented. The training process shown +in Listing 4 involves training each layer once with quantization +afterward. +Listing 4: The training process of the network +for data , target in trainset : +network . train layer1 ( transform ( data ) ) +spyker . quantize (network .conv1. kernel , 0, 0.5 , 1) +for data , target in trainset : +network . train layer2 ( transform ( data ) ) +spyker . quantize (network .conv2. kernel , 0, 0.5 , 1) +D. Inference +The call operator of the network shown in Listing 5 im- +plements the prediction procedure which processes the input +spikes and produces the final network output. +Listing 5: Inference function of the network +def +call +( self , data ) : +data = self .conv1( data ) +data = spyker . fire (data , self . thresh1 ) +data = spyker . pool(data , 2) +data = self .conv2( data ) +data = spyker . fire (data , self . thresh2 ) +data = spyker . pool(data , 3) +return spyker . gather ( data ) . flatten (1) +After training, the output features for every sample in the +training set and the testing set are computed (in the gather + +10 +function). Then, an SVM classifier is trained on the training +set outputs. Finally, predictions are made for the testing set +outputs (shown in Listing 6). +Listing 6: Implementation of the dimension reduction and +classification operations +xtr , +ytr = gather ( network , +transform , +train ) +xte , yte = gather ( network , +transform , +test ) +svm = LinearSVC(C=2.4) . fit ( xtr , +ytr ) +pred = svm. predict ( xte ) +accuracy = ( pred == testy .numpy() ) .mean() +V. DISCUSSION +Our brain has amazing capabilities. It can learn and perform +complicated tasks in a robust manner and with low power +consumption. Artificial neural networks have been created +to mimic the power of the brain processes. Deep neural +networks are ANNs that have had major success in recent +years. However, there are structural differences between these +networks and the brain, and they encounter problems when +it comes to tolerance, energy, and sample efficiency. Spiking +neural networks are the next generation of artificial neural +networks. SNNs are not a new concept. However, they have +been brought to attention recently due to their promising +characteristics. The aim of these networks is to build a better +model of the brain compared to DNNs. +Several well-established simulation tools exist for DNNs. +These tools have allowed DNNs to reach their great success +faster and have helped them to computationally scale up. SNNs +lack such high-performance simulation tools. There have been +some attempts at creating such tools, but they have not been +able to live up to expectations. In this work, we introduced +Spyker, a high-performance library written from scratch using +low-level tools to simulate spiking neural networks on both +CPUs and GPUs. Despite being stand-alone, Spyker has great +flexibility and the ability to integrate with other tools to +create a smooth developing experience. We compared the +performance of this library with SpykeTorch, a simulation tool +built on the PyTorch framework. We showed that Spyker is +multiple times faster compared to this library. Furthermore, +to demonstrate the applicability of SNNs in describing the +underlying neural mechanisms of the brain functions and the +role of Spyker in this field, we compared the similarity of +a spiking neural network implemented with this library with +the similarity of the ResNet model to a macaque monkey +brain. Finally, we illustrated an example implementation to +demonstrate the easy and modern interface of the library. +Strong SNN models can be implemented using the Spyker +library to solve real-world machine learning problems. Fea- +tures like fast processing and having a C++ interface alongside +the Python interface make this library ready for both research +and production. Generalization is an important concept in +machine learning and having neural networks that learn and +run fast are quite desirable. SNNs have the potential to become +state-of-the-art models in machine learning. Other potential +use cases of the library is to study and understand how the +brain processes information using simulations. In other words, +this library enables us to look at neuroscience through the eyes +of a brain-inspired neural network. +Although this library has been shown to be performant, there +is room for more improvements. Spyker has a sparse interface +that runs on the CPU. The sparse interface can be extended to +also run on the GPU, and this can improve the performance +even further. Furthermore, the support for a larger number +of neural models, coding schemes, and learning rules can be +added. This helps the library to cover a great range of SNN +building blocks. When choosing a model to be deployed on +embedded and neuromorphic processors, SNNs are among the +top choices due to their energy efficiency. SNNs are often used +in neuromorphic computing. Another direction that Spyker can +take is in this direction. 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A Faster +Approach to Spiking Deep Convolutional Neural Networks, October +2022. + diff --git a/LtFRT4oBgHgl3EQf2Di0/content/tmp_files/load_file.txt b/LtFRT4oBgHgl3EQf2Di0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..44fc2e331ae07186890a54aca2e7da03f2d97979 --- /dev/null +++ b/LtFRT4oBgHgl3EQf2Di0/content/tmp_files/load_file.txt @@ -0,0 +1,772 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf,len=771 +page_content='1 Spyker: High-performance Library for Spiking Deep Neural Networks Shahriar Rezghi Shirsavar†‡, Mohammad-Reza A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Dehaqani†‡, †School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran {shahriar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='rezghi, dehaqani}@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='ir ‡School of Cognitive Sciences, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran ∗Corresponding author: Mohammad-Reza A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Dehaqani, dehaqani@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='ir Abstract—Spiking neural networks (SNNs) have been recently brought to light due to their promising capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SNNs simulate the brain with higher biological plausibility compared to previous generations of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Learning with fewer samples and consuming less power are among the key features of these networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' However, the theoretical advantages of SNNs have not been seen in practice due to the slowness of simulation tools and the impracticality of the proposed network structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In this work, we implement a high-performance library named Spyker using C++/CUDA from scratch that outperforms its predecessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Several SNNs are implemented in this work with different learning rules (spike-timing-dependent plasticity and reinforcement learning) using Spyker that achieve significantly better runtimes, to prove the practicality of the library in the simulation of large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' To our knowledge, no such tools have been developed to simulate large-scale spiking neural networks with high performance using a modular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Furthermore, a comparison of the represented stimuli extracted from Spyker to recorded electrophysiology data is performed to demonstrate the applicability of SNNs in describing the underlying neural mechanisms of the brain functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The aim of this library is to take a significant step toward uncovering the true potential of the brain computations using SNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Index Terms—Spiking Neural Network, Learning Rules, C++/CUDA, Modular Structure, Biological Plausibility I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' INTRODUCTION The human brain can operate with amazing robustness and energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Artificial neural networks (ANNs) aim at modeling the brain, and three generations of these networks have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Each generation of ANNs improves the quality of the modeling of the brain compared to the last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The first generation of ANNs makes use of the McCulloch-Pitts neurons [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Although these neurons are inspired by biological neurons, time dynamics are not considered in this model, and the learning rules proposed for them lack power and biological plausibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' These neurons were used in multi-layer perceptron (MLPs) [2] and Hopfield [3] networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The second generation of ANNs uses a continuous activa- tion function (ReLU [4] and sigmoid [5], for example) instead of thresholding, which makes them suitable for processing analog signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' They have attracted the attention of researchers in recent years and were able to reach high accuracies [6], [7] (even surpassing humans) and win different challenges [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Despite the success of DNNs, there are structural differences between these networks and the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Lack of temporal dynamics, using analog signals for network propagation and activation functions, learning rules without biological roots, and the need for large amounts of data [9] and energy [10] to achieve acceptable results are among these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The third generation of neural networks is spiking neural networks (SNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The neural models used in these networks simulate biological neurons more accurately, and the coding mechanisms used in these networks are found in neural communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Furthermore, the learning rules used in these networks have been discovered in the brain [11]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Having lower energy consumption, learning with fewer samples, and solving more complicated tasks due to time dynamics (several electrophysiological studies emphasize the role of temporal dynamics in neural coding [14], [15]) are some of the advan- tages of SNNs compared to the second generation of ANNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SNNs can be used to solve machine learning tasks, study and explore brain functionality, and run on specialized hardware with low power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The research being done on these networks aims to address the disadvantages of DNNs with more realistic modeling of the brain functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Several high-performance well-established frameworks like PyTorch [16], TensorFlow [17], and MXNet [18] have been developed for DNNs in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' These libraries have en- abled DNNs to achieve new highs in solving machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SNNs are not yet comparable to DNNs due to the lack of fast simulation tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' There have been some attempts, like SpykeTorch [19] and BindsNet [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SpykeTorch, written on top of the PyTorch framework, is a simulator for large-scale spiking neural networks (SDNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' However, it has a slow runtime, and training even simple networks can take up to days to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' To our knowledge, Spyker is the first toolbox to simulate large-scale networks with high performance, is easy to use, has the flexibility to be used in multiple languages, and has the compatibility to integrate with other commonly used tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In order to fill this need, we have developed Spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spyker is a C++/CUDA library written from scratch with both C++ and Python interfaces and support for dense and sparse structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Although Spyker is a stand-alone library, it has a highly flexible API and can work with PyTorch tensors and Numpy arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Figure 1 shows an overview of the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In order to increase performance, small-sized integers are used alongside floating-point numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' It also uses highly-optimized low-level back-end libraries such as OneDNN and cuDNN to speed up heavy computations such as convolutions and matrix multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spyker can be compiled on various CPUs to be arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='13659v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='CV] 31 Jan 2023 2 optimized locally and take advantage of native CPU-specific instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spiking neural networks are made of different building blocks (see [21] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The first block is the modeling of the biological neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Some examples of this are leaky integrate-and-fire [22], spike-response model [23], and Izhikevich model [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Another building block is neural coding, which can be rate coding [25], temporal coding, phase coding and synchrony coding [26], or other coding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The final building block is the learning mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Examples of these mechanisms are STDP [27], [28], R-STDP [29], backpropagation [30], and conversion from ANNs to SNNs [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spyker has a modular implementation of these three blocks that enables its users to build SNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spyker provides SNN functionality with a high-performance and easy-to-use interface with an open-source and permissive license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' It can run on CPU and CUDA devices and has both dense and sparse interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The library introduces new features and fixes most of the shortcomings of its prede- cessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The improvements include adding batch processing, strided convolutions, internal padding for convolutions, fully connected layers, and the rate coding mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Compared to its predecessor, the interface of the library is simpler, closer to the current API of deep learning libraries, and more straightforward to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In this work, several successful network structures are implemented using this library to prove its operability, its runtime speed is compared to SpykeTorch, and the results indicate Spyker can run up to eight times faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The proposed work is able to reduce the gap between SNNs and DNNs and bring us a step closer to uncovering the true potential of spiking neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' We start with a description of dimensionality of the input arrays and how the spike trains are implemented in the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Afterward, we provide an explanation of different building blocks of SNNs and how they are implemented in Spyker and modeled in the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Then, we implement network struc- tures that have been succesful to prove its operatibility, and we compare the performance of the library to its predecessor on these networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Furthermore, comparison of the represented stimuli extracted from Spyker to recorded electrophisiology data is performed to demonstrate the applicability of SNNs in describing the underlying neural mechanisms of the brain functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Finally, we demonstrate an example usage of the library and discuss the impacts of this work and how it can be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' METHODS The interface of the Spyker can be better explained when the classes and methods of the interface are grouped by building blocks of SNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The categories are feature enhancement, neural coding, neural model, and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In this section, the structure of the input to the network is explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Afterward, the sparse and the dense interfaces are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Finally, the building blocks of the library are discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Network Input Arrays passed through convolutional neural networks that process images are often four-dimensional arrays composed of batch size (B or N), number of channels (C), image height (H), and image width (W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The order can either be BCHW or BHWC (or NCHW or NHWC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SNNs have temporal dynamics, and it is implemented as a dimension that represents time steps in Spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The library implements five-dimensional arrays with BTCHW order (T being the time steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Since DNNs process analog signals, data types used in these net- works are (usually four-byte) floating-point numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This data type can be computationally expensive compared to a small- sized integer type and take up more space in the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Since SNNs process binary signals, Spyker can optionally use eight-bit (or wider) integers alongside floating-point numbers to improve performance further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Dense vs Sparse interface The dense interface of Spyker uses the fully allocated memory buffers that are used in neural network computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' However, the sparse interface only needs to hold the indices of the spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Conversion between dense and sparse interfaces are provided in the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The sparse interface has some advantages compared to the dense interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In the dense interface, the time consumed by each operation is a function of the size of each of the 5 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' However, in the sparse interface, it depends on the number of spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This means both memory and time consumed will be greatly reduced when processing sparser signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Furthermore, since neurons fire at most once when using rank order coding, the increment of the number of time steps will have a smaller effect in the sparse interface compared to the dense interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Feature Enhancement A transformation can be used to enhance features of the in- put signal (image) before the neural coding process [32]–[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This results in highlighted features having higher intensities and appearing in earlier time steps, meaning more excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Feature enhancement is done through filtering the input here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Various filters are supported in Spyker, and they are introduced in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 1) Difference of Gaussian Filter: The first filter is the Dif- ference of Gaussian (DoG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This filter increases the intensities of edges and other details in the image (see Figure 2 for an example) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' It approximates the center-surround properties of the ganglion cells of the retina [36] (see also [37], [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This operation is implemented as spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='DoG(size, filters, pad , device) where size is the size of the width and the height of the filter, filters is a list of DoG filter descriptions (each description takes in two standard deviations), pad is the size of the padding of the image, and device is the device the filter will run on (CPU, GPU or others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 2) Gabor Filter: The following filter is the Gabor filter that determines the presence of specific frequency in content in a specific direction in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Research Indicates [39] that the Gabor filter is used in the human visual cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The Gabor filter is implemented as spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='Gabor(size, filters, pad , device).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The parameters of this class are the same as the DoG class, but the filters are Gabor filter descriptions, and each description takes in sigma, theta, gamma, lambda, and psi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 3 Numpy Array PyTorch Tensor Numpy Array PyTorch Tensor Feature Enhancement Neural Coding Neural Model Learning T=0 T=1 T=2 T=3 A+ A- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 1: Overview of the Spyker library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spyker API supports PyTorch tensors and Numpy arrays as well as a built-in data wrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The output of Spyker operations have the same container type as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The functionality of Spyker can be grouped into subcategories shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 3) Laplacian of Gaussian Filter: The Laplacian of Gaus- sian (LoG) layer is also implemented in Spyker, and it is ap- proximated using two DoG filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' An LoG filter with standard deviation σ can be approximated using two DoG filters with (σ √ 2, σ/ √ 2) and (σ/ √ 2, σ √ 2) standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This filter exists in Spyker as spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='LoG(size, stds, pad, device) where stds are a list of standard deviations needed to describe multiple LoG filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 4) Shape of the Filters: The previously explained filters have kernel size Kc × Kh × Kw, which are square kernels (Kh = Kw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The input can have B × Ci × Hi × Wi shape which corresponds to batch, channels, height, and width of the input, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The output will have B × Co × Ho × Wo shape where: Co = Ci × Kc Ho = Hi + 2 × Ph − Kh + 1 Wo = Wi + 2 × Ph − Kw + 1 (1) and Ph and Pw are height and width padding of the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The Kc filters are applied to each channel separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 5) Zero-phase Component Analysis: Final implemented layer is zero-phase component analysis (ZCA) Whitening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' It has been suggested [34] that this transformation can im- prove the accuracy of SNNs on real-world images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spyker implements an efficient version of ZCA whitening by taking advantage of routines from highly optimized linear algebra libraries (BLAS and LAPACK) that operate on symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This layer is implemented as spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='ZCA class which has a fit(array, epsilon) and a call function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Neural Coding SNNs process spike trains, but the input consists of analog values (for example, images are made of pixel values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In order to make these inputs suitable for the network, a conversion scheme is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The mapping from stimuli to neural re- sponses is called neural coding [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Coding schemes imple- mented in Spyker are explained in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 1) Rate Coding: Out of several coding schemes suggested, rate coding is widely used where the rate of firing of the neurons represents information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In this scheme, the rate of firing is dependent on the intensity of the input value (higher intensity corresponds to faster firing) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The exact time of firing in each neuron is stochastic in nature and may be modeled with a Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' A lengthy window of time is required to transmit the information in this coding, and the spikes are not quite sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 2) Temporal Coding: Another popular coding scheme is temporal coding [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Recordings in the primary visual cortex show [42] that the response latency decreases with the stimulus contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This coding scheme can convey information through the timings of the spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Multiple forms of this scheme have been proposed, including rank order coding [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Instead of computing the exact timing of each spike, the timings are computed relative to one another in rank order coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This relative (instead of exact) timing can increase invariance to changes in the input intensity and contrast [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' It has been suggested [44] that temporal coding might be more efficient in some situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 3) Coding in Spyker: Spyker supports rank order and rate coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The concept of time is implemented with spikes occuring in time steps in this library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Rank order coding maps higher intensities to earlier time steps of a neuron firing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In order to calculate the time step the neuron will fire in, Spyker sorts the intensity values by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This calculates rank order between spikes, and the spikes will be distributed among time steps evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The sorting operation is computationally S P Y K E R4 T=0 T=1 T=2 T=3 B&W Image B&W Image DoG Filtered Gabor Filtered T=0 T=0 T=1 T=1 T=2 T=2 T=3 T=3 Input Image (Gray or HSV) Feature Enhancement Encoded input data ready to be processed by the network Neural Coding Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 2: The figure shows a black and white image being filtered by DoG and Gabor filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The theta parameter of the Gabor filter is set to -15 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Then the images are coded using rank order coding into four time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spikes are shown with white color on a black background through time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spikes carry on from the previous to the current time step (cumulative structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' expensive (specially on GPUs), and optionally, it can be disabled to have runtime improvements (however, accuracy might be affected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Since processing time steps sequantially is inefficient and time-consuming, Spyker processes all the time steps at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' To this end, when a neuron fires in time step ti, it will also fire at time steps ti+1, ti+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=', tn where n is the number of time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' An example of this cumulative structure can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Neural Model Once the input is filtered and coded, it gets processed by the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The network is built using fully connected, convolution, integrate-and-fire (IF) activation, pooling, and padding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' These operations are explained in the follow- ing subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 1) Convolution: The integrate-and-fire mechanism is im- plemented by combining convolution and the IF activation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The internal potentials of the neurons are computed using convolution operation, and the IF activation operation produces spikes where neurons have a potential higher than a specified threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Multiple layers can be assembled and stacked on top of one another to create deeper structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The convolution layer has a kernel with Co×Ci×Kh×Kw shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' the synaptic weights are initialized randomly with a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' It performs two-dimensional convo- lution with support for padding and stride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The input has B ×T ×Ci ×Hi ×Wi shape which corresponds to batch, time steps, channels, height, and width of the input, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The output has B × T × Co × Ho × Wo shape where: Ho = ⌊Hi + 2 × Ph − Kh Sh ⌋ + 1 Wo = ⌊Wi + 2 × Pw − Kw Sw ⌋ + 1 (2) And Ph, Pw, Sh, Sw are the height and width of convolution padding and stride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Padding increases the size of the two- dimensional input before convolution operation by expanding the edges of the input and filling in the new space with a constant value (usually zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Stride is the number of steps the convolution window takes when it moves on the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The output of the convolution layers are internal potentials of neurons that need to be passed through an IF activation layer to become output spike trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This layer is imple- mented with spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='Conv(insize, outsize, kernel, stride, pad, mean, std, device) class in Spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 2) Fully Connected: The fully connected layer is combined with the IF activation to model the IF neurons, much similar to what happens in the convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This layer has a kernel with I × O shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The synaptic weights are initialized 5 randomly with a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The input has B ×T ×I which corresponds to batch, time steps, and input size, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The output has B × T × O shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The fully connected layer is represeneted by spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='FC(insize, outsize, mean, std, device) in the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 3) Pooling: The pooling layer performs two-dimensional max pooling operation with a window size ofLh×Lw, a stride of Sh ×Sw, and a padding of Ph, Pw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The input has B ×T × Ci×Hi×Wi shape and the output has B×T ×Co×Ho×Wo shape where: Ho = ⌊Hi + 2 × Ph − Lh Sh ⌋ + 1 Wo = ⌊Wi + 2 × Pw − Lw Sw ⌋ + 1 (3) The interface of Spyker has the spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='pool(array, kernel, stride, pad, rates) function to run the pooling operation on the input given the kernel, stride, and padding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' rates argument is the rate of firing of the neurons when rate coding is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The pooling operation selects neurons that fire earlier when rank order coding is used, and selects neurons that have a higher firing rate when rate coding is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Learning Learning in the brain happens when the strength of connec- tions change between its neurons, and this change in strength is named synaptic plasticity [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Learning methods that utilize synaptic plasticity have been developed for SNNs [27]–[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 1) Spike-timing-dependent Plasticity: One widely rec- ognized synaptic plasticity learning rule is spike-timing- dependent plasticity (STDP) [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' STDP learning rule op- erates by adjusting synaptic weights and utilizing the timing of the spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' A pre-synaptic neuron firing before (after) the post- synaptic neuron results in a strengthed (weakened) connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' STDP allows the neurons to extract and learn frequent features in the input [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' STDP layer changes synaptic weights with stabilization: ∆Wi,j = � A+ k (Wi,j − Lk)(Uk − Wi,j), tj ≤ ti A− k (Wi,j − Lk)(Uk − Wi,j), tj ≥ ti (4) where A+ k , A− k , Lk, Uk are the positive learning rate, negative learning rate, lower bound, and upper bound of the kth configuration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' If stabilization is not set, then the formula becomes: ∆Wi,j = � A+ k , tj ≤ ti A− k , tj ≥ ti (5) then the weights are computed: W + i,j = max(Lk, min(Uk, ∆Wi,j)) (6) Input, neurons selected by the winner-take-all mechanism (WTA), and the output are passed to a function belonging to fully connected or convolution layers, and the STDP learn- ing rule is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Convolution or fully connected layers in Spyker can have multiple STDP configurations (differ- ent learning rules, weight clipping, enabling/disabling stabi- lizer) implemented as spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='STDPConfig(positive, negative, stabilize, lower, upper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Each winner neuron can be mapped to an STDP configuration, and that neuron will be updated using the learning rates and such that belongs to the selected configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SpykeTorch creates an STDP object for each configuration, and mapping winner neurons to different con- figurations is done by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Compared to SpykeTorch, Spyker provides a more flexible and easy to use API for weight updating and enables batch updating, which improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Samples are processed in mini-batches which increases performance drastically (see the results section), and the batch update rule does not differ from single-sample processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 2) Reward-modulated STDP: Another approach is using the reinforcement (RL) learning rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' One method based on RL is reward-modulated STDP [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' R-STDP adjusts the STDP such that neurons that respond correctly are rewarded, and punished otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' It has been suggested [33] that when the input has non-diagnostic frequent features that are less effective in decision-making, R-STDP is able to discard these features and improve the decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Since con- volution and fully connected layers accept STDP configura- tions as input, R-STDP can be implemented by passing two configurations to a layer (one for rewarding and one for pun- ishing), and mapping each winner neuron to a configuration based on data labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' If one formulates this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' ∆Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='j will be: � � � � � � � � � � � � � � � � A+ r (Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='j − Lr)(Ur − Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' tpre < tpost A− r (Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='j − Lr)(Ur − Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' tpre ≥ tpost ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' if reward � A− p (Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='j − Lp)(Up − Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' tpre < tpost A+ p (Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='j − Lp)(Up − Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' tpre ≥ tpost ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' if punish (7) 3) Winner-take-all and Lateral Inhibition: When a neu- ron fires at a specific location,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' lateral inhibition [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' [48] operation inhibits other neurons belonging to other neural maps from firing in that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Lateral inhibition for the convolution operation can be used with spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='inhibit(array , thershold, inplace) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Winner neurons that STDP weight updating will be performed on are selected by the winner-take-all [49], [50] operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' WTA selects neurons that fire earlier, and if the firing time of neurons is the same, then the one that has a higher internal potential will be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This operation is implemented with spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='fcwta(array, radius , count, threshold) for fully connected and spyker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='convwta( array, radius, count, threshold) for convolution operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' RESULTS In this section, we will test the performance of the library against the SpykeTorch library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Afterward, a comparison of the represented stimuli extracted from Spyker to recorded electro- physiology data is conducted to demonstrate the applicability of SNNs in describing the underlying neural mechanisms of brain functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Library Performance In this section, we compare the performance of the library to SpykeTorch on two networks that classify the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 6 SpykeTorch Spyker Python Spyker Python Alt Spyker C++ Spyker C++ Alt 95 96 97 98 99 100 Accuracy (%) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='72 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='55 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='632 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='502 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='606 Accuracy results for the MNIST dataset SpykeTorch Spyker Python Spyker Python Alt Spyker C++ Spyker C++ Alt 0h 3h 6h 9h 12h 15h 18h 21h Time (s) 21h 17m 4h 21m 3h 21m 4h 7m 3h 8m Runtime results for the MNIST dataset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 3: Comparison plots of the runtime and accuracy of Spyker aganist SpykeTorch on the Mozafari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The plot on the left shows the runtime comparison of Spyker and SpykeTorch implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The plot on the right also compares accuracy of the two implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Comparisons are between SpykeTorch (ST), implementation using Spyker in Python (SP Py), alternative version using Spyker in Python (SPA Py), and their C++ counterparts (SP C++, SPA C++).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The error bars are minimum and maximum values of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 1) R-STDP Network: The first netwrok is the Mozfari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' network [33] which has three convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The first layer is trained two times with STDP, the second layer four times with STDP, and the third layer 680 times with R-STDP on the training set while compuing the test accuracy at each iteration while training the third layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' We made a small change to the structure of the network (named Alt for alternative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' We removed the input padding from the last convolution layer and changed its window size to 4 and the output channels to 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Results can be seen in Figure 3 and Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' All the tests are performed on Inte Core i7-9700k with 64G memory and Nvidia Geforce GTX 1080 Ti with 12G memory (Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In order to compare the results, we test whether the two- sample mean difference confidence interval (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='9%) contains zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The null hypothesis is having the same means, and the alternative is having different means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The test results indicate that the Spyker Python implementation is faster compared to the SpykeTorch implementation (Confidence intervals are TABLE I: Comparisons of the the runtime and accuracy of Spyker aganist SpykeTorch on the Mozafari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Implementation Time Time (S±SD) Accuracy (%±SD) Runs SpykeTorch 21h17m 76,672±916 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='720±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='163 12 Spyker Python 04h49m 15668±52 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='550±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='169 30 Spyker Python Alt 03h31m 12,114±14 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='632±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='112 30 Spyker C++ 03h52m 14,869±50 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='502±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='157 30 [15477, 15859] and [72607, 80737] for Spyker and Spyke- Torch respectively, showing no intersection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Furthermore, the alternative implementation is faster both in the Python implementation with [-3738, -3370] interval and the C++ implementation with [-3828, -3339] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' As expected, the C++ interface is faster compared to the Python interface with [- 1078, -520] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The results for the accuracy comparisons show that there are no significant differences ([96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='932, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='169] and [95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='996, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='444] for Python vs SpykeTorch implementa- tions respectively, showing intersection, [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='89, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='793] for C++ vs Python, [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='649, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='813] for Python alternative vs Python, and [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='763, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='971] for C++ alternative vs C++).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 2) STDP Network: Subsequently, the Kheradpisheh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' network [32] is used for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This network is made of two convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The first layer is trained 2 times with STDP, and the second layer is trained 20 times with STDP on the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The output of the network is classified uing the SVM classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The elapsed time measured consists of the time needed to train the network on the training set and make predictions for the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The time to utilize SVM is not taken into account because the libraries that simulate the neural network portion are compared here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The results can be seen in in Figure 4 and Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' TABLE II: Comparisons of the the runtime and accuracy of Spyker aganist SpykeTorch on the Kheradpisheh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Implementation Time Time (S±SD) Accuracy (%±SD) Runs SpykeTorch GPU 47m30s 2,850±64 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='392±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='093 30 Spyker GPU Single 21m23s 1,283±6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='465±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='095 30 Spyker GPU 05m53s 353±9 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='461±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='079 30 Spyker Sparse 08m16s 496±1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='464±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='091 30 The test results indicate that the Spyker GPU implemen- tation is faster compared to the SpykeTorch implementation (confidence interval [-2728, -2265]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Since the SpykeTorch implementation processes one sample at a time, we also implemented a single sample version on the GPU, and this implementation runs faster compared to the SpykeTorch im- plementation (confidence interval [-1795, -1338]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' There is also an implementation using the sparse interface of the Spyker (that runs on CPU) that is faster than the SpykeTorch implementation on the GPU (confidence interval [-2586, - 2120]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' These results show that the Spyker implementation is faster while the accuracy is not significantly different ([-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='373, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='511] for Spyker GPU, [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='458, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='603] for single-sample, and [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='405, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='549] for sparse implementation, all against the 7 SpykeTorch GPU Spyker GPU Single Spyker GPU Spyker CPU Sparse 95 96 97 98 99 100 Accuracy (%) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='393 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='465 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='462 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='465 Accuracy results for the MNIST dataset SpykeTorch GPU Spyker GPU Single Spyker GPU Spyker CPU Sparse 0m 10m 20m 30m 40m 50m Time (s) 47m 30s 21m 23s 5m 53s 8m 16s Runtime results for the MNIST dataset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 4: Comparison plots of the runtime and accuracy of Spyker against SpykeTorch on the Kheradpisheh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The plot on the left shows shows the runtime com- parison of Spyker and SpykeTorch implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The plot on the right also compares accuracy of the two implementa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Comparisons are between GPU implementation using SpykeTorch (SP GPU), GPU implementation using Spyker with single-sample instead of batch processing (SP Single), GPU implementation using Spyker (SP GPU), and Sparse CPU implementation using Spyker (SP Sparse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The error bars are minimum and maximum values of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SpykeTorch implementation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Analyzing the Underlying Structures of the Brain In order to demonstrate the use case and the importance of the library in neuroscience research, a similarity analysis is done in this section to compare the biological plausibility of an SNN and a deep CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The neural data needed for the analysis is recorded as spiking activity and LFP signals from Inferior Temporal (IT) cortex using a single electrode (169 sessions from two macaque monkeys, the neural data for the monkeys are pooled together) [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The task implemented here is a Rapid Serial Visual Presentation (RSVP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The intervals are 50ms for stimulus and 450ms interstimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Eighy-one greyscale images of real-world objects and Gaussian low-pass filtered and high-pass filtered variations of some are shown during the task (total 155 images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The categories of the stimuli are animal face (AF), human face (HF), animal body part(AB), human body part (HB), natual objects (N), and man-made objects (MM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The SNN used here is structurally similar to the one intro- duced by Shirsavar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The input of the SNN is resized to 32 and passed through 3 LoG fitlers with stds of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='471, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='099, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The window sizes of the filters are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Then, the output is thresholded and coded into 15 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The first convolution layer has 16 output channels with awindow size of 5 and a padding of 2, and the second convolution layer has 32 output channels with a window size of 3 and a padding of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The pooling layers have 2 and 3 window sizes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The layers are trained 20 times on the images, and the learning rates are doubled after each image until they reach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Firing times (divided by number of time steps) of the final layer is used as the network output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The CNN network used here is a ResNet-50 with the classifier layer replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The network is not pretrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The input image is resized to 256 and cropped to 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The network is trained 15 times on the dataset with Adam optimizer and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='0001 learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' using a linear SVM classifier to classify the 6 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' the accuracies for the 6 classes are 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='569 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='240 (SD), 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='623 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='538, and 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='247 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='257 for ResNet- 50, SNN, and an SVM classifier that is used on the average firing rates of the neural recordings of the monkeys between 150ms and 200m from the onset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Figure 5 Shows the results of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The average Kendall’s Tau value for the interval between 125ms and 175ms shown in the figure is tested between the SNN and the ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Using a Mann- Whitney U test with the alpha value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='001 results in a p- value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='028-07, which shows significant difference between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This indicates that the SNN has a closer structure to monkey brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Rate Coding Output In this section, we look at the output of an SNN that uses rate coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The SNN network used here is the Shirsavar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The number of output channels in the convolutional layers are set to 25 and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The training is not changed in that 15 time steps are used with rank order coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' However, the inference is done with 300 time steps and rate coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Afterward, the spike output of 40 neurons are plotted for one testing sample for each class shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The figure also cointains a plot of T-SNE transformed firing rates as output fetures and the recall score for each class for the average of 30 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The accuracy of the 30 runs is 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='635±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='171 on the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' LIBRARY DEMONSTRATION In this section, a sample usage of the library is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The network used here is introduced by Shirsavar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' [52] to classify the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The network has two convolutional layers trained with the STDP learning rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The code shown in this section is only a part of the actual implementation, with the aim of providing a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' For the complete implementation, please visit the GitHub repository of Spyker1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='com/ShahriarRezghi/Spyker 8 Inferior Temporal Cortex 450ms Blank 50ms Stimulus 450ms Blank Time SNN ResNet + + Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 5: Similarity comparison of SNN and ResNet-50 to monkey neural data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The similarity measurement used here is the cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The RDM for the monkey is computed for the 50ms interval after the onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The RDMs are adjusted with histogram equalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The RSA is calculated with 50ms window size and 5ms stride and 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Kendall’s Tau measurement is used for the RSA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The RSA is averaged in the interval between 125ms and 175ms and compared in the plot in the top right with 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 6: Raster plot of an SNN network for the MNIST test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In this figure, 40 neurons are plotted in 300 time steps for 10 samples of the MNIST testing set, each image belonging to one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Transformation The transformation from the input image to the network input consists of feature enhancement and spike coding, shown in Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Here, a module named Transform is defined that performs the transformation when called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This module applies 3 LoG filters with different standard deviations to the input image with padding to keep the original width and height of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The output is stored in 6 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Each channel of this output is then coded into fifteen time steps using rank order coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Listing 1: Implementation of the Transform module class Transform : def init ( self , device ) : std = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='471 , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='099, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='042] self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' f i l t = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='LoG(3 , std , pad=3, device=device ) def call ( self , data ) : data = self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' f i l t ( data ) spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' threshold ( data , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='01) return spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' code( data , 15) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Network The network has two convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Here, a module named Network is defined (shown in Listing 2) to train the neurons and make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Here, the convolution layers are initialized, STDP configurations are set, and the winner selection function is wrapped with a lambda function to keep the hyperparameters in the initialization of the function of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Listing 2: Implementation of the Network module class Network: def init ( self , device ) : self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' thresh1 , self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' thresh2 = 16, 5 self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv1 = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='Conv(6 , 100, 5, pad=2, mean=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='5 , std =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='02, device=device ) self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv2 = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='Conv(100, 200, 3, pad=1, mean=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='5 , std =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='02, device=device ) config1 = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='STDPConfig(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='0004 , −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='0003) config2 = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='STDPConfig(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='0004 , −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='0003) self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' stdpconfig = [ config1 ] self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' stdpconfig = [ config2 ] self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='wta1 = lambda x: spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' convwta(x, 3, 5) self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='wta2 = lambda x: spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' convwta(x, 1, 8) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Learning Training each layer is done in a separate function shown in Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The training of the layers is done in a sequantial order (one layer after another).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Training of the first layer is done in the train layer1 function with the STDP learning rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Here, the output of the first convolution is computed, and lateral inhibition is performed on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Then, winner neurons are selected, and STDP weight updating is performed on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The STDP learning rates in the first layer are multiplied by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='5 every 2000 samples, and the multiplying process stops once the positive learning rate reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The second layer is trained in a similar way in the train layer2 function with the STDP learning rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Listing 3: The code for training of the network layers def train layer1 ( self , data ) : output = self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv1( data ) spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' threshold ( output , self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' thresh1 ) spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' inhibit ( output ) winners = self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='wta1( output ) spikes = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' fire ( output ) self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' stdp ( data , winners , spikes ) def train layer2 ( self , data ) : data = self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv1( data ) data = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' fire (data , self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' thresh1 ) data = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' pool(data , 2) output = self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv2( data ) spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' threshold ( output , self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' thresh2 ) spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' inhibit ( output ) winners = self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='wta2( output ) spikes = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' fire ( output ) self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' stdp ( data , winners , spikes ) After defining the network module, the process of training and classification is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The training process shown in Listing 4 involves training each layer once with quantization afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Listing 4: The training process of the network for data , target in trainset : network .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' train layer1 ( transform ( data ) ) spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' quantize (network .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' kernel , 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='5 , 1) for data , target in trainset : network .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' train layer2 ( transform ( data ) ) spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' quantize (network .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' kernel , 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='5 , 1) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Inference The call operator of the network shown in Listing 5 im- plements the prediction procedure which processes the input spikes and produces the final network output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Listing 5: Inference function of the network def call ( self , data ) : data = self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv1( data ) data = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' fire (data , self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' thresh1 ) data = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' pool(data , 2) data = self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='conv2( data ) data = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' fire (data , self .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' thresh2 ) data = spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' pool(data , 3) return spyker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' gather ( data ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' flatten (1) After training, the output features for every sample in the training set and the testing set are computed (in the gather 10 function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Then, an SVM classifier is trained on the training set outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Finally, predictions are made for the testing set outputs (shown in Listing 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Listing 6: Implementation of the dimension reduction and classification operations xtr , ytr = gather ( network , transform , train ) xte , yte = gather ( network , transform , test ) svm = LinearSVC(C=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' fit ( xtr , ytr ) pred = svm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' predict ( xte ) accuracy = ( pred == testy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='numpy() ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content='mean() V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' DISCUSSION Our brain has amazing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' It can learn and perform complicated tasks in a robust manner and with low power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Artificial neural networks have been created to mimic the power of the brain processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Deep neural networks are ANNs that have had major success in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' However, there are structural differences between these networks and the brain, and they encounter problems when it comes to tolerance, energy, and sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spiking neural networks are the next generation of artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SNNs are not a new concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' However, they have been brought to attention recently due to their promising characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The aim of these networks is to build a better model of the brain compared to DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Several well-established simulation tools exist for DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' These tools have allowed DNNs to reach their great success faster and have helped them to computationally scale up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SNNs lack such high-performance simulation tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' There have been some attempts at creating such tools, but they have not been able to live up to expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In this work, we introduced Spyker, a high-performance library written from scratch using low-level tools to simulate spiking neural networks on both CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Despite being stand-alone, Spyker has great flexibility and the ability to integrate with other tools to create a smooth developing experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' We compared the performance of this library with SpykeTorch, a simulation tool built on the PyTorch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' We showed that Spyker is multiple times faster compared to this library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Furthermore, to demonstrate the applicability of SNNs in describing the underlying neural mechanisms of the brain functions and the role of Spyker in this field, we compared the similarity of a spiking neural network implemented with this library with the similarity of the ResNet model to a macaque monkey brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Finally, we illustrated an example implementation to demonstrate the easy and modern interface of the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Strong SNN models can be implemented using the Spyker library to solve real-world machine learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Fea- tures like fast processing and having a C++ interface alongside the Python interface make this library ready for both research and production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Generalization is an important concept in machine learning and having neural networks that learn and run fast are quite desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SNNs have the potential to become state-of-the-art models in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Other potential use cases of the library is to study and understand how the brain processes information using simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' In other words, this library enables us to look at neuroscience through the eyes of a brain-inspired neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Although this library has been shown to be performant, there is room for more improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Spyker has a sparse interface that runs on the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The sparse interface can be extended to also run on the GPU, and this can improve the performance even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Furthermore, the support for a larger number of neural models, coding schemes, and learning rules can be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' This helps the library to cover a great range of SNN building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' When choosing a model to be deployed on embedded and neuromorphic processors, SNNs are among the top choices due to their energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' SNNs are often used in neuromorphic computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Another direction that Spyker can take is in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' The computational efficiency of the sparse interface of Spyker can be further improved and made compatible with these types of processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' REFERENCES [1] Warren S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' McCulloch and Walter Pitts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' A logical calculus of the ideas immanent in nervous activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' Bulletin of Mathematical Biophysics, 5(4):115–133, December 1943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} +page_content=' [2] F.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf'} diff --git a/Q9E3T4oBgHgl3EQfyQvd/content/tmp_files/2301.04719v1.pdf.txt b/Q9E3T4oBgHgl3EQfyQvd/content/tmp_files/2301.04719v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0c82fa701b6eafa61efe1f25d6d64db29a5ec7c --- /dev/null +++ b/Q9E3T4oBgHgl3EQfyQvd/content/tmp_files/2301.04719v1.pdf.txt @@ -0,0 +1,2519 @@ +How To Optimize My Blockchain? +A Multi-Level Recommendation Approach +Jeeta Ann Chacko +chacko@in.tum.de +Technical University of Munich +Ruben Mayer +mayerr@in.tum.de +Technical University of Munich +Hans-Arno Jacobsen +jacobsen@eecg.toronto.edu +University of Toronto +Abstract +Aside from the conception of new blockchain architectures, +existing blockchain optimizations in the literature primarily fo- +cus on system or data-oriented optimizations within prevailing +blockchains. However, since blockchains handle multiple aspects +ranging from organizational governance to smart contract design, +a holistic approach that encompasses all the different layers of a +given blockchain system is required to ensure that all optimization +opportunities are taken into consideration. In this vein, we define +a multi-level optimization recommendation approach that identi- +fies optimization opportunities within a blockchain at the system, +data, and user level. Multiple metrics and attributes are derived +from a blockchain log and nine optimization recommendations are +formalized. We implement an automated optimization recommen- +dation tool, BlockOptR, based on these concepts. The system is +extensively evaluated with a wide range of workloads covering mul- +tiple real-world scenarios. After implementing the recommended +optimizations, we observe an average of 20% improvement in the +success rate of transactions and an average of 40% improvement in +latency. +ACM Reference Format: +Jeeta Ann Chacko, Ruben Mayer, and Hans-Arno Jacobsen. 2023. How To +Optimize My Blockchain? A Multi-Level Recommendation Approach. In +Proceedings of ACM Conference (Conference’23). ACM, New York, NY, USA, +15 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +Introduction +When blockchains were first introduced, they supported only +simple cryptocurrency exchange transactions [50]. However, over +time blockchains evolved to support complex transactions using +smart contracts, thus entering the arena of decentralized trans- +actional management systems such as distributed databases [64]. +Since blockchains target consensus in a trustless environment, they +cannot easily match the performance of databases [9, 16, 22, 26, 53, +59, 80]. However, with the advent of permissioned blockchains that +offer access control and transaction execution policies, blockchains +strive to improve their performance while still providing at least +partially decentralized trust [3, 5, 28, 48]. +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +Conference’23, June 2023, Seattle, WA, USA +© 2023 Association for Computing Machinery. +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +• Activity reordering +• Process model pruning +• Transaction rate control +User level +• Delta writes +• Smart contract partitioning +• Data model alteration +Data level +• Block size adaptation +• Endorser restructuring +• Client resource boost +System level +Delta keys +Primary key duplication +Primary key alteration +Figure 1: Multi-level blockchain optimization +Apart from the proliferation of new blockchain system designs, +there is highly vibrant and diverse ongoing research in the domain +of system optimizations that focus on performance enhancements +within prevailing permissioned blockchains [13, 27, 36, 37, 41, 54, +65–68]. The vast range of the literature targets control parameter +tuning [13, 41, 68], transaction execution remodeling [27, 37, 66], +and smart contract optimization [54]. However, we notice that a +collective approach that encompasses all these optimization possi- +bilities for a particular blockchain under the same umbrella is miss- +ing. Further, the literature falls short for an end-to-end optimiza- +tion approach that includes not only system-level tuning and data +remodeling but also process model redesign. Since permissioned +blockchains are mainly employed by enterprises, a model-driven ap- +proach is often followed where the setup of the blockchain network, +its transaction regulations, the underlying smart contract, and the +data model are primarily based on a business process model created +specifically for a particular application [21, 40, 56, 63, 69]. Such pro- +cess models may be designed by business domain experts who are +unaware of performance implications. For example, in Hyperledger +Fabric (a.k.a. Fabric) [5], many transaction failures arise due to the +order in which the transactions are executed [13, 65, 67]. Such fail- +ures could be reduced if the client processes that issue the transac- +tions followed a different business logic in the first place. The promi- +nence of data management while executing business processes has +often been highlighted by the database community [11, 20, 34]. We +make a similar argument for the importance of the process view in +blockchains since the aspects covered by blockchains are manifold +and not limited to data alone. +Therefore, given the numerous optimizations possible within +a given blockchain system, their varying influence on a case-by- +case basis [6, 13, 23, 51, 68, 81], and the resulting implementation +efforts, there is a pressing need for a recommendation system that +guides the user in selecting effective optimization strategies suitable +for the blockchain under consideration depending on the specific +arXiv:2301.04719v1 [cs.DC] 11 Jan 2023 + +use-case. Again, we can draw parallels from the exhaustive lit- +erature on parameter tuning and indexing recommendations for +databases [1, 2, 42, 73]. However, since blockchains juggle multiple +factors such as organizational governance [62], database defini- +tions [59], consensus algorithms [46], provenance tracking [60], +and smart contract design [47], a holistic perspective to optimiza- +tion recommendations is desirable, which is currently lacking. +To address this gap, we propose a multi-level optimization rec- +ommendation approach for blockchains that provides to the users +a comprehensive understanding of the different optimization pos- +sibilities for their blockchain system, thus enabling them to make +a well-informed decision. Inspired by the abstraction levels in +databases [45], we define three levels of abstraction for blockchain +optimizations: system, data, and user-level (cf. Figure 1). The system- +level recommendations include identifying ideal system configura- +tions such as the block size or endorsement policy. The data-level +recommendations deal with understanding the data model and op- +timizing smart contracts. The user-level recommendations focus on +business process models and workloads induced by client processes. +For example, we identified two activities in a digital rights man- +agement scenario (cf. Section 5.2) that frequently cause transaction +conflicts and recommend a process model redesign to reduce such +failures. Our approach can also verify compliance with the new +process model. We design and implement a recommendation tool +named BlockOptR that analyzes the blockchain logs from Fabric, +one of the most widely used blockchains by enterprises [61], to +demonstrate the performance improvements yielded by our ap- +proach. +Our contributions can be summarized as follows: +(1) We define a multi-level optimization recommendation approach +that extensively analyzes the blockchain log and recommends opti- +mization possibilities from different perspectives. Our method helps +users gain a comprehensive understanding of their current system +and make educated decisions regarding optimization strategies. +(2) We provide a formal definition for our recommendation strate- +gies based on common attributes, such that any blockchain log +with similar attributes can reuse our approach. We also discuss how +our approach translates to different blockchain platforms, thereby +providing the reader with a technology-independent outlook. +(3) We automate the extraction, preprocessing, and event log gen- +eration techniques for Fabric blockchain data. Thus, our tool Block- +OptR will help to ease further research in the area of log-based +analysis such as process mining in blockchains, since a preprocessed +blockchain log can be directly obtained. +(4) We demonstrate the effectiveness of the optimization recom- +mendations by implementing and evaluating them. Our experi- +ments indicate an average of 20% improvement in the percentage +of successful transactions and an average of 40% improvement in +latency after applying the recommendations by BlockOptR. +(5) We extensively evaluate BlockOptR with three different types +of workloads: A set of 24 synthetic workloads generated with a +wide range of control variables, four widespread use case-based +workloads from the literature, and a real-world event log of a loan +application process. Thus, we cover a wide range of scenarios in +our experimentation that are representative for real blockchain +applications. This aids in overcoming the lack of publicly available +data that restricts current research on process mining in permis- +sioned blockchains. The BlockOptR tool, all the smart contracts, +the workload generation scripts, and all the event logs are released +as open-source to encourage further research in this area [10]. +(6) We further establish the positive effect of our holistic recom- +mendation approach on top of existing blockchain optimizations. +Thus, we highlight that BlockOptR complements existing system- +level blockchain optimization strategies such as FabricSharp [65] +and Fabric++ [67] by adding higher-level optimizations. +2 +Background +2.1 +Hyperledger Fabric +Fabric is an open-source permissioned blockchain system popu- +larly used by enterprises [5]. The main components of Fabric are a +smart contract (called chaincode), a distributed immutable ledger, a +distributed world state database, a set of distributed peers, and an +ordering service. The smart contract defines all the supported trans- +actions on the blockchain. The transaction flow in Fabric follows an +execute-order-validate (EOV) model [70]. The EOV model of Fabric +is comparable to optimistic concurrency control in databases [31] +and is therefore prone to multi-version concurrency control (MVCC) +conflicts, which result in transaction failures. +(1) In the execution phase, transaction proposals are created by +clients and sent to the endorsers. Endorsers are a set of specific peers +that have the authority to execute the smart contract to endorse +a transaction. An endorsement policy is configured to define the +number of required endorsers for a transaction to be deemed valid. +Endorsers generate read-write sets after smart contract execution. +The transaction proposal and the read-write sets are signed by the +endorsers and sent back to the clients. +(2) In the ordering phase, the clients forward these transactions to +the ordering service. The ordering service orders the transactions +into blocks using Raft [55], a crash fault-tolerant consensus algo- +rithm, and sends them to all the peers in the network. Configurable +parameters limit the number of transactions included in a block +(block size) in terms of the number of transactions (block count), a +timeout (block timeout), or the size of transactions in bytes (block +bytes). Blocks are created whenever the buffered set of incoming +transactions satisfies any of the three conditions. +(3) In the validation phase, every peer validates every transaction. +Every peer in the Fabric network has a copy of the distributed ledger +and the world state. Peers validate both the endorser signatures +based on the endorsement policy and the read-write set with the +current world state. If the validation is successful, the world state +is updated. Else, a failure is detected. If the endorsement validation +fails, it is called an endorsement policy failure; if the read-write +set validation fails, it is called an MVCC read conflict. MVCC read +conflicts for range reads are called phantom read conflicts. Regard- +less of the success or failure of the validation, all transactions are +appended to the distributed ledger. Also, in the literature, MVCC +read conflicts are often classified into inter-block and intra-block +failures depending on whether the conflicting transactions reside +in the same block or different blocks in the blockchain [13, 67]. +2.2 +Event Logs and Process Mining +An event log is a record of process executions over time. Pro- +cess mining [75] is the technique of deriving a process model that + +exhibits the most frequent behaviors in an event log. It is mainly +used for process discovery which helps to understand the underlying +process model, conformance checking where deviations between +a predicted process model and the actual behavior of the process +can be identified and model enhancement where bottlenecks are +identified and removed. The minimum data required in an event +log for process mining are: +(1) CaseID: To distinguish different executions of the same pro- +cess. Example: ProductID in a supply chain management +related event log. A complete execution of a process is called +a trace. +(2) Activity name: To identify the different steps in a process. +Example: Ship or Unload activity in a supply chain manage- +ment related event log. +(3) Timestamp: To determine the order of the different activities. +The event log can also have other attributes such as process owner, +resources, and cost. Various algorithms are used to derive the pro- +cess model such as alpha miner [76], heuristics miner [79] and fuzzy +miner [30]. The core concept of all these algorithms is to analyze +the different traces of the set of activities in the log and simplify +the traces through abstraction or aggregation to produce a com- +plete process model. Various open-source and commercial process +mining tools are available (ProM [78], Disco [29], Celonis [12]). +3 +A Process Perspective to Blockchains +Our work posits blockchain optimization as a holistic method- +ology rather than a pure system-level approach by introducing a +process perspective. In this section, we emphasize the necessity and +effectiveness of understanding the dependency between business +processes and the performance of the blockchain through exam- +ples. Further, these examples motivate the need for an optimization +recommender since many process-level optimizations can only be +employed with approval from the decision-making bodies of an +organization and, in most cases, cannot be automatically applied. +Figure 2: Derived process model for SCM scenario +Process model pruning is an example of a process-level opti- +mization that positively affects the system’s performance. Figure 2 +shows the process model derived from the blockchain log of a +supply chain management (SCM) scenario. The highlighted paths +and the traces embedded in the figure identify two unnecessary +branches in the process model. Unless the advanced shipping notice +is pushed (PushASN), one should never execute the Ship activity. +Similarly, the Unload activity should never be executed unless a +product has been shipped. Such illogical activity paths can arise +due to manual errors or transaction failures, and the smart contract +is designed to handle such issues, as we explain in the following +example. +If the Unload transaction executes without a corresponding Ship, +the transaction will only read the state but not modify it. However, +it is up to the smart contract designer to either fail the transac- +tion upon execution or commit the read-only transaction to the +blockchain. Both these designs have their trade-offs. Committing +the transaction adds an immutable record on the blockchain, which +helps to track, for example, individuals or organizations who devi- +ated from the expected process model. In a supply chain manage- +ment scenario specifically, this is critical since the entire pipeline +is distributed, and the primary purpose of the blockchain here is +to provide data provenance among untrusted participants. How- +ever, on the other hand, failing a transaction immediately upon +execution ensures that such unnecessary transactions do not go +through all the time-consuming phases (ordering and validation), +which can improve the system performance. We observe a 27% +improvement in throughput and 19% increase in success rate of +transactions when unnecessary activity paths are pruned in the +smart contract (Section 6.2, Figure 13). The pruning can also be +implemented at the process execution level by enforcing incentive +or penalty measures for organizations or individuals that adhere to +or deviate from the expected process model. This approach ensures +that system performance is not prioritized over data provenance +and hence, combines the advantages of both smart contract designs +we discussed above. +Without activity reordering +Activity order +Activity +Read data, Value +Write data, Value +Validity +1 +PushASN +{ ProductID, 1 } +{ ProductID, 2 } +Success +2 +UpdateAuditInfo +{ ProductID, 1 } +{ AuditID, 001 } +{ AuditID, 002 } +Abort +With activity reordering +Activity order +Activity +Read data, Value +Write data, Value +Validity +1 +UpdateAuditInfo +{ ProductID, 1 } +{ AuditID, 001 } +{ AuditID, 002 } +Success +2 +PushASN +{ ProductID, 1 } +{ ProductID, 2 } +Success +Figure 3: Transaction dependency conflict example +Another cause of failures are transactional dependencies, and +research in serialization algorithms has effectively reduced such fail- +ures through transaction reordering [65, 67]. However, reordering +algorithms are expensive, as they basically need to solve the NP- +hard problem of generating conflict-free dependency graphs [67]. +An increase in endorsement policy failures due to inconsistent +world states and the inability to handle large range queries are +known problems of transaction reordering [13]. A different ap- +proach to the problem of dependency conflicts is to identify re- +orderable and unreorderable [65] activities instead of transactions. +While the literature analyzes the keys accessed by transactions +to understand serializability, the data model needs to be analyzed +for process-level serialization. If two concurrent activities read the +same data element but write to different elements in the data model +then such activities are reorderable. +Figure 4: Derived process model after activity reordering + +QueryProdu +QueryASN +PushASN +UpdateAudi +Shi +Unload43 traces +1.58% of the log +Ship +PushASN +QueryASN +Unload42traces +1.55%of the log +PushASN +QueryProducts +QueryASN +UnloadPushASNFor example, in the same supply chain management scenario, the +UpdateAuditInfo activity reads a productID and writes an auditID, +whereas the PushASN, Ship, and Unload activities read and write +to the productID. Therefore, the pairs {UpdateAuditInfo, PushASN}, +{UpdateAuditInfo, Ship} and {UpdateAuditInfo, Unload} are reorder- +able activities while {PushASN, Ship, Unload} are unreorderable. +Figure 3 shows an example of a reorderable pair of activities where +UpdateAuditInfo can succeed if it is executed either after the com- +mit or before the execution of PushASN. Based on the business +logic, it may be possible to impose procedures to restrict or resched- +ule certain activities to execute only at specific periods. For example, +the corresponding process model in Figure 2 shows that UpdateAu- +ditInfo occurs frequently between PushASN and Ship activities and +therefore, UpdateAuditInfo may be executed before the transactions +of the other two activities commit. However, UpdateAuditInfo is +not a time-critical activity and can be rescheduled to take place only +at specific times when traffic is low on the supply chain. We observe +a 24% increase in throughput and 15% increase in success rate of +transactions after a corresponding redesign where UpdateAuditInfo +and QueryProducts activities are executed after PushASN, Ship, +Unload. The new process model derived from the blockchain log +confirms the adherence to the new design (Figure 4). Thus, by iden- +tifying conflicting activities, the process model can be redesigned +to reduce transaction conflicts before they take place. +4 +Blockchain Optimization Recommender +We introduce an approach to recommend optimizations from +three different abstraction levels: system, data, and user-level. The +primary requirement to design and implement such a multi-level +recommendation system is reliable data on all three levels. Knowl- +edge about the system configurations (e.g., block size) and perfor- +mance (e.g., throughput, transaction failures) is vital for generating +system-level recommendations. Information about the current data +model and access patterns, such as key distribution and dependen- +cies, is essential for data-level recommendations. Lastly, knowledge +concerning the use-case, business processes, and transaction work- +load is necessary for user-level recommendations. It is important to +note that such information is not restricted to a specific level but is +helpful across all levels. For example, the system-level performance +can indicate the need for optimizations at all three levels. +The very definition of a blockchain implies the availability of +a distributed ledger with immutable data regarding every trans- +action executed overtime. If we consider smart contracts, then +every execution of the smart contract results in a transaction that +is logged in the ledger. We consider this data (hereafter referred +to as the blockchain log) as the primary source to derive opti- +mization recommendations since, to our knowledge, such a dis- +tributed ledger consisting of all transactions is available for most +blockchains. Therefore, our transaction-centric approach to de- +riving blockchain optimization recommendations is applicable to +different blockchains. +We preprocess the raw data from the blockchain to create a +blockchain log. Then, we obtain the values for key metrics which +are used to detect multi-level optimization recommendations. Pro- +cess mining strategies are then applied to the blockchain log to +derive the process model. We identify the applicable optimizations +using the recommendations and the derived process model. Figure 5 +Fabric Network +Blockchain Data +Preprocessing +Metrics Derivation +Event Log +Generation +Process Model +Generation +Optimization +Recommendation +BlockOptR +Optimization +Implementation +Figure 5: BlockOptR workflow +illustrates the workflow of our approach. We automated the main +elements of this workflow as a tool, BlockOptR [10], implemented +in Python and Node.js. +4.1 +Blockchain Data Preprocessing +BlockOptR registers as a client on the Fabric network, reads the +entire blockchain and saves it as JSON files. Next, the log is cleaned +by removing the configuration and setup-related transactions that +are not relevant and converted to CSV format. All information re- +garding each transaction executed in the Fabric network is logged +on the blockchain. We extract seven attributes and derive two at- +tributes from this extensive logged data. These attributes enable +the derivation of multiple metrics required to recommend optimiza- +tions. The output of the data preprocessing step is a blockchain log +with the following nine attributes. +(1) Client timestamp: The time at which the client generated +the transaction. +(2) Activity name: The name of the smart contract function +whose execution created the transaction. A(x) defines the +activity name of a transaction x. +(3) Function arguments: The value of the parameters of the +smart contract function. +(4) Endorsers: The set of all endorsers of the transaction. +(5) Invokers: The set of all clients and their respective organi- +zation who invoked the transactions. +(6) Read-write set: The set of keys accessed (read or written) +by the transaction. The separate read set and write set of a +transaction are also kept. RWS(x), RS(x) and WS(x) corre- +spondingly define the read-write set, read set and write set +of a transaction x. +(7) Transaction status: The status of the transaction that can +have the values success, MVCC read conflict (MRC), +phantom read conflict and endorsement policy failure. +ST(x) defines the status of a transaction x. +(8) Transaction type: The type of transaction which is de- +rived from the read-write set. This can have the values read, +write, update, range read and delete. Transaction type +is derived from the read-write set. TT(x) defines the type of +a transaction x. +(9) Commit order: The order of the transactions in the blockchain +log is the order in which transactions were committed to the +blockchain. +4.2 +Event Log Generation +The blockchain log extracted from the Fabric network can be +used as an event log to apply process mining techniques that assist +in recommending user-level optimizations. However, unlike the + +event logs created by process-aware information systems [74], a +CaseID is not directly available in the event log extracted from a +blockchain. Also, in most of the use-cases we observed, no single +attribute is common to all activities that can be directly used as the +CaseID. Therefore, we need to derive a common element for each +use-case based on domain knowledge [4, 8, 17, 19, 44]. Since we are +interested in a transactional perspective of the process model, we +find a common element for all activities by analyzing the function +arguments and read-write sets available in the log. For example, +in the SCM scenario the productKey is a common element for all +activities and is a suitable choice since the use-case is specifically +related to tracking multiple products. This process of extracting the +common element is automated for all the use-cases in this paper +and can be easily extended for other use-cases. Once a common +element is identified, we define a trace as a unique set of activities +with the same value for the common element. We then assign a +new CaseID to every trace. +Further, only the time at which the clients sent the transaction +(client timestamp) is available in our log. However, there is no guar- +antee that the same order in which clients send their transactions +will be maintained when the transactions are committed to the +blockchain. Therefore, to derive the process model accurately, we +use the commit order in place of the timestamp. Thus, with the +generated CaseID and extracted/derived attributes, we have a com- +plete event log. Now, any process mining technique can be applied +to the event log to derive a process model. For example, we used +the Alpha algorithm to derive the process models shown in Figure 2 +and 4 [76]. +4.3 +Metrics +We define a set of metrics by scrutinizing multiple blockchain +logs and analyzing metrics from the literature. +(1) Rate metrics: BlockOptR calculates the average transaction +rate as well as the transaction rate distribution over time intervals +from the event log. Transaction rate (Tr) is the average rate at +which transactions are sent from the clients and is derived from the +total transactions in the log and the client timestamps. Transaction +rate distribution (Trdi) is the transaction rate at a specific interval +i derived from the log. A user-configurable interval size (ins) in +seconds is used to calculate this metric. Usage: Transaction rate is a +useful metric to understand the performance. The rate distribution +provides insights regarding periods of high or low traffic. +(2) Failure metrics: Similar to Tr, the total failure rate (TFr) as +well as the rates of each type of failure (MVCC read conflicts, phan- +tom read conflicts, endorsement policy failures) are derived from +the log. The failure rate distribution (Frdi) is calculated similar to +Trd. Usage: Failure metrics help to detect times of high transac- +tion failures. Optimizations such as transaction rate control can be +applied based on the failure metrics. +(3) Block size: The user-configured block count (Bcount) and block +timeout (Btimeout) are extracted from the log. The average num- +ber of transactions in a block (Bsizeavg) is also derived from the +log. Bsizeavg is equivalent to the average block size and can also +be defined as min{Bcount, Tr ∗ Btimeout}. Usage: Bsizeavg along with +the rate metrics helps a user understand the effectiveness of their +block size configurations. For example, if Tr is 500, Bcount is 100, +Btimeout is 1 and Bsizeavg is 100, then 100 transactions are packed +into a block when 500 transactions are actually available every +second. This means more blocks than necessary are being created +which is inefficient, as block creation is expensive. Similarly, if Tr +is 100, Bcount is 500, Btimeout is 2, and Bsizeavg is 200, then blocks +are created only every 2 seconds and transactions are queued up +for a waiting period before being put into blocks. Both scenarios +lead to performance degradation. So, based on the value of Bsizeavg, +the user can update Bcount and Btimeout to efficiently handle the +transaction rate. +(4) Endorser significance (EDsig) defines the number of transac- +tions endorsed by each endorsing peer. Usage: This metric helps +in identifying endorser bottlenecks. Suppose a limited number of +endorsers always carry out the endorsements. In that case, the user +can consider distributing the transactions more evenly among the +endorsers or expanding the set of endorsers. +(5) Invoker significance (IVsig) defines the number of transac- +tions invoked by each client. Usage: This metric helps to identify +clients and the corresponding organizations that invoke a majority +of the transactions. Client resource allocation decisions of such +organizations can be made based on this metric. +(6) Key frequency (Kfreq) is defined as the number of failed trans- +actions that access a specific key. Key significance (Ksig) is defined +as the number of activities that access a specific key. HK defines +the set of hotkeys that have high key frequency based on user- +configurable thresholds. Usage: Identifying the hotkeys assists the +users to identify optimization possibilities in their smart contracts, +and key significance helps to detect the exact activities (that cor- +respond to smart contract functions) that access the hotkeys. For +example, if several functions access the same key, then the different +functions could be separated into multiple smart contracts. Every +smart contract executes on a different world state, thereby reducing +failures (see example in Section 5). +(7) Data-value correlation (corDV) defines that two transactions +are correlated if both access a same set of keys and one of them +fails. Usage: Data-value correlation helps to identify transaction +dependencies. Such dependent transactions are the root cause of +MVCC read conflicts [13]. Various optimization strategies, such as +process model redesign and transaction rate control, can be applied +to these correlated transactions to mitigate failures. +(8) Proximity correlation (corP) defines the distance between +two transactions that have a high data value correlation. For ex- +ample, if corP(x, y) == 1 then transaction y happened immediately +after x with no transactions in between. Further, we also derive +the activity-based proximity correlation (corPA) which defines +the distance between transactions of the same activity. Usage: Ana- +lyzing if the proximity correlation is “less than the block size” or +“greater than the block size” can reveal useful insights regarding +inter- and intra-block failures. If intra-block failures are very high, +smaller block sizes can potentially reduce failures [13]. This metric +also helps to choose between inter- or intra-block transaction re- +ordering strategies offered by different Fabric optimizations [65, 67]. +4.4 +Optimization Recommendations +We use a multi-level approach to utilize the defined attributes and +metrics for recommending blockchain-specific optimization strate- +gies. The optimization recommendation techniques explained in +this section include configurable thresholds. We define appropriate + +Table 1: Formalization of optimization recommendations +Recommendations +Necessary conditions +Activity +reordering +if corDV (x, y) == 1 ∧ WS(x) ∩ WS(y) == ∅ +Process model +pruning +if A(x) = A(y) ∧ TT (x) ≠ TT (y) +Transaction rate +control +if (Trdi ≥ Rt1) ∧ (Frdi ≥ Trdi ∗ Rt2) +Delta writes +if corPA(x, y) == 1 ∧ ST (x) == MRC ∧ +|WS(x) | == |WS(y) | == 1 ∧ WS(x) ± 1 == WS(y) +Smart contract +partitioning +if Ksig(HKi) > 1 +Data model +alteration +if (Ksig(HKi) == 1) ∨ ( |HK | == 1) +Block size +adaptation +if (Tr ≥ Bsizeavg ∗ Bt) ∨ (Tr < Bsizeavg ∗ Bt) +Endorser +restructuring +if EDsig(e) > |TX | ∗ Et +Client resource +boost +if IVsig(c) > |TX | ∗ It +where x, y ∈ TX, e ∈ E, c ∈ I, HKi ∈ HK +TX, E, I, HK are set of all transactions, endorsers, invokers and hotkeys +Rt1, Rt2, Bt, Et, It are user configurable thresholds +default values for these thresholds based on our analysis of multiple +logs, but the user can adapt these default values to fine-tune the +detection strategies. The necessary condition to recommend each +optimization strategy is formalized in Table 1. +4.4.1 +User Level Recommendations +At the user level, it is essential to focus on the actual workload of the +running application. The rate and order in which the transactions +are generated and committed to the blockchain has a vital impact +on performance. We analyze the rate, dependencies, and type of +the transactions to recommend optimizations at the user level. +(1) Activity reordering: Reorderable pairs of transactions can be +identified by using the data value correlation and the read-write set. +BlockOptR identifies the activities corresponding to such transac- +tion pairs and recommends a process model redesign. The redesign +should ensure that the identified activities are restructured to re- +duce conflicts (cf. Section 3). +(2) Process model pruning: If activities deviate from an expected +behavior, then process model pruning is recommended. The trans- +action type of all transactions related to an activity is analyzed to +identify anomalies. Comparing the traces in the event log and the +derived process model with the identified anomalies helps to detect +model pruning opportunities (cf. Section 3). +(3) Transaction rate control: BlockOptR evaluates the transac- +tion rate distribution over time and identifies times when the rate is +very high. It then checks the failure rates in the same time interval. +If the failure rate is also very high, rate control is recommended. +Two configurable thresholds are used to tune the tolerance level of +transaction rate and failures. +4.4.2 +Data Level Recommendations +For data-level recommendations, we focus on identifying the spe- +cific areas in the data model that can be optimized by analyzing +transaction failures, proximity correlation, read-write sets, and key +significance. This aids the user in altering the smart contract and +thereby the underlying data model to improve performance. +(4) Delta writes: Update transactions that only perform increment +or decrement operations can be converted to delta-writes. Delta +writes enable writing to multiple unique delta keys, which can be +aggregated whenever the current value is required. Reading the +key before each write is also not required. Thus, update transac- +tions are converted to write-only transactions that write to unique +keys. This helps to reduce transaction dependency-related failures. +Delta writes are recommended when a single key is incremented +or decremented by a failed transaction. +(5) Smart contract partitioning: A possibility to reduce transac- +tion dependencies is to split a smart contract into multiple ones. +Each smart contract will access separate world states, thereby avoid- +ing conflicts. The functionality of the original smart contract will +not change because it is possible to invoke functions between the +two smart contracts if interaction is required. +For example, in a music rights management scenario, if a key +MusicID is found to be hot and multiple functions such as Play() +and viewMetaData() access this same key, then one can separate +the functions into two different smart contracts. In other words, the +underlying database is split into two by duplicating the primary +key (MusicID) across both and having different secondary keys in +each. The play count of MusicID is recorded in one and metadata is +read from the other (cf. Section 6.2). This is analogous to designing +the table layout in relational databases. The smart contract needs +to be analyzed and updated to implement this optimization. Smart +contract partitioning is recommended if multiple activities access a +hotkey. +(6) Data model alteration: If activities have a dependency on +themselves, then a data model alteration can be beneficial to reduce +transaction conflicts. For example, in a digital voting scenario, if +a key ElectionID is found to be hot and is only accessed by the +function Vote(), then a possible optimization is to use another +primary key such as VoterID. Then, instead of updating all the +votes together, the votes can be updated per voter (cf. Section 6.2). +Further, if only a single hotkey is detected then it is beneficial to +analyze the data model to understand the reason for the skewed +access to this specific data element (cf. Section 6.3). Data model +alteration is recommended if a hotkey is accessed only by a single +activity or if a single hotkey is detected. +4.4.3 +System Level Recommendations +At the system level, we focus on two crucial system configuration +settings that can significantly affect the performance of Fabric: the +endorsement policy and the block size [13, 68]. Further, we also +identify client bottlenecks to aid in resource allocation decisions. We +use the endorser significance, invoker significance, transaction rate, +and actual block size metrics to derive system-level optimization +recommendations. Since these recommendations are based on the +blockchain log generated by the running application, it helps the +user to identify ideal configuration settings based on their current +use-case and workload, leading to performance improvements. +(7) Block size adaptation: The average transaction rate (Tr), the +average block size (Bsizeavg) and a configurable threshold (Bt) are +used to recommend block size adaptation. The literature recom- +mends smaller block sizes when transaction rates are lower and +larger block sizes when the rates are higher [13, 68]. If the block +size is too small, too many blocks are created, and block creation be- +comes a bottleneck. If the block size is too large, the block creation +is delayed by waiting for sufficient transactions. Therefore, if block +size adaptation is recommended, then set Btimeout and Bcount such + +Fabric Network +Automated +Workflow Engines +ü Activity reordering +ü Activity reordering +ü Transaction rate control +ü Client resource boost +Clients +Smart contract updates +ü Delta writes +ü Smart contract partitioning +ü Data model alteration +ü Process model pruning +Configuration updates +ü Block size adaptation +ü Endorser restructuring +Optimization +Recommendations +BlockOptR +Figure 6: Optimization implementation on a live Fabric network +that min{Bcount, Tr ∗ Btimeout} is equal to Tr. We do not provide +recommendations for block bytes adaptation since it is difficult to +accurately derive the size of a transaction (that can include the +transaction payload, endorser identities and other metadata) from +the log. +(8) Endorser restructuring: For every Fabric transaction gener- +ated by the clients, the corresponding smart contract function is +executed by the endorsers defined in the endorsement policy. Smart +contract execution is a time and resource-consuming action. If the +same endorsers receive a higher load of transactions while others +remain idle, this indicates a bottleneck or load imbalance. Such load +imbalances can occur when the endorsement policy explicitly de- +fines an endorsement as mandatory from a specific set of endorsers. +For example, the endorsement policy And(Org1,OR(Org2,Org3)) +implies that an endorsement from Org1 is mandatory. As a conse- +quence, Org1 could become an endorsement bottleneck. We detect +endorser bottlenecks by identifying endorsers that endorse more +transactions than a user-specified threshold. The default thresh- +old values detect whether all the endorsers participate equally in +the endorsement process. The threshold values can be adapted to +increase or decrease the sensitivity to imbalances. +(9) Client resource boost: Multiple time-consuming tasks are +performed by the clients in a Fabric network, including but not +limited to transaction proposal invocation, endorser response veri- +fication, packing of endorser responses as a transaction, transaction +submission to the ordering service, and collection of peer commit +responses. The invoker significance metrics identify the clients +and the corresponding organizations that invoke a majority of the +transactions. This identification can assist in resource allocation +decisions, such as increasing the number and size of clients regis- +tered to the identified organization. It could also point to problems +in the underlying business process. +4.5 +Implementation of Optimizations +The recommended optimizations can be implemented in several +ways. Figure 6 visualizes where the different recommendations +can be implemented on a live Fabric network. Here, we show an +automated workflow engine that triggers transactions based on a +process model. These transactions are sent via the clients to the +Fabric network. The logs of the Fabric network are analyzed by +BlockOptR to generate optimization recommendations. Each of the +recommended optimizations can be implemented at different levels +as shown in the figure. +Activity reordering can be implemented by modifying the un- +derlying process model in the workflow engine such that activities +follow a conflict-free order. Alternatively, one can monitor the +transactions on the clients and reorder either per client or across +all clients using a client manager. Process model pruning can be +implemented via organizational measures such as incentives or +penalties to ensure that activities adhere to their expected behavior +(not shown in the figure). However, pruning can also be imple- +mented directly in the smart contract by early aborting anomalous +transactions during the endorsement phase. Transaction rate con- +trol can be implemented in multiple ways. Each client can monitor +their own transaction rate and perform load shedding or queuing. +The same can be done across clients using a central monitor. A +third approach is to monitor the transaction rate in the ordering +service and apply load shedding there. Smart contract revisions are +required to implement all the data-level optimizations. In Fabric, +smart contract upgrades are possible on the fly without restarting +the system [72]. Block size can be adapted either by changing the +configuration file or by using a configuration update transaction in +Fabric [71]. Endorser restructuring can be implemented by altering +the endorsement policy. The policy can be changed in the Fabric +configuration file or using a configuration update transaction [71]. +Based on the transaction load per client identified by BlockOptR, +client resources can be scaled if the current allocation appears in- +sufficient to handle the load and the new clients can be dynamically +registered to the Fabric network. +Our implementations. Although all optimizations can be ap- +plied in a live system on the fly, since our evaluation runs in an +experimental environment, we restart the Fabric network after +every experiment. We use the Caliper benchmarking system [35] +which has a client manager that can be configured to order the +transactions across clients and control the rate of transactions gen- +erated, thus emulating activity reordering and transaction rate +control. The number of clients can also be scaled to demonstrate +a client resource boost. Process model pruning and all data-level +optimizations are implemented by analyzing and modifying the +smart contract. Block size and endorsement policies are updated in +the Fabric configuration file. +5 +Experimental Methodology +We used version 2.0 of HyperledgerLab [13], which is an au- +tomated testbed for Hyperledger Fabric 2.2 integrated with the +Caliper benchmarking system. We set up a Kubernetes cluster of +1 master and 5 worker nodes over which all the Fabric network +components as well as Caliper components are distributed as Ku- +bernetes pods. Each node runs on a Ubuntu Focal (20.04) virtual +machine with 4 vCPUs and 9.8 GB RAM. We use 10 Caliper work- +ers for our experiments. For every experiment, we measure the +success rate which is the percentage of successful transactions out + +Table 2: Control variables +Control Variable +Values (Default in bold) +Workload type +Uniform, +Read-heavy, +Insert-heavy, Update-heavy, +RangeRead-heavy +Endorsement policy +P1, P2, P3, P4 +Endorser distribution skew +0, 6 +Key distribution skew +1, 2 +Number of organizations +2, 4 +Block count +50, 300, 1000 +Send rate +50, 300, 1000 +Transaction dist skew +0, 70% +of the total number of transactions, the average latency and the +throughput of all successful transactions. +5.1 +Workload Generation +The content of the distributed ledger, which is used as the in- +put to our tool, is a direct result of the workload executed on the +blockchain. Therefore, we extensively evaluate BlockOptR by using +three different types of workload. Also, after implementing the rec- +ommendations generated by BlockOptR, we rerun the experiments +with the same workloads to analyze the effect of the optimization. +5.1.1 +Synthetic workloads +We use an extended version of a synthetic workload generator +that can generate synthetic workloads based on a set of control +variables for a generic smart contract genChain [13]. We use a range +of values for these control variables described in Table 2 to generate +multiple workloads of 10,000 transactions each. The endorsement +policies used in our experiments are: +P1: And(Org1, Or(Org2,Org3,Org4)) +P2: And(Or(Org1,Org2), Or(Org3,Org4)) +P3: Majority(Org1,...,OrgN) +P4: OutOf(2,Org1,Org2,Org3,Org4) +By generating synthetic workloads, we ensure that multiple +realistic scenarios are covered in our experiments. We then evaluate +BlockOptR with each of these workloads to generate optimization +recommendations. Further, we implement each of the recommended +optimizations to evaluate the performance improvement. +5.1.2 +Use-case based workloads +Secondly, we use extended versions of four popular use-case +based smart contracts from the literature [13] and generate work- +loads. BlockOptR is then used to generate optimization recommen- +dations with these workloads. The four smart contracts we use are +as follows. +Supply Chain Management (SCM): This smart contract defines +the operations of a logistics network that includes sending an ad- +vanced shipping notice, shipping a product, reading the shipping +notice and unloading the product (in this order). There is also a +query operation to query the information of the different products +(queryProducts) and a updateAuditInfo function that updates +an audit entry with the product details. These can happen at any +point in time. We generated a workload of 10,000 transactions based +on these assumptions by sending in order the transactions pushASN, +ship, queryASN and unload while the transactions queryProducts +and updateAuditInfo are sent randomly. +Digital Rights Management (DRM): This smart contract manages +the rights of artists in the music industry. The smart contract in- +cludes a Play function that is executed whenever a piece of music +is played by any user. The other smart contract functions include +Table 3: Experiments with the synthetic workload +Experiment +Number +Control variable +Value +Optimizations recommended +1 +Endorsement +P1 +Endorser restructuring +Policy +Activity reordering +2 +Endorsement Policy / +P2 / 6 +Endorser restructuring +Endorser dist skew +Activity reordering +3 +No: of orgs +4 +Transaction rate control +4 +Workload +Read-heavy +Activity reordering +5 +Workload +Update-heavy +Transaction rate control +6 +Workload +Insert-heavy +Activity reordering +7 +Workload +RangeRead-heavy +Activity reordering +Transaction rate control +8 +Key +Activity reordering +distribution skew +2 +Smart contract partitioning +Block size adaptation +9 +Block count +50 +Activity reordering +Transaction rate control +10 +Block count +300 +Activity reordering +Transaction rate control +11 +Block count +1000 +Activity reordering +12 +Send rate +50 +Activity reordering +13 +Send rate +300 +Activity reordering +Block size adaptation +Transaction rate control +14 +Send rate +1000 +Activity reordering +Transaction rate control +15 +Transaction +70% +Activity reordering +distribution skew +Client resource boost +adding a new piece of music, querying the rights, viewing the meta- +data and calculating the revenue of the right holders. In a realistic +scenario, the Play transaction would be executed far more fre- +quently than all the other transactions. Therefore, we create a Play +heavy workload for this use-case. We generate 10,000 transactions +randomly where 70% of the transactions are Play. The remaining +30% comprise all the other transactions generated uniformly at +random. +Electronic Health Records (EHR): In this smart contract, patients +can provide or revoke access rights to medical institutes as well as +research institutes to query their medical records. We assume that +the number of patients would be more than the other participants +and generate a 70% update-heavy workload of 10,000 transactions. +Digital Voting (DV): This smart contract includes a function to +vote in an election, query the parties, query the results as well as end +the election. We can assume that during an actual election there will +be periods of high traffic while the voting is taking place. Therefore, +we generate a workload which initially sends 1,000 queryParties +transactions at a rate of 100 TPS, then 5,000 Vote transactions +at a rate of 300 TPS and finally 1 seeResults and endElection +transaction each. +5.1.3 +Loan Application Process (LAP) +Thirdly, we created a smart contract and workload using a real- +life event log of the loan application process of a Dutch financial +institute which is available publicly [77] together with the corre- +sponding process flow [57]. We extracted all the events of the first +2,000 loan applications and created 20,000 corresponding transac- +tions. We then created a smart contract where every activity in the +loan application process flow has a corresponding smart contract +function. The event log contains an employeeID for every employee +in the bank handling loan applications and an applicationID for +every loan application processed by the bank. The smart contract +we implemented uses the employeeID as the key and the value of +the key is an array of structures where every structure includes + +Table 4: Settings to implement optimization +Optimizations +Settings +Recommended +Activity reordering +Reorder workload generation +Transaction rate control +Set send rate to 100 TPS +Process model pruning +Delta writes +Update smart contract +Smart contract partitioning +Data model alteration +Block size adaptation +Set block count to derived transaction rate +Endorser restructuring +Set endorsement policy to P4 +Client resource boost +Double clients for recommended organization +the applicationID, loan type, loan amount and loan status. +Therefore, querying a specific employeeID will easily provide all +the applications processed by that employee. We then executed the +20,000 transactions on the smart contract at a low rate of 10 TPS to +simulate a real world scenario where manually processing the loan +applications takes a long time. We also ran the same experiment at +a higher rate of 300 TPS to emulate an automated loan application +and validation process. We use BlockOptR to generate optimiza- +tion recommendations which help to improve the smart contract +implementation and thereby the performance. +Though the LAP event logs are from a database setting, this +is a realistic use-case for blockchains as an automated loan ap- +plication system requires security and decentralized trust (e.g., +micro-loans, decentralized loan applications, and more generally +DeFi [33, 58, 82, 83]). Consequently, this experiment demonstrates +the utility of BlockOptR in a realistic scenario. In the use-case based +experiments, all the transactions followed the expected order based +on the assumptions we defined. In contrast, with this real event log, +we evaluate the real order in which the transactions are executed +which can deviate from the process model. +6 +Experimental Results +We exhaustively evaluate our recommendation approach with +a wide range of workloads and smart contracts. Please note that, +whenever transaction rate control is implemented there is an ex- +pected decrease in the throughput. However, clients benefit heav- +ily from higher success rates, and the apparent decrease in the +throughput is just closer to the sustainable throughput of the sys- +tem. In all our experiments the default value for the thresholds are +Et = 0.5, Rt1 = 300, Rt2 = 0.3, Bt = 0.6 and It = 0.5. All the settings +including the control variable values changed to implement each +recommended optimization is shown in Table 4. +6.1 +Synthetic Workloads +Due to space restrictions, we present 15 workloads in Table 3. +The full list of experiments and results can be seen in our reposi- +tory [10]. The control variable that is tuned for each experiment +is shown along with its value. All the other control variables have +the default value shown in Table 2. Experiments 1 to 15 are con- +ducted with no optimizations applied and then BlockOptR is used +to derive optimization recommendations. The recommendations +generated by BlockOptR are also shown in Table 3. Since the syn- +thetic smart contract has a simple logic with no branches, incre- +ment/decrement operations or complex data model, process model +pruning, delta writes and data model alterations are not recom- +mended here. Next, we implement the recommended optimizations +and re-execute all the experiments. The results of the experiments +are grouped based on the optimization recommendations and can +be seen in Figures 7, 8, 9, 10, 11 and 12. We also explain how the +thresholds are set for our experiments and how they can be tuned +by users. +6.1.1 +Endorser restructuring: The effect of endorser restruc- +turing can be seen in Figure 7. When the endorsement policy +is P1, all the clients must send their transactions to Org1 due +to the specific endorsement policy and hence, an endorsement +bottleneck is detected for Org1. Since the endorsement policy re- +quires signatures from two organizations, we change the policy +to OutOf(2,Org1,Org2,Org3,Org4) so that the clients can distrib- +ute the transactions evenly among all endorsers. This optimization +leads to a 29% increase in throughput (Figure 7). In Experiment 2, +since the endorser distribution is skewed, the clients send transac- +tions unevenly and therefore two of the organizations endorse far +more often than the other two. We re-executed the experiment with +an even distribution of transactions to the endorsers and observe +a 26% increase in throughput (Figure 7). The main impact of this +optimization is on throughput and latency as it reduces transaction +queuing on few specific peers and instead distributes them evenly. +We set the thresholds for this recommendation such that we ex- +pect an even distribution of transactions to all endorsers, i.e., even +minor bottlenecks are detected. This can be tuned to detect only +severe bottlenecks. Further, since these are synthetic experiments, +changing the endorsement policy is not critical. In real scenarios, +consultation with the governing bodies of an enterprise is required +before changing the policy. Still, the recommendations by Block- +OptR help to highlight bottlenecks which in turn can convince the +management to change the policy. +6.1.2 +Client resource boost: Figure 8 shows the effect of client +resource scaling. After increasing the number of clients, we observe +a 75% decrease in latency, a 15% increase in throughput, and a +7% increase in success rate. The thresholds are set such that this +optimization is recommended when more than 50% of transactions +are invoked by the same organization. This can be fine-tuned to +detect less severe bottlenecks. +6.1.3 +Block size adaptation: The effect of block size adaptation +can be seen in Figure 9. In our experiments, we use the default +block time out of 1s. Therefore, we make the block count equal +to the transaction rate whenever the block size adaptation is rec- +ommended. After changing the block size, we observe up to 93% +improvement in throughput and 85% improvement in success rate +(Figure 9; Block count: 50). The thresholds are set such that this op- +timization is recommended whenever the average block size is 60% +larger or smaller than the transaction send rate derived from the +log. The thresholds can be decreased to make the recommendation +more sensitive to transaction rate changes. +6.1.4 +Transaction rate control: The effect of transaction rate +control is shown in Figure 10. In these experiments, periods of +high traffic (around 300 TPS) were also identified as periods of +high failure rates. We then lowered the transaction send rate to 100 +TPS on the clients and re-executed the experiments. We observe +significant improvement of up to 87% in latency and 36% in success +rate (Figure 10; Send rate: 1000). We set the thresholds for this +recommendation at 300 TPS which is the default send rate of our +experiments. This means that we consider the current traffic of + +NewEnd Pol +107.1 +151.4 +103.4 +141.1 +16.8 +10.4 +19.2 +12.3 +87.5 +89.4 +77.4 +87.9 +1.0 +10.0 +100.0 +1000.0 +W/O +W +W/O +W +Endorsement policy: P1 +Endorsement policy: P2 +Endorser dist skew: 6 +Control Variables +Success throughput (tps) +Average latency (s) +Percentage of success (%) +Figure 7: Endorser restructuring +Transaction dist skew: 70% +160.8 +190.6 +3.3 +0.8 +59.9 +64.4 +0.1 +1.0 +10.0 +100.0 +1000.0 +W/O +W +Transaction dist skew: 70% +Control Variable +Success throughput (tps) +Average latency (s) +Percentage of success (%) +Figure 8: Client resource boost +New +New Block size +14.8 +217.9 +43.6 +217.9 +189.1 +199.1 +182.8 +227.3 +3.3 +4.9 +6.8 +4.4 +11.4 +11.2 +12.5 +10.0 +13.8 +92.8 +37.6 +92.6 +63.3 +65.7 +79.0 +84.5 +1.0 +10.0 +100.0 +1000.0 +W/O +W +W/O +W +W/O +W +W/O +W +Block count: 50 +Block count: 100 +Send rate: 1000 +Send rate: 500, 1000 +Control Variables +Success throughput (tps) +Average latency (s) +Percentage of success (%) +Figure 9: Block size adaptation +New +New Rate control +121.9 +88.6 +117.7 +90.1 +179.4 +95.3 +99.3 +40.6 +173.3 +97.0 +204.1 +95.7 +211.6 +95.7 +155.7 +94.9 +189.1 +96.7 +182.8 +95.6 +160.8 +73.4 +16.1 +4.8 +16.7 +4.3 +6.1 +2.2 +2.9 +1.2 +8.1 +1.4 +6.7 +1.6 +6.3 +2.0 +13.3 +1.9 +11.4 +1.4 +12.5 +1.9 +3.3 +1.1 +84.7 +97.3 +84.9 +97.4 +83.5 +97.0 +37.7 +41.3 +81.6 +99.1 +91.8 +99.1 +91.9 +98.7 +85.4 +98.9 +63.3 +99.2 +79.0 +98.8 +59.9 +74.0 +1.0 +10.0 +100.0 +1000.0 +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +Endorsement +policy: P3 +No: of orgs: 4 +Workload: +Update-heavy +Key +distribution +skew: 2 +Block count: +300 +Block count: +500 +Block count: +1000 +Send rate: 500 +Send rate: +1000 +Send rate: +500, 1000 +Transaction +dist skew: 70% +Control Variables +Success throughput (tps) +Average latency (s) +Percentage of success (%) +Figure 10: Transaction rate control +New +New Reordering +107.1 +198.2 +103.4 +152.3 +231.8 +243.9 +208.1 +239.0 +12.4 +36.2 +99.3 +172.1 +14.8 +19.2 +173.3 +221.7 +211.6 +239.6 +49.2 +49.6 +174.4 +188.2 +189.1 +200.6 +160.8 +217.8 +16.8 +7.1 +19.2 +9.5 +4.3 +3.9 +6.4 +4.1 +27.3 +22.7 +2.9 +2.0 +3.3 +2.3 +8.1 +5.0 +6.3 +3.7 +1.5 +1.1 +7.3 +6.8 +11.4 +10.4 +3.3 +2.1 +87.5 +92.1 +77.4 +91.5 +95.2 +96.2 +97.2 +97.9 +11.5 +27.8 +37.7 +67.8 +13.8 +18.4 +81.6 +92.7 +91.9 +94.4 +99.4 +99.7 +90.9 +92.1 +63.3 +64.6 +59.9 +77.8 +1.0 +10.0 +100.0 +1000.0 +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +Endorsement +policy: P1 +Endorsement +policy: P2 +Endorser dist +skew: 6 +Workload: +Read-heavy +Workload: +Insert-heavy +Workload: +RangeRead- +heavy +Key dist + skew: 2 +Block count: 50 +Block count: +300 +Block count: +1000 +Send rate: 50 +Send rate: 300 +Send rate: +1000 +Transaction dist +skew: 70% +Control Variables +Success throughput (tps) +Average latency (s) +Percentage of success (%) +Figure 11: Activity reordering +New +New All optimizations +107.1 +159.3 +103.4 +152.1 +99.3 +67.2 +14.8 +230.6 +173.3 +97.1 +211.6 +97.5 +189.1 +95.7 +160.8 +85.8 +16.8 +11.8 +19.2 +12.2 +2.9 +1.2 +3.3 +3.6 +8.1 +1.3 +6.3 +1.6 +11.4 +1.7 +3.3 +0.8 +87.5 +89.8 +77.4 +85.0 +37.7 +68.5 +13.8 +93.6 +81.6 +99.3 +91.9 +99.1 +63.3 +98.9 +59.9 +86.6 +0.1 +1.0 +10.0 +100.0 +1000.0 +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +W/O +W +Endorsement policy: +P1 +EndorsementPolicy:P2 +Endorser dist skew: 6 +Key dist skew: 2 +Block count: 50 +Block count: 300 +Block count: 1000 +Send rate: 1000 +Transaction dist skew: +70% +Control Variables +Success throughput (tps) +Average latency (s) +Percentage of success (%) +Figure 12: All recommended optimizations combined +the system as high and want to detect periods of failure. Users can +adjust this threshold based on what is considered high (more than +the sustainable traffic rate) for their Fabric network installation. +6.1.5 +Activity reordering: The effect of activity reordering can +be seen in Figure 11. We observe that BlockOptR recommends ac- +tivity reordering for all experiments except Experiments 3 and 5 +(Table 3). Reordering was suggested for two activities (Read and +Update) which conflict with each other. We updated the configura- +tion of the client manager to generate read transactions before all +other transactions. This implementation emulates a scenario where +organizational measures were applied to enforce activity reordering. +We then re-executed the experiments and observe a performance +improvement in all the experiments. There is up to 65% increase in +throughput and 58% increase in success rate (Figure 11; Workload: +RangeReadheavy). We have set the thresholds such that if 40% of +the MVCC failures are caused by activities that can be reordered, +this strategy is recommended. This can be made more lenient by +increasing the threshold such that reordering is suggested only in +very significant cases. For Experiments 3 and 5, less than 40% of +MVCC conflicts are caused by the two activities where reordering +is possible. For example, the activity Update has a dependency on +itself which cannot be removed by reordering. +6.1.6 +Combined optimizations: We also executed the experi- +ments after applying all the recommended optimizations together. +We observe up to a 93% improvement in throughput and 85% im- +provement in the success rate (Figure 12: Block count: 50). In all the +experiments, the performance obtained by applying all the optimiza- +tions is comparable to the performance yielded by the optimization +with the highest improvement. +Further remarks. Though smart contract partitioning is recom- +mended for Experiment 8, this optimization requires understanding +the functionality of the smart contract. Unfortunately, for the syn- +thetically generated smart contract that includes only generic read, +update and insert functions, we cannot redesign the smart contract. + +207.48 +98.18 +275.31 +286.62 +96.76 +7.28 +1.1 +2.59 +1.87 +3.79 +79.83 +99.47 +94.22 +99.04 +97.73 +1.00 +10.00 +100.00 +1000.00 +Without +optimization +Transaction rate +control +Activity +reordering +Process model +pruning +All optimizations +Success throughput (tps) +Average latency (s) +Success rate (%) +SimpleSCM +Figure 13: SCM use-case +35.1 +60.7 +81.4 +53.4 +110.7 +14.0 +18.1 +11.7 +10.5 +6.0 +20.1 +49.7 +47.6 +27.3 +82.6 +1.0 +10.0 +100.0 +1000.0 +Without +optimization +Delta-write +Activity +reordering +Smart +contract +partition +All +optimizations +Success throughput (tps) +Average latency (s) +Success rate (%) +DRM +Figure 14: DRM use-case +NewEHR +55.57 +64.34 +135.96 +99.56 +75.97 +6.4 +1.78 +3.57 +2.31 +1.77 +19.70 +65.39 +57.94 +35.01 +78.85 +1.00 +10.00 +100.00 +1000.00 +Without +optimization +Transaction +rate control +Activity +reordering +Process +model +pruning +All +optimizations +Success throughput (tps) +Average latency (s) +Success rate (%) +Figure 15: EHR use-case +DV +4.2 +4.7 +54.3 +46.3 +4.6 +3.7 +4.1 +2.3 +10.2 +11.3 +100.0 +100.0 +1.0 +10.0 +100.0 +Without +optimization +Transaction rate +control +Data model +alteration +All optimizations +Success throughput (tps) +Average latency (s) +Success rate (%) +Figure 16: Digital voting use-case +3.2 +6.6 +18.7 +63.3 +14.2 +24.4 +1.5 +1.2 +2.0 +1.4 +1.1 +1.6 +31.8 +66.0 +7.0 +22.0 +14.4 +24.9 +1.0 +10.0 +100.0 +Without +optimization +Data model +alteration +Without +optimization +Data model +alteration +Transaction rate +control +All optimizations +Send rate: 10tps +Send rate: 300tps +Success throughput (tps) +Average latency (s) +Success rate (%) +LAP +Figure 17: Loan application process use-case +6.2 +Use-case based Workloads +Supply Chain Management (SCM): With the SCM use-case, three +optimizations are recommended by BlockOptR: activity reorder- +ing, process model pruning and transaction rate control (Figure 13). +After implementing reordering for the reorderable activities (query- +Products and UpdateAuditInfo), we observe a 24% increase in +throughput and 15% increase in success rate. Pruning was recom- +mended for the Ship activities that occur without or before the +PushASN activity. It was also recommended to prune Unload activi- +ties that occur without or before the Ship activity. We adapted the +smart contract to implement the pruning recommendation. This +resulted in a 27% improvement in throughput and 19% increase in +success rate. Transaction rate control and applying all recommen- +dations together also improves the performance. +Digital Rights Management (DRM): With the DRM use-case, three +optimizations are recommended by BlockOptR: activity reordering, +delta-writes and smart contract partitioning. Figure 14 shows the +results of applying these optimizations. To implement the delta +write recommendation, we observed that the Play function in the +smart contract has an increment operation to count the number of +times a piece of music was played. We converted this into a delta +write and the delta-keys are aggregated whenever the calcRevenue +function is invoked (since it requires the play count). With this +optimization, we can observe a significant improvement of 42% in +throughput and 50% in success rate. However, the average latency +increases in this case because the calcRevenue function now takes +up more time for aggregation. Since calcRevenue is not executed as +frequently as Play, the overall performance is not affected though. +Activity reordering was recommended for calcRevenue and +queryRightHolders functions and we reconfigured the clients to +send these activities after all other activities. This emulates a sce- +nario where an organization restricts specific transactions to spe- +cific time periods. We observe more than 50% increase in both +throughput and success rate with this optimization. +Hot keys were detected and frequently used by four activities. +We analysed the smart contract and discovered that, though all four +functions have a dependency on the same key, the functionalities +are different. Play and calcRevenue need only the play count, +while viewMetaData and queryRightHolders need metadata and +not the play count of a piece of music. Therefore, we split the +smart contract into two, where one smart contract has the Play +and calcRevenue functions and the second smart contract has the +FabricSharp +100.92 +103.56 +96.56 +99.16 +93.36 +62.32 +2.09 +2.07 +2.04 +1.90 +3.54 +1.42 +94.14 +96.56 +90.08 +92.50 +87.17 +99.47 +1.00 +10.00 +100.00 +1000.00 +Without +optimization +Endorser +restructuring +Without +optimization +Endorser +restructuring +Without +optimization +Transaction rate +control +Endorsement policy: P1 +Endorsement policy: P2 +Endorser dist skew: 6 +Workload: Insert-heavy +Control Variables +Success throughput (tps) +Average latency (s) +Success rate (%) +Generator +Figure 18: Synthetic workloads with FabricSharp +other two functions. The create function is included in both smart +contracts, and invocation of the first smart contract invokes the +same function in the second smart contract. We observe a 35% +increase in throughput and a 26% increase in success rate with this +optimization. Applying all the optimizations together improves the +performance by more than 50%. +Electronic Health Records (EHR): In this use-case, three optimiza- +tions were recommended: activity reordering, process model prun- +ing and transaction rate control (Figure 15). Activity reordering for +the read activities resulted in a 60-65% improvement in throughput +and success rate. When the smart contract was updated to prune +illogical paths (revoke access to records without granting access), +we observe around 43% increase in throughput and success rate. +After applying transaction rate control, a 69% increase in success +rate was observed. All optimizations applied together also improve +the performance. +Digital Voting (DV): In this use-case, two optimizations were +recommended: transaction rate control and data model alteration. +The results are shown in Figure 16. High failure rates were detected +for periods when the Vote transactions were frequent. After ap- +plying transaction rate control, a slight improvement of 11% in +throughput was observed. The hotkeys were detected and most +frequently used by the Vote function resulting in a recommenda- +tion to alter the data model. We analysed the smart contract and +observed that partyID was used as the key for the vote function +which is invoked by multiple voters during the voting phase. We +redesigned the smart contract such that voterID is assigned as the +primary key. Since voters are restricted to a single vote, we observe +100% success rate with this new smart contract because there are no +more transaction dependencies. We also observe an improvement +in the performance when both optimizations are applied together. +6.3 +Loan Application Process (LAP) +The optimization recommended for the LAP use-case was data +model alteration (Figure 17). The employeeID 1 had a high key + +Fabric++ +Generator UH, RH and RRH +106.27 +57.56 +159.47 +69.41 +144.61 +69.02 +194.22 +83.70 +95.78 +56.28 +213.47 +83.92 +3.62 +1.33 +3.13 +1.57 +2.58 +1.56 +2.87 +1.10 +10.36 +1.01 +1.85 +1.02 +41.04 +59.22 +61.87 +71.37 +53.70 +70.36 +77.49 +85.02 +45.57 +57.14 +78.24 +85.33 +1.00 +10.00 +100.00 +1000.00 +Without +optimization +Transaction +rate control +Activity +reordering +All +optimizations +Without +optimization +Transaction +rate control +Activity +reordering +All +optimizations +Without +optimization +Transaction +rate control +Activity +reordering +All +optimizations +Workload: Update-heavy +Workload: Read-heavy +Workload: RangeRead-heavy +Control Variables +Success throughput (tps) +Average latency (s) +Success rate (%) +Figure 19: Synthetic workloads with Fabric++ +frequency since this employee processed the highest number of +loan applications. We then re-implemented our smart contract and +assigned applicationID as the key and modeled the value as a +structure that includes employeeID, loan amount, loan type and +loan status. This new implementation helped to remove the hot +key and yielded more than 50% improvement in throughput and +success rate for both the lower and higher send rates. +6.4 +Fabric Extensions +As a holistic recommendation approach, our work lies orthogo- +nal to existing Fabric optimizations in the literature. In this section, +we demonstrate how our approach works on top of two optimized +extensions of Fabric: FabricSharp [65] and Fabric++ [67]. Both im- +plement different transaction reordering strategies that mitigate +MVCC read conflicts. The Fabric++ scheduler is integrated in the +FabricSharp implementation [25] and we use this for our exper- +iments. We executed the synthetic workloads on both and then +used BlockOptR to generate recommendations. The literature says +FabricSharp increases endorsement policy failures and is less per- +formant for insert-heavy workloads while Fabric++ is least per- +formant with an update-heavy, read-heavy and range-read-heavy +workloads [13]. Therefore, we execute these specific experiments +shown in Figures 18 and 19 with the synthetic workloads. Activ- +ity reordering, transaction rate control and endorser restructuring +were recommended and by implementing these recommendations, +we observe up to a 55% increase in throughput and 46% increase in +success rate (Figure 19: RangeRead-heavy workload). Our experi- +ments with these Fabric extensions show that even with effective +system-level optimizations, Fabric can still benefit from optimiza- +tions at all levels of abstraction. +7 +Lessons Learned and Limitations +We demonstrated that BlockOptR is capable of effectively recom- +mending suitable optimization strategies. Further, we also explained +how to implement these optimizations and quantified the perfor- +mance improvements after implementation. This section discusses +the insights we gained from our experiments. +User level optimizations. Activity reordering was one of the +most frequently recommended optimizations in our experiments. +We highlight use-cases such as SCM where such reordering can be +applicable. Our model pruning recommendation emphasizes that +identifying incompetencies in the process model can lead not only +to efficient process execution but also improve the performance of +the underlying system. Load shedding or queuing is often employed +when systems cannot handle the workload. Using our recommen- +dations, specific activities and time periods can be identified where +such rate control techniques are most effective. For example, rate +control is recommended for the Vote activity in the digital vot- +ing use-case. Therefore, instead of system-wide rate control, only +the specific clients that deal with the identified activities need to +employ rate control techniques. +Data level optimizations. These optimizations show how the +design of the smart contract and the data model significantly in- +fluence the performance. The smart contract is initially designed +with a specific process model in mind. However, we understand +how the smart contract is being used in practice by analyzing the +blockchain logs. BlockOptR pinpoints functions and keys that cause +bottlenecks which in turn helps the smart contract developer to +make appropriate modifications. +System level optimizations. Setting the endorsement policy +is a management decision that often excludes discussions with the +technical team designing the blockchain. Our recommendations +highlight the need to bring together management and technical +discussions to decide optimal configuration settings. Further, we +also demonstrate the need to verify whether the policy is being used +effectively. For example, even if the policy defines the equal dis- +tribution of endorsements, the clients may send their transactions +in a skewed manner. In such instances, we recommend enforcing +a management measure, such as dividing the endorsers equally +among the clients such that clients of one organization only send +transactions to specific endorsers. The compliance with such mea- +sures can also be checked by BlockOptR. Block size optimization +is frequently discussed in the literature and associated with the +transaction rate of a system [13, 36, 68]. Instead of system-level +changes such as using transaction rate monitors, we derive the +transaction rate and the actual block size from the log. This helps +to understand traffic patterns over time and find reasonable block +size settings. While the literature mainly focuses on optimizing +the peers and ordering service components of Fabric [27, 65, 67], +our client-related recommendations highlight the need to focus on +client-side optimizations as well. +Technology Independence. Our multi-level recommendation +approach is demonstrated using the Fabric blockchain. Technology +independence is difficult to attain due to the vast implementation +variations between the numerous blockchain systems and the cor- +responding differences in the contents of the distributed ledger. +However, we draw attention to specific examples which can guide +future researchers to translate our approach to other blockchain +systems. In Quorum, the block time or mining frequency has a lin- +early proportional influence on the transaction latencies [7] which +is analogous to our block size adaptation recommendation strat- +egy. Also, Corda has the concept of notaries to attest transactions +where distributing the transactions over multiple notaries is ex- +pected to improve the throughput [18]. This is again comparable to +our endorsement restructuring recommendation. Further, there are +numerous gas-fee reduction and vulnerability detection strategies +for Ethereum smart contracts in the literature [54] which translate + +to our recommendations at the data level. Tools like Lorikeet and +Caterpiller automate the conversion and execution of process mod- +els as Ethereum smart contracts, which would make it easier to +implement the user-level optimizations that we recommend [43, 69]. +Limitations. The optimizations recommended by BlockOptR +need to be manually implemented by the user. A self-adaptive +system with a feedback loop that automatically implements the +recommendations is possible. However, in an enterprise scenario, +for many of the optimizations such as endorser restructuring, ac- +tivity reordering, and process model pruning, management level +approvals might be required before implementation. Additionally, +for applications that do not follow a specific process model, the +event logs can be misleading. In such scenarios, user-level optimiza- +tions such as activity reordering and process model pruning are +not relevant. Therefore, domain knowledge about the use-case is +required for implementing the recommended optimizations appro- +priately. Further, our implementation of some of the optimizations +such as transaction rate control are trivial in such benchmarking +scenarios and do not account for real-world overheads. However, +the implementations are mainly for demonstrative purposes. Our +work focuses on the multi-level recommendation approach used +by BlockOptR rather than the implementation of the optimizations. +Finally, our experiments without and with the recommended opti- +mizations are done on similar workloads generated with the same +input parameters, i.e., we assume a continued trend in the pattern +of the workload after the optimizations are applied. However, in +scenarios where the workload fluctuates or the optimization imple- +mentation is delayed, BlockOptR may need to be re-executed to +generate new recommendations. +8 +Related Work +The literature proposes various Fabric optimization strategies +such as transaction reordering [13, 65, 67], block size optimiza- +tions [13, 36], CRDTs [52], and parallelizing various components [27]. +Our work lies orthogonal to such optimization strategies and fo- +cuses on an optimization recommendation approach. We demon- +strate how our recommendations can be used along with two of the +literature’s optimization strategies to improve performance further. +There is also extensive research in the database community on +index and query optimizations that include self-tuning systems +as well as recommendation systems [2, 14, 15, 38, 73]. Though +we can draw parallels from these research, our work focuses on +blockchain-specific optimization recommendations. Different con- +figuration settings (such as block size and endorsement policy) and +the concept of smart contracts introduce new dimensions to the +recommendation approach, which are not required for databases. +There is ongoing research on applying process mining tech- +niques on blockchains to derive process-level insights [24, 32, 39, +49]. Klinkmüller et al. [39] and Mühlberger et al. [49] describe +different approaches to extract process data from the Ethereum +blockchain. Hobeck et al. [32] use process mining on an Ethereum- +based betting application to identify shortcomings in the appli- +cation. Process mining on blockchains currently only focuses on +permissionless blockchains as they are publicly accessible. How- +ever, deriving and studying the process model is equally critical for +private blockchains, and therefore, our work contributes to this less +explored area of research. Further, unlike the related work, we focus +on using process mining for recommending blockchain optimiza- +tion strategies. We only found a single paper that uses permissioned +blockchains, where Duchmann et al. [24] extract process data from +Fabric and detect semantic errors in a smart contract. Though our +work is comparable, we extract not only the process data but also +blockchain-specific attributes from Fabric, derive multiple metrics, +and recommend optimization strategies. +There is extensive research in the database community in the +domain of data-aware business processes that encourage a business +process perspective to database management systems [11, 20, 34]. +Calvanese et al. [11] comprehensively survey the contributions in +this realm and catalog contributions from various fields, including +database theory and process management. These works were an +important motivation for us to view blockchains from a business +process perspective. However, our work brings new contributions +since blockchains deal with several other elements apart from data, +such as smart contracts and endorsement policies. +9 +Conclusions +This paper showcases the necessity and effectiveness of having +a holistic perspective on blockchain optimizations. We define a +multi-level recommendation approach based on several metrics and +attributes derived from the blockchain log. We define a total of nine +optimizations at the system, data, and user-level of a blockchain. +We implement an automated optimization recommendation tool, +BlockOptR, based on these concepts. Further, we demonstrate how +such optimizations can be implemented to improve the system per- +formance. After implementing the recommended optimizations, we +observe an average of 20% improvement in the success rate and +an average of 40% improvement in latency. We extensively evalu- +ate the system with a wide range of workloads covering multiple +real-world scenarios. We hope to inspire enterprises to use our +contributions to detect blockchain optimization strategies and to +contribute their live blockchain (anonymized) logs for further re- +search in this domain. The BlockOptR tool, all the smart contracts, +the workload generation scripts, and all the event logs are available +as open-source [10]. We also plan to extend our tool to include +more optimization recommendations. +In terms of future work, we are currently developing a ProM plu- +gin which would provide a user-friendly interface for BlockOptR. +Presently, the threshold settings of BlockOptR depend on the busi- +ness network setup. For example, the rate threshold for our setup +was 300 TPS as higher rates led to instabilities, but this can vary for +other deployments. Therefore, tuning these thresholds automati- +cally in BlockOptR could be a future extension. Another interesting +extension is to define additional attributes that applications can log, +thereby providing more data for optimization recommendations. +Further, investigating the effect of workload fluctuations and delay +in applying the recommendations is another challenging future +direction. +Acknowledgments +This work is funded in part by the Deutsche Forschungsgemein- +schaft (DFG, German Research Foundation) - 392214008, and by +the Bavarian Cooperative Research Program of the Free State of +Bavaria - DIK-2002-0013//DIK0114/02. + +References +[1] Parinaz Ameri. 2016. Challenges of index recommendation for databases: With +specific evaluation on a NoSQL database. In dalam 28th GI-Workshop on Founda- +tions of Databases (Grundlagen von Datenbaken), Nörten-Hardenberg, Germany. +[2] Parinaz Ameri. 2016. On a self-tuning index recommendation approach for +databases. 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Association for Computing Machinery, New York, NY, +USA, 121–126. https://doi.org/10.1145/3327960.3332395 +[83] Dirk A Zetzsche, Douglas W Arner, and Ross P Buckley. 2020. +Decen- +tralized Finance. +Journal of Financial Regulation 6, 2 (09 2020), 172–203. +https://doi.org/10.1093/jfr/fjaa010 arXiv:https://academic.oup.com/jfr/article- +pdf/6/2/172/37064506/fjaa010.pdf + diff --git a/Q9E3T4oBgHgl3EQfyQvd/content/tmp_files/load_file.txt b/Q9E3T4oBgHgl3EQfyQvd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5062d1a129e654815140cb6ce6b4c3d116883e25 --- /dev/null +++ b/Q9E3T4oBgHgl3EQfyQvd/content/tmp_files/load_file.txt @@ -0,0 +1,1972 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf,len=1971 +page_content='How To Optimize My Blockchain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' A Multi-Level Recommendation Approach Jeeta Ann Chacko chacko@in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='de Technical University of Munich Ruben Mayer mayerr@in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='de Technical University of Munich Hans-Arno Jacobsen jacobsen@eecg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='edu University of Toronto Abstract Aside from the conception of new blockchain architectures, existing blockchain optimizations in the literature primarily fo- cus on system or data-oriented optimizations within prevailing blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, since blockchains handle multiple aspects ranging from organizational governance to smart contract design, a holistic approach that encompasses all the different layers of a given blockchain system is required to ensure that all optimization opportunities are taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In this vein, we define a multi-level optimization recommendation approach that identi- fies optimization opportunities within a blockchain at the system, data, and user level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Multiple metrics and attributes are derived from a blockchain log and nine optimization recommendations are formalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We implement an automated optimization recommen- dation tool, BlockOptR, based on these concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The system is extensively evaluated with a wide range of workloads covering mul- tiple real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' After implementing the recommended optimizations, we observe an average of 20% improvement in the success rate of transactions and an average of 40% improvement in latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' ACM Reference Format: Jeeta Ann Chacko, Ruben Mayer, and Hans-Arno Jacobsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' How To Optimize My Blockchain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' A Multi-Level Recommendation Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In Proceedings of ACM Conference (Conference’23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' ACM, New York, NY, USA, 15 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='nnnnnnn 1 Introduction When blockchains were first introduced, they supported only simple cryptocurrency exchange transactions [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, over time blockchains evolved to support complex transactions using smart contracts, thus entering the arena of decentralized trans- actional management systems such as distributed databases [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Since blockchains target consensus in a trustless environment, they cannot easily match the performance of databases [9, 16, 22, 26, 53, 59, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, with the advent of permissioned blockchains that offer access control and transaction execution policies, blockchains strive to improve their performance while still providing at least partially decentralized trust [3, 5, 28, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Conference’23, June 2023, Seattle, WA, USA © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='nnnnnnn Activity reordering Process model pruning Transaction rate control User level Delta writes Smart contract partitioning Data model alteration Data level Block size adaptation Endorser restructuring Client resource boost System level Delta keys Primary key duplication Primary key alteration Figure 1: Multi-level blockchain optimization Apart from the proliferation of new blockchain system designs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' there is highly vibrant and diverse ongoing research in the domain of system optimizations that focus on performance enhancements within prevailing permissioned blockchains [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 54,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 65–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The vast range of the literature targets control parameter tuning [13, 41, 68], transaction execution remodeling [27, 37, 66], and smart contract optimization [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, we notice that a collective approach that encompasses all these optimization possi- bilities for a particular blockchain under the same umbrella is miss- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, the literature falls short for an end-to-end optimiza- tion approach that includes not only system-level tuning and data remodeling but also process model redesign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Since permissioned blockchains are mainly employed by enterprises, a model-driven ap- proach is often followed where the setup of the blockchain network, its transaction regulations, the underlying smart contract, and the data model are primarily based on a business process model created specifically for a particular application [21, 40, 56, 63, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Such pro- cess models may be designed by business domain experts who are unaware of performance implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, in Hyperledger Fabric (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Fabric) [5], many transaction failures arise due to the order in which the transactions are executed [13, 65, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Such fail- ures could be reduced if the client processes that issue the transac- tions followed a different business logic in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The promi- nence of data management while executing business processes has often been highlighted by the database community [11, 20, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We make a similar argument for the importance of the process view in blockchains since the aspects covered by blockchains are manifold and not limited to data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, given the numerous optimizations possible within a given blockchain system, their varying influence on a case-by- case basis [6, 13, 23, 51, 68, 81], and the resulting implementation efforts, there is a pressing need for a recommendation system that guides the user in selecting effective optimization strategies suitable for the blockchain under consideration depending on the specific arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='04719v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='DC] 11 Jan 2023 use-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Again, we can draw parallels from the exhaustive lit- erature on parameter tuning and indexing recommendations for databases [1, 2, 42, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, since blockchains juggle multiple factors such as organizational governance [62], database defini- tions [59], consensus algorithms [46], provenance tracking [60], and smart contract design [47], a holistic perspective to optimiza- tion recommendations is desirable, which is currently lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' To address this gap, we propose a multi-level optimization rec- ommendation approach for blockchains that provides to the users a comprehensive understanding of the different optimization pos- sibilities for their blockchain system, thus enabling them to make a well-informed decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Inspired by the abstraction levels in databases [45], we define three levels of abstraction for blockchain optimizations: system, data, and user-level (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The system- level recommendations include identifying ideal system configura- tions such as the block size or endorsement policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The data-level recommendations deal with understanding the data model and op- timizing smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The user-level recommendations focus on business process models and workloads induced by client processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, we identified two activities in a digital rights man- agement scenario (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2) that frequently cause transaction conflicts and recommend a process model redesign to reduce such failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our approach can also verify compliance with the new process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We design and implement a recommendation tool named BlockOptR that analyzes the blockchain logs from Fabric, one of the most widely used blockchains by enterprises [61], to demonstrate the performance improvements yielded by our ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our contributions can be summarized as follows: (1) We define a multi-level optimization recommendation approach that extensively analyzes the blockchain log and recommends opti- mization possibilities from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our method helps users gain a comprehensive understanding of their current system and make educated decisions regarding optimization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (2) We provide a formal definition for our recommendation strate- gies based on common attributes, such that any blockchain log with similar attributes can reuse our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We also discuss how our approach translates to different blockchain platforms, thereby providing the reader with a technology-independent outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (3) We automate the extraction, preprocessing, and event log gen- eration techniques for Fabric blockchain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Thus, our tool Block- OptR will help to ease further research in the area of log-based analysis such as process mining in blockchains, since a preprocessed blockchain log can be directly obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (4) We demonstrate the effectiveness of the optimization recom- mendations by implementing and evaluating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our experi- ments indicate an average of 20% improvement in the percentage of successful transactions and an average of 40% improvement in latency after applying the recommendations by BlockOptR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (5) We extensively evaluate BlockOptR with three different types of workloads: A set of 24 synthetic workloads generated with a wide range of control variables, four widespread use case-based workloads from the literature, and a real-world event log of a loan application process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Thus, we cover a wide range of scenarios in our experimentation that are representative for real blockchain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This aids in overcoming the lack of publicly available data that restricts current research on process mining in permis- sioned blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The BlockOptR tool, all the smart contracts, the workload generation scripts, and all the event logs are released as open-source to encourage further research in this area [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (6) We further establish the positive effect of our holistic recom- mendation approach on top of existing blockchain optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Thus, we highlight that BlockOptR complements existing system- level blockchain optimization strategies such as FabricSharp [65] and Fabric++ [67] by adding higher-level optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 Hyperledger Fabric Fabric is an open-source permissioned blockchain system popu- larly used by enterprises [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The main components of Fabric are a smart contract (called chaincode), a distributed immutable ledger, a distributed world state database, a set of distributed peers, and an ordering service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The smart contract defines all the supported trans- actions on the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The transaction flow in Fabric follows an execute-order-validate (EOV) model [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The EOV model of Fabric is comparable to optimistic concurrency control in databases [31] and is therefore prone to multi-version concurrency control (MVCC) conflicts, which result in transaction failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (1) In the execution phase, transaction proposals are created by clients and sent to the endorsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Endorsers are a set of specific peers that have the authority to execute the smart contract to endorse a transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' An endorsement policy is configured to define the number of required endorsers for a transaction to be deemed valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Endorsers generate read-write sets after smart contract execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The transaction proposal and the read-write sets are signed by the endorsers and sent back to the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (2) In the ordering phase, the clients forward these transactions to the ordering service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The ordering service orders the transactions into blocks using Raft [55], a crash fault-tolerant consensus algo- rithm, and sends them to all the peers in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Configurable parameters limit the number of transactions included in a block (block size) in terms of the number of transactions (block count), a timeout (block timeout), or the size of transactions in bytes (block bytes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Blocks are created whenever the buffered set of incoming transactions satisfies any of the three conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (3) In the validation phase, every peer validates every transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Every peer in the Fabric network has a copy of the distributed ledger and the world state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Peers validate both the endorser signatures based on the endorsement policy and the read-write set with the current world state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If the validation is successful, the world state is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Else, a failure is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If the endorsement validation fails, it is called an endorsement policy failure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' if the read-write set validation fails, it is called an MVCC read conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' MVCC read conflicts for range reads are called phantom read conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Regard- less of the success or failure of the validation, all transactions are appended to the distributed ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Also, in the literature, MVCC read conflicts are often classified into inter-block and intra-block failures depending on whether the conflicting transactions reside in the same block or different blocks in the blockchain [13, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 Event Logs and Process Mining An event log is a record of process executions over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Pro- cess mining [75] is the technique of deriving a process model that exhibits the most frequent behaviors in an event log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' It is mainly used for process discovery which helps to understand the underlying process model, conformance checking where deviations between a predicted process model and the actual behavior of the process can be identified and model enhancement where bottlenecks are identified and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The minimum data required in an event log for process mining are: (1) CaseID: To distinguish different executions of the same pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Example: ProductID in a supply chain management related event log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' A complete execution of a process is called a trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (2) Activity name: To identify the different steps in a process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Example: Ship or Unload activity in a supply chain manage- ment related event log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (3) Timestamp: To determine the order of the different activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The event log can also have other attributes such as process owner, resources, and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Various algorithms are used to derive the pro- cess model such as alpha miner [76], heuristics miner [79] and fuzzy miner [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The core concept of all these algorithms is to analyze the different traces of the set of activities in the log and simplify the traces through abstraction or aggregation to produce a com- plete process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Various open-source and commercial process mining tools are available (ProM [78], Disco [29], Celonis [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 3 A Process Perspective to Blockchains Our work posits blockchain optimization as a holistic method- ology rather than a pure system-level approach by introducing a process perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In this section, we emphasize the necessity and effectiveness of understanding the dependency between business processes and the performance of the blockchain through exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, these examples motivate the need for an optimization recommender since many process-level optimizations can only be employed with approval from the decision-making bodies of an organization and, in most cases, cannot be automatically applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Figure 2: Derived process model for SCM scenario Process model pruning is an example of a process-level opti- mization that positively affects the system’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Figure 2 shows the process model derived from the blockchain log of a supply chain management (SCM) scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The highlighted paths and the traces embedded in the figure identify two unnecessary branches in the process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Unless the advanced shipping notice is pushed (PushASN), one should never execute the Ship activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Similarly, the Unload activity should never be executed unless a product has been shipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Such illogical activity paths can arise due to manual errors or transaction failures, and the smart contract is designed to handle such issues, as we explain in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If the Unload transaction executes without a corresponding Ship, the transaction will only read the state but not modify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, it is up to the smart contract designer to either fail the transac- tion upon execution or commit the read-only transaction to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Both these designs have their trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Committing the transaction adds an immutable record on the blockchain, which helps to track, for example, individuals or organizations who devi- ated from the expected process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In a supply chain manage- ment scenario specifically, this is critical since the entire pipeline is distributed, and the primary purpose of the blockchain here is to provide data provenance among untrusted participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' How- ever, on the other hand, failing a transaction immediately upon execution ensures that such unnecessary transactions do not go through all the time-consuming phases (ordering and validation), which can improve the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We observe a 27% improvement in throughput and 19% increase in success rate of transactions when unnecessary activity paths are pruned in the smart contract (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2, Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The pruning can also be implemented at the process execution level by enforcing incentive or penalty measures for organizations or individuals that adhere to or deviate from the expected process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This approach ensures that system performance is not prioritized over data provenance and hence, combines the advantages of both smart contract designs we discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Without activity reordering Activity order Activity Read data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Value Write data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Value Validity 1 PushASN { ProductID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 1 } { ProductID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 2 } Success 2 UpdateAuditInfo { ProductID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 1 } { AuditID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 001 } { AuditID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 002 } Abort With activity reordering Activity order Activity Read data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Value Write data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Value Validity 1 UpdateAuditInfo { ProductID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 1 } { AuditID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 001 } { AuditID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 002 } Success 2 PushASN { ProductID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 1 } { ProductID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 2 } Success Figure 3: Transaction dependency conflict example Another cause of failures are transactional dependencies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' and research in serialization algorithms has effectively reduced such fail- ures through transaction reordering [65,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, reordering algorithms are expensive, as they basically need to solve the NP- hard problem of generating conflict-free dependency graphs [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' An increase in endorsement policy failures due to inconsistent world states and the inability to handle large range queries are known problems of transaction reordering [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' A different ap- proach to the problem of dependency conflicts is to identify re- orderable and unreorderable [65] activities instead of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' While the literature analyzes the keys accessed by transactions to understand serializability, the data model needs to be analyzed for process-level serialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If two concurrent activities read the same data element but write to different elements in the data model then such activities are reorderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Figure 4: Derived process model after activity reordering QueryProdu QueryASN PushASN UpdateAudi Shi Unload43 traces 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='58% of the log Ship PushASN QueryASN Unload42traces 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='55%of the log PushASN QueryProducts QueryASN UnloadPushASNFor example, in the same supply chain management scenario, the UpdateAuditInfo activity reads a productID and writes an auditID, whereas the PushASN, Ship, and Unload activities read and write to the productID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, the pairs {UpdateAuditInfo, PushASN}, {UpdateAuditInfo, Ship} and {UpdateAuditInfo, Unload} are reorder- able activities while {PushASN, Ship, Unload} are unreorderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Figure 3 shows an example of a reorderable pair of activities where UpdateAuditInfo can succeed if it is executed either after the com- mit or before the execution of PushASN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Based on the business logic, it may be possible to impose procedures to restrict or resched- ule certain activities to execute only at specific periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, the corresponding process model in Figure 2 shows that UpdateAu- ditInfo occurs frequently between PushASN and Ship activities and therefore, UpdateAuditInfo may be executed before the transactions of the other two activities commit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, UpdateAuditInfo is not a time-critical activity and can be rescheduled to take place only at specific times when traffic is low on the supply chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We observe a 24% increase in throughput and 15% increase in success rate of transactions after a corresponding redesign where UpdateAuditInfo and QueryProducts activities are executed after PushASN, Ship, Unload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The new process model derived from the blockchain log confirms the adherence to the new design (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Thus, by iden- tifying conflicting activities, the process model can be redesigned to reduce transaction conflicts before they take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4 Blockchain Optimization Recommender We introduce an approach to recommend optimizations from three different abstraction levels: system, data, and user-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The primary requirement to design and implement such a multi-level recommendation system is reliable data on all three levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Knowl- edge about the system configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=', block size) and perfor- mance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=', throughput, transaction failures) is vital for generating system-level recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Information about the current data model and access patterns, such as key distribution and dependen- cies, is essential for data-level recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Lastly, knowledge concerning the use-case, business processes, and transaction work- load is necessary for user-level recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' It is important to note that such information is not restricted to a specific level but is helpful across all levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, the system-level performance can indicate the need for optimizations at all three levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The very definition of a blockchain implies the availability of a distributed ledger with immutable data regarding every trans- action executed overtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If we consider smart contracts, then every execution of the smart contract results in a transaction that is logged in the ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We consider this data (hereafter referred to as the blockchain log) as the primary source to derive opti- mization recommendations since, to our knowledge, such a dis- tributed ledger consisting of all transactions is available for most blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, our transaction-centric approach to de- riving blockchain optimization recommendations is applicable to different blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We preprocess the raw data from the blockchain to create a blockchain log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Then, we obtain the values for key metrics which are used to detect multi-level optimization recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Pro- cess mining strategies are then applied to the blockchain log to derive the process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We identify the applicable optimizations using the recommendations and the derived process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Figure 5 Fabric Network Blockchain Data Preprocessing Metrics Derivation Event Log Generation Process Model Generation Optimization Recommendation BlockOptR Optimization Implementation Figure 5: BlockOptR workflow illustrates the workflow of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We automated the main elements of this workflow as a tool, BlockOptR [10], implemented in Python and Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='js.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 Blockchain Data Preprocessing BlockOptR registers as a client on the Fabric network, reads the entire blockchain and saves it as JSON files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Next, the log is cleaned by removing the configuration and setup-related transactions that are not relevant and converted to CSV format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' All information re- garding each transaction executed in the Fabric network is logged on the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We extract seven attributes and derive two at- tributes from this extensive logged data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' These attributes enable the derivation of multiple metrics required to recommend optimiza- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The output of the data preprocessing step is a blockchain log with the following nine attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (1) Client timestamp: The time at which the client generated the transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (2) Activity name: The name of the smart contract function whose execution created the transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' A(x) defines the activity name of a transaction x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (3) Function arguments: The value of the parameters of the smart contract function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (4) Endorsers: The set of all endorsers of the transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (5) Invokers: The set of all clients and their respective organi- zation who invoked the transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (6) Read-write set: The set of keys accessed (read or written) by the transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The separate read set and write set of a transaction are also kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' RWS(x), RS(x) and WS(x) corre- spondingly define the read-write set, read set and write set of a transaction x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (7) Transaction status: The status of the transaction that can have the values success, MVCC read conflict (MRC), phantom read conflict and endorsement policy failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' ST(x) defines the status of a transaction x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (8) Transaction type: The type of transaction which is de- rived from the read-write set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This can have the values read, write, update, range read and delete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Transaction type is derived from the read-write set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' TT(x) defines the type of a transaction x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (9) Commit order: The order of the transactions in the blockchain log is the order in which transactions were committed to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 Event Log Generation The blockchain log extracted from the Fabric network can be used as an event log to apply process mining techniques that assist in recommending user-level optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, unlike the event logs created by process-aware information systems [74], a CaseID is not directly available in the event log extracted from a blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Also, in most of the use-cases we observed, no single attribute is common to all activities that can be directly used as the CaseID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, we need to derive a common element for each use-case based on domain knowledge [4, 8, 17, 19, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Since we are interested in a transactional perspective of the process model, we find a common element for all activities by analyzing the function arguments and read-write sets available in the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, in the SCM scenario the productKey is a common element for all activities and is a suitable choice since the use-case is specifically related to tracking multiple products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This process of extracting the common element is automated for all the use-cases in this paper and can be easily extended for other use-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Once a common element is identified, we define a trace as a unique set of activities with the same value for the common element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We then assign a new CaseID to every trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, only the time at which the clients sent the transaction (client timestamp) is available in our log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, there is no guar- antee that the same order in which clients send their transactions will be maintained when the transactions are committed to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, to derive the process model accurately, we use the commit order in place of the timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Thus, with the generated CaseID and extracted/derived attributes, we have a com- plete event log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Now, any process mining technique can be applied to the event log to derive a process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, we used the Alpha algorithm to derive the process models shown in Figure 2 and 4 [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 Metrics We define a set of metrics by scrutinizing multiple blockchain logs and analyzing metrics from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (1) Rate metrics: BlockOptR calculates the average transaction rate as well as the transaction rate distribution over time intervals from the event log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Transaction rate (Tr) is the average rate at which transactions are sent from the clients and is derived from the total transactions in the log and the client timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Transaction rate distribution (Trdi) is the transaction rate at a specific interval i derived from the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' A user-configurable interval size (ins) in seconds is used to calculate this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Usage: Transaction rate is a useful metric to understand the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The rate distribution provides insights regarding periods of high or low traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (2) Failure metrics: Similar to Tr, the total failure rate (TFr) as well as the rates of each type of failure (MVCC read conflicts, phan- tom read conflicts, endorsement policy failures) are derived from the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The failure rate distribution (Frdi) is calculated similar to Trd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Usage: Failure metrics help to detect times of high transac- tion failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Optimizations such as transaction rate control can be applied based on the failure metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (3) Block size: The user-configured block count (Bcount) and block timeout (Btimeout) are extracted from the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The average num- ber of transactions in a block (Bsizeavg) is also derived from the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Bsizeavg is equivalent to the average block size and can also be defined as min{Bcount, Tr ∗ Btimeout}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Usage: Bsizeavg along with the rate metrics helps a user understand the effectiveness of their block size configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, if Tr is 500, Bcount is 100, Btimeout is 1 and Bsizeavg is 100, then 100 transactions are packed into a block when 500 transactions are actually available every second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This means more blocks than necessary are being created which is inefficient, as block creation is expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Similarly, if Tr is 100, Bcount is 500, Btimeout is 2, and Bsizeavg is 200, then blocks are created only every 2 seconds and transactions are queued up for a waiting period before being put into blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Both scenarios lead to performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' So, based on the value of Bsizeavg, the user can update Bcount and Btimeout to efficiently handle the transaction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (4) Endorser significance (EDsig) defines the number of transac- tions endorsed by each endorsing peer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Usage: This metric helps in identifying endorser bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Suppose a limited number of endorsers always carry out the endorsements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In that case, the user can consider distributing the transactions more evenly among the endorsers or expanding the set of endorsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (5) Invoker significance (IVsig) defines the number of transac- tions invoked by each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Usage: This metric helps to identify clients and the corresponding organizations that invoke a majority of the transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Client resource allocation decisions of such organizations can be made based on this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (6) Key frequency (Kfreq) is defined as the number of failed trans- actions that access a specific key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Key significance (Ksig) is defined as the number of activities that access a specific key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' HK defines the set of hotkeys that have high key frequency based on user- configurable thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Usage: Identifying the hotkeys assists the users to identify optimization possibilities in their smart contracts, and key significance helps to detect the exact activities (that cor- respond to smart contract functions) that access the hotkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, if several functions access the same key, then the different functions could be separated into multiple smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Every smart contract executes on a different world state, thereby reducing failures (see example in Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (7) Data-value correlation (corDV) defines that two transactions are correlated if both access a same set of keys and one of them fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Usage: Data-value correlation helps to identify transaction dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Such dependent transactions are the root cause of MVCC read conflicts [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Various optimization strategies, such as process model redesign and transaction rate control, can be applied to these correlated transactions to mitigate failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (8) Proximity correlation (corP) defines the distance between two transactions that have a high data value correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For ex- ample, if corP(x, y) == 1 then transaction y happened immediately after x with no transactions in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, we also derive the activity-based proximity correlation (corPA) which defines the distance between transactions of the same activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Usage: Ana- lyzing if the proximity correlation is “less than the block size” or “greater than the block size” can reveal useful insights regarding inter- and intra-block failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If intra-block failures are very high, smaller block sizes can potentially reduce failures [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This metric also helps to choose between inter- or intra-block transaction re- ordering strategies offered by different Fabric optimizations [65, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 Optimization Recommendations We use a multi-level approach to utilize the defined attributes and metrics for recommending blockchain-specific optimization strate- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The optimization recommendation techniques explained in this section include configurable thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We define appropriate Table 1: Formalization of optimization recommendations Recommendations Necessary conditions Activity reordering if corDV (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' y) == 1 ∧ WS(x) ∩ WS(y) == ∅ Process model pruning if A(x) = A(y) ∧ TT (x) ≠ TT (y) Transaction rate control if (Trdi ≥ Rt1) ∧ (Frdi ≥ Trdi ∗ Rt2) Delta writes if corPA(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' y) == 1 ∧ ST (x) == MRC ∧ |WS(x) | == |WS(y) | == 1 ∧ WS(x) ± 1 == WS(y) Smart contract partitioning if Ksig(HKi) > 1 Data model alteration if (Ksig(HKi) == 1) ∨ ( |HK | == 1) Block size adaptation if (Tr ≥ Bsizeavg ∗ Bt) ∨ (Tr < Bsizeavg ∗ Bt) Endorser restructuring if EDsig(e) > |TX | ∗ Et Client resource boost if IVsig(c) > |TX | ∗ It where x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' y ∈ TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' e ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' c ∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' HKi ∈ HK TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' HK are set of all transactions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' endorsers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' invokers and hotkeys Rt1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Rt2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Bt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' It are user configurable thresholds default values for these thresholds based on our analysis of multiple logs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' but the user can adapt these default values to fine-tune the detection strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The necessary condition to recommend each optimization strategy is formalized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 User Level Recommendations At the user level, it is essential to focus on the actual workload of the running application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The rate and order in which the transactions are generated and committed to the blockchain has a vital impact on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We analyze the rate, dependencies, and type of the transactions to recommend optimizations at the user level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (1) Activity reordering: Reorderable pairs of transactions can be identified by using the data value correlation and the read-write set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' BlockOptR identifies the activities corresponding to such transac- tion pairs and recommends a process model redesign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The redesign should ensure that the identified activities are restructured to re- duce conflicts (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (2) Process model pruning: If activities deviate from an expected behavior, then process model pruning is recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The trans- action type of all transactions related to an activity is analyzed to identify anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Comparing the traces in the event log and the derived process model with the identified anomalies helps to detect model pruning opportunities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (3) Transaction rate control: BlockOptR evaluates the transac- tion rate distribution over time and identifies times when the rate is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' It then checks the failure rates in the same time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If the failure rate is also very high, rate control is recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Two configurable thresholds are used to tune the tolerance level of transaction rate and failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 Data Level Recommendations For data-level recommendations, we focus on identifying the spe- cific areas in the data model that can be optimized by analyzing transaction failures, proximity correlation, read-write sets, and key significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This aids the user in altering the smart contract and thereby the underlying data model to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (4) Delta writes: Update transactions that only perform increment or decrement operations can be converted to delta-writes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Delta writes enable writing to multiple unique delta keys, which can be aggregated whenever the current value is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Reading the key before each write is also not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Thus, update transac- tions are converted to write-only transactions that write to unique keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This helps to reduce transaction dependency-related failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Delta writes are recommended when a single key is incremented or decremented by a failed transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (5) Smart contract partitioning: A possibility to reduce transac- tion dependencies is to split a smart contract into multiple ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Each smart contract will access separate world states, thereby avoid- ing conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The functionality of the original smart contract will not change because it is possible to invoke functions between the two smart contracts if interaction is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, in a music rights management scenario, if a key MusicID is found to be hot and multiple functions such as Play() and viewMetaData() access this same key, then one can separate the functions into two different smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In other words, the underlying database is split into two by duplicating the primary key (MusicID) across both and having different secondary keys in each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The play count of MusicID is recorded in one and metadata is read from the other (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This is analogous to designing the table layout in relational databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The smart contract needs to be analyzed and updated to implement this optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Smart contract partitioning is recommended if multiple activities access a hotkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (6) Data model alteration: If activities have a dependency on themselves, then a data model alteration can be beneficial to reduce transaction conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, in a digital voting scenario, if a key ElectionID is found to be hot and is only accessed by the function Vote(), then a possible optimization is to use another primary key such as VoterID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Then, instead of updating all the votes together, the votes can be updated per voter (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, if only a single hotkey is detected then it is beneficial to analyze the data model to understand the reason for the skewed access to this specific data element (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Data model alteration is recommended if a hotkey is accessed only by a single activity or if a single hotkey is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 System Level Recommendations At the system level, we focus on two crucial system configuration settings that can significantly affect the performance of Fabric: the endorsement policy and the block size [13, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, we also identify client bottlenecks to aid in resource allocation decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We use the endorser significance, invoker significance, transaction rate, and actual block size metrics to derive system-level optimization recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Since these recommendations are based on the blockchain log generated by the running application, it helps the user to identify ideal configuration settings based on their current use-case and workload, leading to performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (7) Block size adaptation: The average transaction rate (Tr), the average block size (Bsizeavg) and a configurable threshold (Bt) are used to recommend block size adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The literature recom- mends smaller block sizes when transaction rates are lower and larger block sizes when the rates are higher [13, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If the block size is too small, too many blocks are created, and block creation be- comes a bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If the block size is too large, the block creation is delayed by waiting for sufficient transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' if block size adaptation is recommended,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' then set Btimeout and Bcount such ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Fabric Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Automated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Workflow Engines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Transaction rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Client resource boost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Clients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Smart contract updates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Delta writes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Smart contract partitioning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Data model alteration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Process model pruning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Configuration updates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Block size adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='ü Endorser restructuring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Recommendations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='BlockOptR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Figure 6: Optimization implementation on a live Fabric network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='that min{Bcount,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Tr ∗ Btimeout} is equal to Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We do not provide recommendations for block bytes adaptation since it is difficult to accurately derive the size of a transaction (that can include the transaction payload, endorser identities and other metadata) from the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (8) Endorser restructuring: For every Fabric transaction gener- ated by the clients, the corresponding smart contract function is executed by the endorsers defined in the endorsement policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Smart contract execution is a time and resource-consuming action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' If the same endorsers receive a higher load of transactions while others remain idle, this indicates a bottleneck or load imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Such load imbalances can occur when the endorsement policy explicitly de- fines an endorsement as mandatory from a specific set of endorsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, the endorsement policy And(Org1,OR(Org2,Org3)) implies that an endorsement from Org1 is mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' As a conse- quence, Org1 could become an endorsement bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We detect endorser bottlenecks by identifying endorsers that endorse more transactions than a user-specified threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The default thresh- old values detect whether all the endorsers participate equally in the endorsement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The threshold values can be adapted to increase or decrease the sensitivity to imbalances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' (9) Client resource boost: Multiple time-consuming tasks are performed by the clients in a Fabric network, including but not limited to transaction proposal invocation, endorser response veri- fication, packing of endorser responses as a transaction, transaction submission to the ordering service, and collection of peer commit responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The invoker significance metrics identify the clients and the corresponding organizations that invoke a majority of the transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This identification can assist in resource allocation decisions, such as increasing the number and size of clients regis- tered to the identified organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' It could also point to problems in the underlying business process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='5 Implementation of Optimizations The recommended optimizations can be implemented in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Figure 6 visualizes where the different recommendations can be implemented on a live Fabric network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Here, we show an automated workflow engine that triggers transactions based on a process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' These transactions are sent via the clients to the Fabric network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The logs of the Fabric network are analyzed by BlockOptR to generate optimization recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Each of the recommended optimizations can be implemented at different levels as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Activity reordering can be implemented by modifying the un- derlying process model in the workflow engine such that activities follow a conflict-free order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Alternatively, one can monitor the transactions on the clients and reorder either per client or across all clients using a client manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Process model pruning can be implemented via organizational measures such as incentives or penalties to ensure that activities adhere to their expected behavior (not shown in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, pruning can also be imple- mented directly in the smart contract by early aborting anomalous transactions during the endorsement phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Transaction rate con- trol can be implemented in multiple ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Each client can monitor their own transaction rate and perform load shedding or queuing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The same can be done across clients using a central monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' A third approach is to monitor the transaction rate in the ordering service and apply load shedding there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Smart contract revisions are required to implement all the data-level optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In Fabric, smart contract upgrades are possible on the fly without restarting the system [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Block size can be adapted either by changing the configuration file or by using a configuration update transaction in Fabric [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Endorser restructuring can be implemented by altering the endorsement policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The policy can be changed in the Fabric configuration file or using a configuration update transaction [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Based on the transaction load per client identified by BlockOptR, client resources can be scaled if the current allocation appears in- sufficient to handle the load and the new clients can be dynamically registered to the Fabric network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Although all optimizations can be ap- plied in a live system on the fly, since our evaluation runs in an experimental environment, we restart the Fabric network after every experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We use the Caliper benchmarking system [35] which has a client manager that can be configured to order the transactions across clients and control the rate of transactions gen- erated, thus emulating activity reordering and transaction rate control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The number of clients can also be scaled to demonstrate a client resource boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Process model pruning and all data-level optimizations are implemented by analyzing and modifying the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Block size and endorsement policies are updated in the Fabric configuration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 5 Experimental Methodology We used version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 of HyperledgerLab [13], which is an au- tomated testbed for Hyperledger Fabric 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 integrated with the Caliper benchmarking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We set up a Kubernetes cluster of 1 master and 5 worker nodes over which all the Fabric network components as well as Caliper components are distributed as Ku- bernetes pods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Each node runs on a Ubuntu Focal (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='04) virtual machine with 4 vCPUs and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='8 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We use 10 Caliper work- ers for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For every experiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' we measure the success rate which is the percentage of successful transactions out Table 2: Control variables Control Variable Values (Default in bold) Workload type Uniform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Read-heavy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Insert-heavy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Update-heavy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' RangeRead-heavy Endorsement policy P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' P2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' P4 Endorser distribution skew 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6 Key distribution skew 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 2 Number of organizations 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 4 Block count 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 1000 Send rate 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 1000 Transaction dist skew 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 70% of the total number of transactions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' the average latency and the throughput of all successful transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 Workload Generation The content of the distributed ledger, which is used as the in- put to our tool, is a direct result of the workload executed on the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, we extensively evaluate BlockOptR by using three different types of workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Also, after implementing the rec- ommendations generated by BlockOptR, we rerun the experiments with the same workloads to analyze the effect of the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 Synthetic workloads We use an extended version of a synthetic workload generator that can generate synthetic workloads based on a set of control variables for a generic smart contract genChain [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We use a range of values for these control variables described in Table 2 to generate multiple workloads of 10,000 transactions each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The endorsement policies used in our experiments are: P1: And(Org1, Or(Org2,Org3,Org4)) P2: And(Or(Org1,Org2), Or(Org3,Org4)) P3: Majority(Org1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=',OrgN) P4: OutOf(2,Org1,Org2,Org3,Org4) By generating synthetic workloads, we ensure that multiple realistic scenarios are covered in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We then evaluate BlockOptR with each of these workloads to generate optimization recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, we implement each of the recommended optimizations to evaluate the performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 Use-case based workloads Secondly, we use extended versions of four popular use-case based smart contracts from the literature [13] and generate work- loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' BlockOptR is then used to generate optimization recommen- dations with these workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The four smart contracts we use are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Supply Chain Management (SCM): This smart contract defines the operations of a logistics network that includes sending an ad- vanced shipping notice, shipping a product, reading the shipping notice and unloading the product (in this order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' There is also a query operation to query the information of the different products (queryProducts) and a updateAuditInfo function that updates an audit entry with the product details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' These can happen at any point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We generated a workload of 10,000 transactions based on these assumptions by sending in order the transactions pushASN, ship, queryASN and unload while the transactions queryProducts and updateAuditInfo are sent randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Digital Rights Management (DRM): This smart contract manages the rights of artists in the music industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The smart contract in- cludes a Play function that is executed whenever a piece of music is played by any user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The other smart contract functions include ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Table 3: Experiments with the synthetic workload ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Experiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Control variable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Optimizations recommended ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Endorsement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Endorser restructuring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Policy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Endorsement Policy / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='P2 / 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Endorser restructuring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Endorser dist skew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='No: of orgs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Workload ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Read-heavy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Workload ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Update-heavy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Workload ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Insert-heavy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Workload ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='RangeRead-heavy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='distribution skew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Smart contract partitioning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Block size adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Block count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Block count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Block count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Send rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Send rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Block size adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Send rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='70% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='distribution skew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Client resource boost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='adding a new piece of music,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' querying the rights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' viewing the meta- data and calculating the revenue of the right holders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In a realistic scenario, the Play transaction would be executed far more fre- quently than all the other transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, we create a Play heavy workload for this use-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We generate 10,000 transactions randomly where 70% of the transactions are Play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The remaining 30% comprise all the other transactions generated uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Electronic Health Records (EHR): In this smart contract, patients can provide or revoke access rights to medical institutes as well as research institutes to query their medical records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We assume that the number of patients would be more than the other participants and generate a 70% update-heavy workload of 10,000 transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Digital Voting (DV): This smart contract includes a function to vote in an election, query the parties, query the results as well as end the election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We can assume that during an actual election there will be periods of high traffic while the voting is taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, we generate a workload which initially sends 1,000 queryParties transactions at a rate of 100 TPS, then 5,000 Vote transactions at a rate of 300 TPS and finally 1 seeResults and endElection transaction each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 Loan Application Process (LAP) Thirdly, we created a smart contract and workload using a real- life event log of the loan application process of a Dutch financial institute which is available publicly [77] together with the corre- sponding process flow [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We extracted all the events of the first 2,000 loan applications and created 20,000 corresponding transac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We then created a smart contract where every activity in the loan application process flow has a corresponding smart contract function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The event log contains an employeeID for every employee in the bank handling loan applications and an applicationID for every loan application processed by the bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The smart contract ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='we implemented uses the employeeID as the key and the value of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='the key is an array of structures where every structure includes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Table 4: Settings to implement optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Optimizations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Settings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Recommended ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Reorder workload generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Set send rate to 100 TPS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Process model pruning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Delta writes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Update smart contract ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Smart contract partitioning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Data model alteration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Block size adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Set block count to derived transaction rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Endorser restructuring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Set endorsement policy to P4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Client resource boost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Double clients for recommended organization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='the applicationID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' loan type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' loan amount and loan status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, querying a specific employeeID will easily provide all the applications processed by that employee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We then executed the 20,000 transactions on the smart contract at a low rate of 10 TPS to simulate a real world scenario where manually processing the loan applications takes a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We also ran the same experiment at a higher rate of 300 TPS to emulate an automated loan application and validation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We use BlockOptR to generate optimiza- tion recommendations which help to improve the smart contract implementation and thereby the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Though the LAP event logs are from a database setting, this is a realistic use-case for blockchains as an automated loan ap- plication system requires security and decentralized trust (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=', micro-loans, decentralized loan applications, and more generally DeFi [33, 58, 82, 83]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Consequently, this experiment demonstrates the utility of BlockOptR in a realistic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In the use-case based experiments, all the transactions followed the expected order based on the assumptions we defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In contrast, with this real event log, we evaluate the real order in which the transactions are executed which can deviate from the process model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6 Experimental Results We exhaustively evaluate our recommendation approach with a wide range of workloads and smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Please note that, whenever transaction rate control is implemented there is an ex- pected decrease in the throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, clients benefit heav- ily from higher success rates, and the apparent decrease in the throughput is just closer to the sustainable throughput of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In all our experiments the default value for the thresholds are Et = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='5, Rt1 = 300, Rt2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3, Bt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='6 and It = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' All the settings including the control variable values changed to implement each recommended optimization is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 Synthetic Workloads Due to space restrictions, we present 15 workloads in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The full list of experiments and results can be seen in our reposi- tory [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The control variable that is tuned for each experiment is shown along with its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' All the other control variables have the default value shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Experiments 1 to 15 are con- ducted with no optimizations applied and then BlockOptR is used to derive optimization recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The recommendations generated by BlockOptR are also shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Since the syn- thetic smart contract has a simple logic with no branches, incre- ment/decrement operations or complex data model, process model pruning, delta writes and data model alterations are not recom- mended here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Next, we implement the recommended optimizations and re-execute all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The results of the experiments are grouped based on the optimization recommendations and can be seen in Figures 7, 8, 9, 10, 11 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We also explain how the thresholds are set for our experiments and how they can be tuned by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 Endorser restructuring: The effect of endorser restruc- turing can be seen in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' When the endorsement policy is P1, all the clients must send their transactions to Org1 due to the specific endorsement policy and hence, an endorsement bottleneck is detected for Org1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Since the endorsement policy re- quires signatures from two organizations, we change the policy to OutOf(2,Org1,Org2,Org3,Org4) so that the clients can distrib- ute the transactions evenly among all endorsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This optimization leads to a 29% increase in throughput (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In Experiment 2, since the endorser distribution is skewed, the clients send transac- tions unevenly and therefore two of the organizations endorse far more often than the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We re-executed the experiment with an even distribution of transactions to the endorsers and observe a 26% increase in throughput (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The main impact of this optimization is on throughput and latency as it reduces transaction queuing on few specific peers and instead distributes them evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We set the thresholds for this recommendation such that we ex- pect an even distribution of transactions to all endorsers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=', even minor bottlenecks are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This can be tuned to detect only severe bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, since these are synthetic experiments, changing the endorsement policy is not critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In real scenarios, consultation with the governing bodies of an enterprise is required before changing the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Still, the recommendations by Block- OptR help to highlight bottlenecks which in turn can convince the management to change the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 Client resource boost: Figure 8 shows the effect of client resource scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' After increasing the number of clients, we observe a 75% decrease in latency, a 15% increase in throughput, and a 7% increase in success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The thresholds are set such that this optimization is recommended when more than 50% of transactions are invoked by the same organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This can be fine-tuned to detect less severe bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 Block size adaptation: The effect of block size adaptation can be seen in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In our experiments, we use the default block time out of 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, we make the block count equal to the transaction rate whenever the block size adaptation is rec- ommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' After changing the block size, we observe up to 93% improvement in throughput and 85% improvement in success rate (Figure 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Block count: 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The thresholds are set such that this op- timization is recommended whenever the average block size is 60% larger or smaller than the transaction send rate derived from the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The thresholds can be decreased to make the recommendation more sensitive to transaction rate changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 Transaction rate control: The effect of transaction rate control is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In these experiments, periods of high traffic (around 300 TPS) were also identified as periods of high failure rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We then lowered the transaction send rate to 100 TPS on the clients and re-executed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We observe significant improvement of up to 87% in latency and 36% in success rate (Figure 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Send rate: 1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We set the thresholds for this recommendation at 300 TPS which is the default send rate of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This means that we consider the current traffic of NewEnd Pol 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 W/O W W/O W Endorsement policy: P1 Endorsement policy: P2 Endorser dist skew: 6 Control Variables Success throughput (tps) Average latency (s) Percentage of success (%) Figure 7: Endorser 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dist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='skew: 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Block count: 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Block count: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Block count: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Send rate: 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Send rate: 300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Send rate: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction dist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='skew: 70% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Control Variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Success throughput (tps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Average latency (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Percentage of success (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Figure 11: Activity reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='New All optimizations ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 W/O W W/O W W/O W W/O W W/O W W/O W W/O W W/O W Endorsement policy: P1 EndorsementPolicy:P2 Endorser dist skew: 6 Key dist skew: 2 Block count: 50 Block count: 300 Block count: 1000 Send rate: 1000 Transaction dist skew: 70% Control Variables Success throughput (tps) Average latency (s) Percentage of success (%) Figure 12: All recommended optimizations combined the system as high and want to detect periods of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Users can adjust this threshold based on what is considered high (more than the sustainable traffic rate) for their Fabric network installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='5 Activity reordering: The effect of activity reordering can be seen in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We observe that BlockOptR recommends ac- tivity reordering for all experiments except Experiments 3 and 5 (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Reordering was suggested for two activities (Read and Update) which conflict with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We updated the configura- tion of the client manager to generate read transactions before all other transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This implementation emulates a scenario where organizational measures were applied to enforce activity reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We then re-executed the experiments and observe a performance improvement in all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' There is up to 65% increase in throughput and 58% increase in success rate (Figure 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Workload: RangeReadheavy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We have set the thresholds such that if 40% of the MVCC failures are caused by activities that can be reordered, this strategy is recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This can be made more lenient by increasing the threshold such that reordering is suggested only in very significant cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For Experiments 3 and 5, less than 40% of MVCC conflicts are caused by the two activities where reordering is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, the activity Update has a dependency on itself which cannot be removed by reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='6 Combined optimizations: We also executed the experi- ments after applying all the recommended optimizations together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We observe up to a 93% improvement in throughput and 85% im- provement in the success rate (Figure 12: Block count: 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In all the experiments, the performance obtained by applying all the optimiza- tions is comparable to the performance yielded by the optimization with the highest improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Though smart contract partitioning is recom- mended for Experiment 8, this optimization requires understanding the functionality of the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Unfortunately, for the syn- thetically generated smart contract that includes only generic read, update and insert functions, we cannot redesign the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='48 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='18 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='31 286.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 Without optimization Transaction rate control Activity reordering Process model pruning All optimizations Success throughput (tps) Average latency (s) Success rate (%) SimpleSCM Figure 13: SCM use-case 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 Without optimization Delta-write Activity reordering Smart contract partition All optimizations Success throughput (tps) Average latency (s) Success rate (%) DRM Figure 14: DRM use-case NewEHR 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='57 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='34 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='96 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='56 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='94 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='01 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 Without optimization Transaction rate control Activity reordering Process model pruning All optimizations Success throughput (tps) Average latency (s) Success rate (%) Figure 15: EHR use-case DV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='7 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 Without optimization Transaction rate control Data model alteration All optimizations Success throughput (tps) Average latency (s) Success rate (%) Figure 16: Digital voting use-case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='0 Without optimization Data model alteration Without optimization Data model alteration Transaction rate control All optimizations Send rate: 10tps Send rate: 300tps Success throughput (tps) Average latency (s) Success rate (%) LAP Figure 17: Loan application process use-case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='2 Use-case based Workloads Supply Chain Management (SCM): With the SCM use-case, three optimizations are recommended by BlockOptR: activity reorder- ing, process model pruning and transaction rate control (Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' After implementing reordering for the reorderable activities (query- Products and UpdateAuditInfo), we observe a 24% increase in throughput and 15% increase in success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Pruning was recom- mended for the Ship activities that occur without or before the PushASN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' It was also recommended to prune Unload activi- ties that occur without or before the Ship activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We adapted the smart contract to implement the pruning recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This resulted in a 27% improvement in throughput and 19% increase in success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Transaction rate control and applying all recommen- dations together also improves the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Digital Rights Management (DRM): With the DRM use-case, three optimizations are recommended by BlockOptR: activity reordering, delta-writes and smart contract partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Figure 14 shows the results of applying these optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' To implement the delta write recommendation, we observed that the Play function in the smart contract has an increment operation to count the number of times a piece of music was played.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We converted this into a delta write and the delta-keys are aggregated whenever the calcRevenue function is invoked (since it requires the play count).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' With this optimization, we can observe a significant improvement of 42% in throughput and 50% in success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, the average latency increases in this case because the calcRevenue function now takes up more time for aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Since calcRevenue is not executed as frequently as Play, the overall performance is not affected though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Activity reordering was recommended for calcRevenue and queryRightHolders functions and we reconfigured the clients to send these activities after all other activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This emulates a sce- nario where an organization restricts specific transactions to spe- cific time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We observe more than 50% increase in both throughput and success rate with this optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Hot keys were detected and frequently used by four activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We analysed the smart contract and discovered that, though all four functions have a dependency on the same key, the functionalities are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Play and calcRevenue need only the play count, while viewMetaData and queryRightHolders need metadata and not the play count of a piece of music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, we split the smart contract into two, where one smart contract has the Play and calcRevenue functions and the second smart contract has the FabricSharp 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='92 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='56 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='56 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='16 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='36 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='42 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='14 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='56 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='08 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='50 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='17 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 Without optimization Endorser restructuring Without optimization Endorser restructuring Without optimization Transaction rate control Endorsement policy: P1 Endorsement policy: P2 Endorser dist skew: 6 Workload: Insert-heavy Control Variables Success throughput (tps) Average latency (s) Success rate (%) Generator Figure 18: Synthetic workloads with FabricSharp other two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The create function is included in both smart contracts, and invocation of the first smart contract invokes the same function in the second smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We observe a 35% increase in throughput and a 26% increase in success rate with this optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Applying all the optimizations together improves the performance by more than 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Electronic Health Records (EHR): In this use-case, three optimiza- tions were recommended: activity reordering, process model prun- ing and transaction rate control (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Activity reordering for the read activities resulted in a 60-65% improvement in throughput and success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' When the smart contract was updated to prune illogical paths (revoke access to records without granting access), we observe around 43% increase in throughput and success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' After applying transaction rate control, a 69% increase in success rate was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' All optimizations applied together also improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Digital Voting (DV): In this use-case, two optimizations were recommended: transaction rate control and data model alteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The results are shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' High failure rates were detected for periods when the Vote transactions were frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' After ap- plying transaction rate control, a slight improvement of 11% in throughput was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The hotkeys were detected and most frequently used by the Vote function resulting in a recommenda- tion to alter the data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We analysed the smart contract and observed that partyID was used as the key for the vote function which is invoked by multiple voters during the voting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We redesigned the smart contract such that voterID is assigned as the primary key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Since voters are restricted to a single vote, we observe 100% success rate with this new smart contract because there are no more transaction dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We also observe an improvement in the performance when both optimizations are applied together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3 Loan Application Process (LAP) The optimization recommended for the LAP use-case was data model alteration (Figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The employeeID 1 had a high key Fabric++ Generator UH, RH and RRH 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='27 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='56 159.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Without ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='All ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='optimizations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Without ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='All ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='optimizations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Without ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Transaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='rate control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Activity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='All ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='optimizations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Workload: Update-heavy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Workload: Read-heavy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Workload: RangeRead-heavy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Control Variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Success throughput (tps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Average latency (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Success rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='Figure 19: Synthetic workloads with Fabric++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='frequency since this employee processed the highest number of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='loan applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We then re-implemented our smart contract and assigned applicationID as the key and modeled the value as a structure that includes employeeID, loan amount, loan type and loan status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This new implementation helped to remove the hot key and yielded more than 50% improvement in throughput and success rate for both the lower and higher send rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='4 Fabric Extensions As a holistic recommendation approach, our work lies orthogo- nal to existing Fabric optimizations in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In this section, we demonstrate how our approach works on top of two optimized extensions of Fabric: FabricSharp [65] and Fabric++ [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Both im- plement different transaction reordering strategies that mitigate MVCC read conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The Fabric++ scheduler is integrated in the FabricSharp implementation [25] and we use this for our exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We executed the synthetic workloads on both and then used BlockOptR to generate recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The literature says FabricSharp increases endorsement policy failures and is less per- formant for insert-heavy workloads while Fabric++ is least per- formant with an update-heavy, read-heavy and range-read-heavy workloads [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, we execute these specific experiments shown in Figures 18 and 19 with the synthetic workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Activ- ity reordering, transaction rate control and endorser restructuring were recommended and by implementing these recommendations, we observe up to a 55% increase in throughput and 46% increase in success rate (Figure 19: RangeRead-heavy workload).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our experi- ments with these Fabric extensions show that even with effective system-level optimizations, Fabric can still benefit from optimiza- tions at all levels of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 7 Lessons Learned and Limitations We demonstrated that BlockOptR is capable of effectively recom- mending suitable optimization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, we also explained how to implement these optimizations and quantified the perfor- mance improvements after implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This section discusses the insights we gained from our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' User level optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Activity reordering was one of the most frequently recommended optimizations in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We highlight use-cases such as SCM where such reordering can be applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our model pruning recommendation emphasizes that identifying incompetencies in the process model can lead not only to efficient process execution but also improve the performance of the underlying system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Load shedding or queuing is often employed when systems cannot handle the workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Using our recommen- dations, specific activities and time periods can be identified where such rate control techniques are most effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, rate control is recommended for the Vote activity in the digital vot- ing use-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, instead of system-wide rate control, only the specific clients that deal with the identified activities need to employ rate control techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Data level optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' These optimizations show how the design of the smart contract and the data model significantly in- fluence the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The smart contract is initially designed with a specific process model in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, we understand how the smart contract is being used in practice by analyzing the blockchain logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' BlockOptR pinpoints functions and keys that cause bottlenecks which in turn helps the smart contract developer to make appropriate modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' System level optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Setting the endorsement policy is a management decision that often excludes discussions with the technical team designing the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our recommendations highlight the need to bring together management and technical discussions to decide optimal configuration settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, we also demonstrate the need to verify whether the policy is being used effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, even if the policy defines the equal dis- tribution of endorsements, the clients may send their transactions in a skewed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In such instances, we recommend enforcing a management measure, such as dividing the endorsers equally among the clients such that clients of one organization only send transactions to specific endorsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The compliance with such mea- sures can also be checked by BlockOptR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Block size optimization is frequently discussed in the literature and associated with the transaction rate of a system [13, 36, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Instead of system-level changes such as using transaction rate monitors, we derive the transaction rate and the actual block size from the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This helps to understand traffic patterns over time and find reasonable block size settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' While the literature mainly focuses on optimizing the peers and ordering service components of Fabric [27, 65, 67], our client-related recommendations highlight the need to focus on client-side optimizations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Technology Independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our multi-level recommendation approach is demonstrated using the Fabric blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Technology independence is difficult to attain due to the vast implementation variations between the numerous blockchain systems and the cor- responding differences in the contents of the distributed ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, we draw attention to specific examples which can guide future researchers to translate our approach to other blockchain systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In Quorum, the block time or mining frequency has a lin- early proportional influence on the transaction latencies [7] which is analogous to our block size adaptation recommendation strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Also, Corda has the concept of notaries to attest transactions where distributing the transactions over multiple notaries is ex- pected to improve the throughput [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' This is again comparable to our endorsement restructuring recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, there are numerous gas-fee reduction and vulnerability detection strategies for Ethereum smart contracts in the literature [54] which translate to our recommendations at the data level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Tools like Lorikeet and Caterpiller automate the conversion and execution of process mod- els as Ethereum smart contracts, which would make it easier to implement the user-level optimizations that we recommend [43, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The optimizations recommended by BlockOptR need to be manually implemented by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' A self-adaptive system with a feedback loop that automatically implements the recommendations is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, in an enterprise scenario, for many of the optimizations such as endorser restructuring, ac- tivity reordering, and process model pruning, management level approvals might be required before implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Additionally, for applications that do not follow a specific process model, the event logs can be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In such scenarios, user-level optimiza- tions such as activity reordering and process model pruning are not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, domain knowledge about the use-case is required for implementing the recommended optimizations appro- priately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, our implementation of some of the optimizations such as transaction rate control are trivial in such benchmarking scenarios and do not account for real-world overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, the implementations are mainly for demonstrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our work focuses on the multi-level recommendation approach used by BlockOptR rather than the implementation of the optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Finally, our experiments without and with the recommended opti- mizations are done on similar workloads generated with the same input parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=', we assume a continued trend in the pattern of the workload after the optimizations are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, in scenarios where the workload fluctuates or the optimization imple- mentation is delayed, BlockOptR may need to be re-executed to generate new recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 8 Related Work The literature proposes various Fabric optimization strategies such as transaction reordering [13, 65, 67], block size optimiza- tions [13, 36], CRDTs [52], and parallelizing various components [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Our work lies orthogonal to such optimization strategies and fo- cuses on an optimization recommendation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We demon- strate how our recommendations can be used along with two of the literature’s optimization strategies to improve performance further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' There is also extensive research in the database community on index and query optimizations that include self-tuning systems as well as recommendation systems [2, 14, 15, 38, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Though we can draw parallels from these research, our work focuses on blockchain-specific optimization recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Different con- figuration settings (such as block size and endorsement policy) and the concept of smart contracts introduce new dimensions to the recommendation approach, which are not required for databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' There is ongoing research on applying process mining tech- niques on blockchains to derive process-level insights [24, 32, 39, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Klinkmüller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' [39] and Mühlberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' [49] describe different approaches to extract process data from the Ethereum blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Hobeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' [32] use process mining on an Ethereum- based betting application to identify shortcomings in the appli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Process mining on blockchains currently only focuses on permissionless blockchains as they are publicly accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' How- ever, deriving and studying the process model is equally critical for private blockchains, and therefore, our work contributes to this less explored area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, unlike the related work, we focus on using process mining for recommending blockchain optimiza- tion strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We only found a single paper that uses permissioned blockchains, where Duchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' [24] extract process data from Fabric and detect semantic errors in a smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Though our work is comparable, we extract not only the process data but also blockchain-specific attributes from Fabric, derive multiple metrics, and recommend optimization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' There is extensive research in the database community in the domain of data-aware business processes that encourage a business process perspective to database management systems [11, 20, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Calvanese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' [11] comprehensively survey the contributions in this realm and catalog contributions from various fields, including database theory and process management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' These works were an important motivation for us to view blockchains from a business process perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' However, our work brings new contributions since blockchains deal with several other elements apart from data, such as smart contracts and endorsement policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 9 Conclusions This paper showcases the necessity and effectiveness of having a holistic perspective on blockchain optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We define a multi-level recommendation approach based on several metrics and attributes derived from the blockchain log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We define a total of nine optimizations at the system, data, and user-level of a blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We implement an automated optimization recommendation tool, BlockOptR, based on these concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, we demonstrate how such optimizations can be implemented to improve the system per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' After implementing the recommended optimizations, we observe an average of 20% improvement in the success rate and an average of 40% improvement in latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We extensively evalu- ate the system with a wide range of workloads covering multiple real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We hope to inspire enterprises to use our contributions to detect blockchain optimization strategies and to contribute their live blockchain (anonymized) logs for further re- search in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' The BlockOptR tool, all the smart contracts, the workload generation scripts, and all the event logs are available as open-source [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' We also plan to extend our tool to include more optimization recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' In terms of future work, we are currently developing a ProM plu- gin which would provide a user-friendly interface for BlockOptR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Presently, the threshold settings of BlockOptR depend on the busi- ness network setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' For example, the rate threshold for our setup was 300 TPS as higher rates led to instabilities, but this can vary for other deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Therefore, tuning these thresholds automati- cally in BlockOptR could be a future extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Another interesting extension is to define additional attributes that applications can log, thereby providing more data for optimization recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Further, investigating the effect of workload fluctuations and delay in applying the recommendations is another challenging future direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Acknowledgments This work is funded in part by the Deutsche Forschungsgemein- schaft (DFG, German Research Foundation) - 392214008, and by the Bavarian Cooperative Research Program of the Free State of Bavaria - DIK-2002-0013//DIK0114/02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' References [1] Parinaz Ameri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Challenges of index recommendation for databases: With specific evaluation on a NoSQL database.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 121–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1145/3327960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='3332395 [83] Dirk A Zetzsche, Douglas W Arner, and Ross P Buckley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Decen- tralized Finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' Journal of Financial Regulation 6, 2 (09 2020), 172–203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='1093/jfr/fjaa010 arXiv:https://academic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='oup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='com/jfr/article- pdf/6/2/172/37064506/fjaa010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} +page_content='pdf' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf'} diff --git a/RtAyT4oBgHgl3EQfVPc7/content/tmp_files/2301.00139v1.pdf.txt b/RtAyT4oBgHgl3EQfVPc7/content/tmp_files/2301.00139v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..81d146a8cba5849e258604d510130a1fbebad7f1 --- /dev/null +++ b/RtAyT4oBgHgl3EQfVPc7/content/tmp_files/2301.00139v1.pdf.txt @@ -0,0 +1,7916 @@ +Submitted to the Annals of Statistics +ON HIGH DIMENSIONAL POISSON MODELS WITH MEASUREMENT +ERROR: HYPOTHESIS TESTING FOR NONLINEAR NONCONVEX +OPTIMIZATION +BY FEI JIANG1, YEQING ZHOU2,*, JIANXUAN LIU3,† AND YANYUAN MA4,‡ +1Department of Epidemiology and Biostatistics, The University of California, San Francisco, fei.jiang@ucsf.edu +2School of Mathematical Sciences, Tongji University, *zhouyeqing@tongji.edu.cn +3Department of Mathematics, Syracuse University, †jliu193@syr.edu +4Department of Statistics, Pennsylvania State University, ‡yzm63@psu.edu +We study estimation and testing in the Poisson regression model with +noisy high dimensional covariates, which has wide applications in analyz- +ing noisy big data. Correcting for the estimation bias due to the covariate +noise leads to a non-convex target function to minimize. Treating the high +dimensional issue further leads us to augment an amenable penalty term to +the target function. We propose to estimate the regression parameter through +minimizing the penalized target function. We derive the L1 and L2 conver- +gence rates of the estimator and prove the variable selection consistency. We +further establish the asymptotic normality of any subset of the parameters, +where the subset can have infinitely many components as long as its cardi- +nality grows sufficiently slow. We develop Wald and score tests based on the +asymptotic normality of the estimator, which permits testing of linear func- +tions of the members if the subset. We examine the finite sample performance +of the proposed tests by extensive simulation. Finally, the proposed method +is successfully applied to the Alzheimer’s Disease Neuroimaging Initiative +study, which motivated this work initially. +1. Introduction. +Count data are routinely encountered in practice. For example, cog- +nitive scores in a neuroscience study, the number of deaths in an infectious disease study, +and the number of clicks on a particular product on an e-commerce platform, are all count +data. Because most of the count data are concentrated on a few small discrete values rather +than expanded on the entire real line and because the distribution of count variables is often +skewed, the familiar linear model becomes less ideal to capture these features. In the liter- +ature, Poisson regression (McCullagh & Nelder 2019) is arguably the most popular model +to describe count outcomes, because it naturally models the skewed distribution for posi- +tive outcomes. On the other hand, together with the count data, a large number of covariates +are often collected thanks to the ever advancing capability of modern technologies. However, +these covariates are often contaminated with errors due to imperfect data acquisition and pro- +cessing procedures. Ignoring these errors can produce biased results, which can finally lead +to misleading statistical inference on the model parameters (Carroll et al. 2006) that explain +the association between covariates and outcomes. Our goal is to develop rigorous statistical +inference procedures to test linear hypotheses in the high dimensional Poisson model with +noisy covariates. Such inference tools will enable explaining the association between the +count outcome and the individual covariate or combination of covariate, quantifying the un- +MSC2020 subject classifications: Primary 00X00, 00X00; secondary 00X00. +Keywords and phrases: High dimension Inference, Measurement Error, Non-convex optimization, Poisson +model. +1 +arXiv:2301.00139v1 [math.ST] 31 Dec 2022 + +2 +certainties of the estimated association, and controlling the false discovery rate when testing +scientifically important hypotheses. +Let Y be the count outcome and X be its associated covariate vector. In the Poisson model, +Y is related to X as +prpY | Xq “ e´exppβTXqtexppβTXquY {Y !. +(1) +We study the testing problem in (1) under the situation that the covariate vector X is both +high dimensional and contaminated with noise. When X is accurately observed, the testing +problem has been extensively discussed in the literature (Ning & Liu 2017, Zhang & Cheng +2017, Van de Geer et al. 2014, Shi et al. 2019). However, when X is not accurately observed, +it is unclear that any of the existing proposed tests are applicable, and testing in the high +dimensional noisy Poisson regression model has not been explored. The major obstacles +in constructing valid hypothesis testing procedures are as follows. 1) The existing lasso- +type penalized Poisson estimator (Jiang & Ma 2021) does not enjoy the variable selection +consistency when the number of parameters is much larger than the sample size. 2) The +asymptotic normality of the estimator has not been established. We develop Wald and score +tests targeting at linear hypothesis on the parameters of interest in (1). To overcome obstacle +1), we improve the penalized Poisson estimator proposed in Jiang & Ma (2021) by using +a class of “amenable” penalty functions first defined in Loh & Wainwright (2015, 2017) in +combination with a modified log-likelihood function to construct estimators. We establish +the estimation consistency and variable selection consistency of the resulting estimators. To +bypass obstacle 2), we derive the asymptotic linear form of the estimators, and establish +the asymptotic normality. The asymptotic normal estimator has a wider range of applications +than the lasso type estimator does, because it facilitates subsequent inference procedures such +as constructing hypothesis testing procedures. +Even after establishing the asymptotic normal properties, it is still challenging to gener- +alize Wald and score tests to the high dimensional setting for Poisson regression with noisy +data. This is because under the amenable penalties (Loh & Wainwright 2015, 2017), the +asymptotic normality of the estimators is built on a minimal signal condition, which requires +the nonzero elements in β to be at least of order λ. Here λ is the penalty parameter which +goes to zero when sample size increases. Now consider testing the null hypothesis β1 “ 0 +versus the alternative β1 “ hn, where β1 is the first element of β. The minimal signal condi- +tion implies that the test will have no power in testing the local alternative when |hn|2 ăă λ. +To resolve this issue, we remove the penalties on the subvector of the parameters involved +in the test. However, it is still unclear how fast the dimension of the subvector can grow +while still ensuring sufficient power. To this end, we derive the convergence property of the +estimators, which provides the explicit rate at which the dimension of the subvector is al- +lowed to grow with the sample size in order to achieve consistency, asymptotic normality, +and sufficient power in testing. Furthermore, to implement the score test, we need to esti- +mate the regression parameters under the null hypothesis, which involves optimization under +linear equality constraints. This type of constrained parameter estimation for noisy Poisson +model has not yet been developed. To fill this gap, we develop a general procedure for pa- +rameter estimation under linear constraints. The constraints include inequality constraints for +the parameter estimation under general Poisson model and an additional equality constraint +imposed by the null hypothesis, which leads to great challenge in establishing the convexity. +Incorporating inequality constraints is practically important because it allows to incorporate +additional parameter information, which will reduce the estimation variation and in turn the +sample size needed to achieve satisfactory estimation accuracy. +We briefly summarize our contributions as follows. First, we develop a new estimation pro- +cedure of the Poisson model with amenable regularization for noisy data. Second, we show + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +3 +the variable selection consistency and the consistency of the resulting estimator. We provide +explicit convergence rate of the estimator. Third, we derive the asymptotic normality of the +estimator for the nonzero parameters and the parameters to test. Fourth, we propose the Wald +and score test procedures by constructing the corresponding test statistics. Fifth, we derive +the asymptotic distributions of the Wald and score test statistics. These five essential elements +combined together finally allow us to perform hypothesis testing for Poisson model with high +dimensional noisy covariates, which allows us to answer important questions such as “if the +left inferior temporal gyrus has a significant impact on the development of Alzheimer’s dis- +ease”. These estimation and inference tasks are not straightforward to achieve, they require +building up a series of theoretical properties first, which involves techniques related to analyz- +ing conditional sub-Gaussian distribution tails, utilizing and modifying various concentration +inequalities, constructing the prime-dual equivalence, carefully bounding various quantities, +linking different vector and matrix norms, and establishing a Lyapunov-type bound (Bentkus +2005) on the probability distribution to derive the asymptotic distribution of proposed test +statistics. All these analyses are performed under the unusual constraints involving both lin- +ear equality constraints and parameter restrictions. We also modify the alternating direction +method of multipliers (ADMM) algorithm to solve a regularized optimization problem un- +der linear constraint in constrained parameter space. Although each individual technique in +its basic form has been used in the literatures of mathematical analysis, statistics, combina- +torics, operations research and computer science, a seamless combination of all these into a +general tool to solve the problem under study is very challenging and difficult. +Count data occur frequently in practice, and it is a rule rather than exception that the co- +variates can be contaminated. In modern data collection mechanism, covariates are almost +always high dimensional. Hence, estimation and inference in Poisson regression with high +dimensional noisy covariates is a general problem with wide applications. A direct motiva- +tion of this work is the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study, which +is a multi-site longitudinal study investigating early detection of Alzheimer’s disease (AD) +and tracking disease progression biomarkers (Weiner et al. 2017). Recently, the advent of +tau-targeted positron emission tomography (PET) tracers such as flortaucipir (18F-AV-1451) +has made it possible to investigate the relative (to patient’s body weight) tissue radioactivity +concentration of the tracers, quantified as standardized uptake value ratio (SUVR), in rela- +tionship to the cognitive function. Therefore, we aim to study the association between cog- +nitive scores and SUVRs from PEG image data. We extract Montreal Cognitive Assessment +(MoCa) scores (Y ) and SUVRs (X) from the PET image in the ADNI study taken within 14 +days of the cognitive tests from 196 subjects in the ADNI phase 3 study. We first perform a +linear lasso regression between the logarithm of MoCa score and the 218 covariates including +age, gender, SUVRs, and volumes of whole brain ROIs. Figure 1 shows the density of the +residuals from the lasso regression, which suggests that the residuals are skewed and hence +the linear lasso regression does not provide a satisfactory fit for the data. This motivates us to +consider Poisson regression. We utilize the Poisson high dimensional hypothesis testing pro- +cedure developed in Shi et al. (2019) to examine which SUVRs are significantly associated +with the MoCa scores. For each covariate of interest, we test the hypothesis that the corre- +sponding coefficient is greater than zero. We plot the logarithm of the p-values from the score +and Wald tests proposed in Shi et al. (2019) for the coefficients of the SUVRs at cortical ROIs. +Figure 1 shows that if using 0.05/218 as a cut off for the p-value, both the Wald and score test +identify the SUVRs at all cortical ROIs as significant predictors, which contradicts the fact +that the cognitive functions are controlled by a subset of brain ROIs (Leisman et al. 2016). +This unsatisfactory result likely attributes to the fact the Shi et al. (2019)’s method relies on +the assumption that the expectation of the exponential of the distance between outcome and +regression function is bounded (Condition (A3) in (Shi et al. 2019)) while neuroimage data + +4 +−1.0 +0.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Density +−0.5 +Density +Residuals +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +GG +G +G +G +G +G +G +G +G +G +G +G +G +G +G +0 +10 +20 +50 +60 +70 +−147.0 +−146.5 +−146.0 +−145.5 +−145.0 +score test +30 +40 +log of pvalues +covariate index +G +G +G +G +G +G +GG +G +G +G +G +G +G +G +G +G +G +GG +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +GG +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +G +GG +G +0 +10 +20 +50 +60 +70 +−120 +−100 +−80 +−60 +−40 +−20 +Wald test +30 +40 +log of pvalues +covariate index +FIG 1. Left: The density of the residuals from lasso regression. The lasso regression does not provide a satisfactory +fit for the data. Middle and Right: The logarithm of the p values from the Wald and score tests proposed by Shi +et al. (2019) for testing whether the SUVR from each cortical regions is significant predictor for the cognitive +score. The Wald and score tests suggest that the SUVRs at all the cortical regions have significant association +with the cognitive score. +are often subject to data acquisition and processing errors, which likely leads to violation +of this assumption. This motivating example demonstrates the necessity of developing novel +statistical inference procedure to test linear hypothesis in the high dimensional Poisson model +with noisy data. +The rest of the paper is organized as follows. Section 2 discusses related work. In Section +3, we describe our model assumptions and the overall estimation strategy. We further detail +the estimation with and without the null constraint, and the construction of the test statistics. +The fundamental theoretical developments are provided in Section 4, where we establish +convergence rates, the asymptotic normal results, and the properties of the test procedures. +We study the practical implementation and the numerical performances in Section 5, where +a detailed algorithm is provided, extensive simulations are carried out, and a ADNI data set +is analyzed. We conclude the paper in Section 6. The main mathematical proofs are provided +in an Appendix given in a Supplementary Document. +2. Related Works and Notations. +Nonlinear models with high dimensional noisy data +are in general hard problems to work with, partly because existing treatments usually lead +to non-convex optimization, which violates standard requirements in the high dimensional +data analysis literature. Thus, only linear models, which are the simplest in all noisy data +problems, have received relatively thorough investigation (Loh & Wainwright 2012, Belloni +& Rosenbaum 2016, Datta & Zou 2017, Belloni, Rosenbaum & Tsybakov 2017, Belloni, +Chernozhukov & Kaul 2017, Li et al. 2021). Expanding the research framework to the Pois- +son regression context is difficult because the link function in the Poisson model is nonlinear. +Subsequently, it is not easy to construct noise adjusted quantities such as a noise adjusted +Hessian matrix like in the linear case. In addition, the Hessian matrix involves heavy tailed +random variables due to the exponential link, even if all the covariates are sub-Gaussian in +their original scale. These difficulties require additional restrictions on the moments of the +covariate distribution as well as on the parameter searching space, which complicates all the +subsequent computation and analysis. Indeed, the only works we are aware of in the high +dimensional Poisson model with noisy data are Jiang & Ma (2021), Sørensen et al. (2015, +2018), Brown et al. (2019), while only the estimator in Jiang & Ma (2021) has been shown to +be consistent. However, because all these methods use lasso-type L1 penalty in the estima- +tion, the resulting estimators do not enjoy variable selection consistency and their asymptotic +distribution results are not established. + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +5 +There is extensive literature on the linear hypothesis testing under high dimensional noise +free setting. (Ning & Liu 2017) introduced a decorrelated score function to construct confi- +dence regions for low dimensional components in high dimensional models. Zhang & Cheng +(2017) used the desparsifying lasso estimator (Van de Geer et al. 2014) to propose a maximal- +type statistic allowing the number of parameters that are involved in the test to grow with the +sample size. Moreover, Shi et al. (2019) proposed a partial penalized likelihood ratio test, a +score test, and a Wald test for testing the linear hypothesis of the parameters in high dimen- +sional generalized linear models. +Notations. +We introduce some general notation that will be used throughout the text. For +a matrix M, let }M}max be the matrix maximum norm, }M}8 be the L8 norm and }M}p +be the Lp norm. Let Fpβq be the σ-field generated by Xi,βTWi,i “ 1,...,n. Further, let +Fx be the sigma-field generated by Xi,i “ 1,...,n. For a general vector a, let }a}8 be +the vector sup-norm, }a}p be the vector lp-norm. Let ej be the unit vector with 1 on its +jth entry. For a vector v “ pv1,...,vpqT, let supppvq be the set of indices with vi ‰ 0 and +}v}0 “ |supppvq|, where |U| stands for the cardinality of the set U. For a vector v P Rp and a +subset S Ď p1,...,pq, we use vS P RS to denote the vector obtained by restricting v on the +set S. +Following Fletcher & Watson (1980), for an arbitrary norm } ¨ }A and its dual normal +} ¨ }D, we define B}x}A as the set pv : }x}A “ vTx,}v}D ď 1q. Thus, for an arbitrary +vector x “ px1,...,xpqT, B}x}1 “ tv “ pv1,...,vpqT : vj “ signpxjq if xj ‰ 0, and |vj| ď +1 if xj “ 0u, and B}x}2 “ tv “ pv1,...,vpqT : vj “ xj{}x}2u. +3. Model, Estimation and Test Statistics. +3.1. Problem Formulation: High dimensional Poisson model with noisy data. +Let Xi be +a p-dimensional covariate, for example the image features, and let Yi be a count random +response variable, for example the MoCa score from the ADNI data. We model the rela- +tionship between Yi and Xi (i “ 1,...,n) through a Poisson model prpYi “ y | Xi “ xq “ +e´exppβT +t xqtexppβT +t xquy{y!. Here, βt is a p-dimensional sparse parameter vector. We allow +the number of nonzero entries in βt to grow with the sample size. We consider Poisson model +here because our response is a count, and Poisson model is arguably the most standard model +for count data. Indeed, Poisson model has been widely used to model the distribution of cog- +nitive scores (Katz et al. 2021, Fallah et al. 2011, Mitnitski et al. 2014). We use eβT +t x to +model the conditional mean of the Poisson model to ensure the positiveness of the mean, and +to allow possible skewness in the distribution (McCullagh & Nelder 2019). We assume βt +to be sparse because it often happens that only a few covariates have effect on the outcome. +For example, in the ADNI data, because the cognitive functions are controlled by a subset of +brain ROIs (Leisman et al. 2016), only a subset of brain features contributes to the cognitive +function. +Furthermore, we assume the covariate Xi is not precisely observed and instead, a contam- +inated version of Xi, denoted Wi, is observed, where Wi “ Xi ` Ui, and Ui is the noise +that is independent of both Xi and Yi. For example, in the ADNI data, Xi can be the true +image features, while Wi represent the observed image features which can deviate from the +truth due to imperfect data collection and processing procedure. Without loss of generality, +assume that EpXiq “ 0, which can always be achieved by centering the observed covariates +in practice. Furthermore, we assume Ui is a normally distributed random noise vector with +mean zero and known covariance matrix Ω. The normal assumption for Ui is the common +assumption at the state of the art in the Poisson measurement error literature and allows to +derive analytic form of the loss function, which is the only setting that we can directly ex- + +6 +amine the convexity of the loss function. The known Ω assumption is only for convenience +of presentation. In practice, it is often replaced by an estimated version based on multiple +observations, validation data or other standard instruments under both low and high dimen- +sional settings (Carroll et al. 2006, Loh & Wainwright 2012), and the corresponding analysis +is routine. Let pXi,Wi,Yi,Uiq be independent and identically distributed (iid) and assume +pWi,Yiq,i “ 1,...,n are the iid observations. In this work, we devise estimation procedures +for β and establishing theoretical properties of the estimator, we further aim at performing +inference, such as conduct hypothesis testing. Throughout, we allow the covariate dimension +to be much higher than the number of observations, i.e. p ąą n. We assume βt is in the fea- +sible set: tβ : }β}0 ď k, }β}2 ď b0u, which is practically sensible. A vector β in the feasible +set automatically satisfies }β}1 ď b0 +? +k. +3.2. General Estimation Strategy. +If the true covariates Xi can be observed and the di- +mension p is fixed, this is a standard regression model and we routinely estimate β by mini- +mizing the negative loglikelihood, which is proportional to +´n´1 +nÿ +i“1 +tYiXT +i β ´ exppβTXiqu. +Here we use exppβTXiq to model the mean of Yi because it preserves the positiveness of +the mean estimate, and it is a standard choice in the generalized linear model (McCullagh & +Nelder 2019). It is useful to note that for normal noise Ui, we have the relation +EtexppβT +t Wi ´ βT +t Ωβt{2q | Xiu “ exppβT +t Xiq, +(2) +EtexppβT +t Wi ´ βT +t Ωβt{2qpWi ´ Ωβtq | Xiu “ exppβT +t XiqXi, +(3) +ErexppβT +t Wi ´ βT +t Ωβt{2qtpWi ´ Ωβtqb2 ´ Ωu | Xis “ exppβT +t XiqXb2 +i . +(4) +Due to the conditional independence of Wi and Yi given Xi, (2) leads to +EtYiWT +i βt ´ exppβT +t Wi ´ βT +t Ωβt{2q | Xi,Yiu “ YiXT +i βt ´ exppβT +t Xiq. +Consequently, it is a reasonable practice to estimate β by minimizing the loss function +Lpβq “ ´n´1 +nÿ +i“1 +tYiWT +i β ´ exppβTWi ´ βTΩβ{2qu, +(5) +which has the same mean as the negative log-likelihood function when Xi is accurately ob- +served. When n ą p, the estimator for β can be obtained by minimizing Lpβq using the +standard gradient descent method. However, when n ă p, without addition regularization, +optimizing (5) is an ill-posed mathematical problem because it does not have a unique solu- +tion. To take into account the ultra-high dimension nature of the model, using the fact that β +is sparse, we propose to estimate β through solving the following constrained minimization +problem +min +}β}1ďR1,}β}2ďR2 +tLpβq ` ρλpβqu +(6) +at suitable R1,R2, where ρλpβq is a suitable penalty function. For convenience, define the +set tβ : }β}1 ď R1,}β}2 ď R2u as the feasible set (Fletcher & Watson 1980). Here R1,R2 +can be any constants that are greater than the true }β}1 and }β}2, respectively. The condition +}β}1 ď R1 is imposed to guarantee that the objective function satisfies the restricted eigen- +value condition discussed in Loh & Wainwright (2012) and therefore the objective function +is convex in the feasible set, while the condition }β}2 ď R2 is imposed to avoid the explo- +sion of the mean function exppβTWi ´ βTΩβ{2q. In practice, we often set R1,R2 to be + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +7 +a constant times the L1, L2 norms of the initial estimators of β. Here, with a slight abuse +of notation, we use the same symbol ρλ to denote both multivariate and univariate penalty +functions and let ρλpβq “ ř}β}0 +j“1 ρλpβjq, where βj is the jth element of β and }β}0 is the +number of nonzero elements in β. +3.3. Estimation under Hypotheses. +Consider testing the hypothesis that CβtM “ t`hn +for some hn P Rr, where C is a r ˆ m matrix with r ď m, βtM is a m-dimensional sub- +vector of β with index set M. The null hypothesis holds when hn “ 0, while the alternative +hypothesis holds when hn ‰ 0. For example if t “ 0, hn “ 1, C “ p1,0q, M contains the +index of the first element in βt, then testing CβtM “ t ` hn is testing the null hypothesis +that βt1 “ 0 versus the alterative that βt1 “ 1. Similarly, we can test βt1 ´ βt2 “ 0 versus +βt1 ´ βt2 ‰ 0 by choosing C “ p1,´1q, t “ 0, hn “ 0 or nonzero, and M “ t1,2u. In +summary, by varying C, t, hn, and M, we can generate different linear hypotheses to test. +Without loss of generality, we assume βM contains the first m elements of β. Further, let +βc +M be the vector containing elements that are not in M, i.e. the last p ´ m components of +β. Let S Ď Mc be the index set of the nonzero elements of βtMc. We assume βtMc to be k +sparse, i.e. |S| “ k. Note that k is allowed to diverge with n. Without loss of generality, we +assume the first k elements in βtMc are none zero. +Suppose we are interested in testing whether CβtM “ t or not. Under the null hypothesis +that H0 : CβtM “ t, we modify the general estimation strategy slightly and consider the +estimator resulting from the equality and inequality constrained minimization: +pβ “ argmin}β}1ďR1,}β}2ďR2 tLpβq ` ρλpβMcqu, s.t. CβM “ t +(7) +for suitable R1,R2. Without assuming the null hypothesis, we consider a similar estimator +resulting from the inequality constrained minimization: +pβa “ argmin}β}1ďR1,}β}2ďR2 tLpβq ` ρλpβMcqu. +(8) +Note that here, both (7) and (8) are slightly different from the general strategy in (6), in that +we do not place the penalty ρλ on the parameters in M, which are to be tested for the linear +relation CβtM “ t. This special treatment is to avoid the situation that the penalty forces +some components in βM to be zero, and therefore the null hypothesis CβtM “ t is affected +not only by the data but also by our penalization. +3.4. Test statistics. +We define +Qpβq ” EtexppβTXqXXTu, +(9) +define the covariance of the residuals +Σpβq ” ErtYiWi ´ exppβTWi ´ βTΩβ{2qpWi ´ Ωβqub2s, +and define +ΨpΣ,Q,βq ” pCrImˆm,0mˆksQ´1 +MYS,MYSpβqΣMYS,MYSpβqQ´1 +MYS,MYSpβqrImˆm,0mˆksTCTq. +Furthermore, let pΣpβq and pQpβq be a sample estimator of Σpβq and Qpβq, respectively. To +test CβM “ t, we introduce two statistics, the Wald statistic +TW “ npC pβaM ´ tqTΨppΣ, pQ, pβaq´1pC pβaM ´ tq, +(10) +and the score statistic +TS “ n +# +BLp pβq +BβT ++ +MYS +pCrImˆm,0mˆks pQ´1 +MYS,MYSp pβqqT + +8 +ˆΨ´1ppΣ, pQ, pβqCrImˆm,0mˆks pQ´1 +MYS,MYSp pβq +# +BLp pβq +Bβ ++ +MYS +. +(11) +As we will show later in Section 4.4 that TW and TS are both asymptotically chi-square +distributed with r degrees of freedom under the null hypothesis. Therefore, to control the +false discovery rate at level α, we reject the null hypothesis if TW ą χ2 +1´αprq when we +perform Wald test, or if TS ą χ2 +1´αprq when we perform score test. Here χ2 +1´αprq is the +1 ´ α quantile of the chi-square distribution. +4. Theoretical Properties. +Define +qβM ” βtM ´ CTpCCTq´1hn, +and let qβ “ p qβT +tM,βT +tMcqT. Thus, the last p ´ m components of qβ, i.e. qβMc, and the last +p´m components of β0, i.e. β0Mc, are identical. However, the first m components of qβ and +β are different, in that C qβM “ t under both null and alternative, while CβtM “ t under the +null alone. Under some conditions, we first show that the inequality and equality constrained +estimator pβ is a consistent estimator of qβ regardless the null or the alternative holds, and +when }hn}2 vanishes, pβ is also consistent as an estimator of the true parameter βt. Further- +more, we show that pβa is a consistent estimator of βt regardless the null or the alternative +holds. We then establish the asymptotic linear form of the estimators of a subvector pβ and +a subvector of pβa, which are formed by components of βt that are either to be tested or +nonzero. Finally, using the asymptotic linear forms, we construct test statistics and prove the +convergence properties of these test statistics under both null and alternative. +4.1. Conditions. +Before we proceed with the specific results, we first list a set of assump- +tions on the univariate penalty function ρλ which are similar to those in Loh & Wainwright +(2015) and Loh & Wainwright (2017) . +(A1) The function ρλptq satisfies ρλp0q “ 0 and is symmetric around zero. +(A2) On the nonnegative real line t ě 0, the function ρλptq is nondecreasing. Furthermore, +ρλptq is subadditive, i.e. ρλpt1 ` t2q ď ρλpt1q ` ρλpt2q for all t1,t2 ě 0. +(A3) For t ą 0, the function ρλptq{t is non-increasing in t. +(A4) The function ρλptq is differentiable at all t ‰ 0 and sub-differentiable at t “ 0, with +limtÑ0` ρ1 +λptq “ λ, where ρ1ptq denotes the derivative of ρptq. Together with the symmet- +ric Condition in (A1), this leads to limtÑ0´ ρ1 +λptq “ ´λ. +(A5) There exists µ ą 0 so that ρλptq ` µt2{2 is convex. +(A6) There exists a γ P p0,`8q such that ρ1 +λptq “ 0 for all t ě γλ. +Conditions (A1)–(A3) are some general requirements as discussed in Zhang et al. (2012). +Condition (A4) restricts the class of penalties by excluding regularizers that are not differ- +entiable at 0, for example, the lasso penalty is excluded. Condition (A5) is known as weak +convexity (Vial 1982, Chen & Gu 2014) and is a type of curvature constraint that controls the +level of nonconvexity of ρλ. Condition (A6) is imposed to allow penalty to be zero if the esti- +mator is γλ away from zero, which removes the estimation bias for the nonzero parameters. +We say ρλ is µ-amenable if Conditions (A1)–(A5) hold, and we name ρλ pµ,γq-amenable +if Conditions (A1)–(A6) hold. The pµ,γq-amenable penalty includes the smoothly clipped +absolute deviation (SCAD) and the minimax concave penalty (Loh & Wainwright 2017). +We need some additional regularity conditions to support the theoretical development. +These conditions impose upper and lower bounds on various quantities to ensure that the up- +per bounds are finite and the lower bounds are positive. They also restrict the relation between + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +9 +the sample size and parameter number so that logppq{n Ñ 0 in a slow rate of 1{tlogpnqu2. To +save space, we only provide a discussion of these conditions here, while provide the details +in the supplementary material. Specifically, Condition (C1) (a) is a standard assumption used +in noisy data problem such as that used in Sentürk & Müller (2005) and is usually satisfied in +practice. Condition (C1) (b) guarantees the boundedness and the invertibility of the Hessian +matrix (4), i.e. the second derivative of the noise free log likelihood. Conditions (C2) and (C3) +bound the total variability of both the response Y and the noise U marginally and condition- +ally on the covariates X. Similar requirement is also assumed in Loh & Wainwright (2012). +Condition (C4) shows that the dimension of the covariate can grow exponentially faster than +the sample size. Finally, Jiang & Ma (2021) have discussed the Conditions (C5)–(C7) and +provided examples showing that the conditions are usually satisfied in practice. +4.2. Consistency. +We first show that the equality and inequality constrained estimator pβ +is a consistent estimator of qβ in Theorems 1 and 2, which is the same as the true parameter +βt, except that the first m components are adjusted to ensure that H0 holds for qβ. +Theorem 1. +Define +α1 ” +min +}β}1ďR1,}β}2ďR2 +αminrEtexppβTXiqXiXT +i us{2. +Assume }C´1 +r Cm´r}2 “ Op1q, ρλ satisfies Conditions (A1) – (A6) and Conditions (C1) – +(C6) in the supplementary material hold. Assume α1 ą 3{4µ, and β is in the feasible set. Let +λ satisfy +4max +! +}BLp qβq{Bβ}8,α1plogppq{nq1{4) +ď λ ď α1 +6R1 +and n ě logppqmaxp16R4 +1τ 4 +1 {α4 +1,64R4 +1τ 2 +1 {α2 +1q. Write t1 ” ?r}C´1 +r Cm´r}2 ` ?m ´ r and +t ” p6λ +? +k ` 2λt1qp4α1 ´ 3µq´1. Then the local minimum of (7) satisfies the error bounds +} pβ ´ qβ}2 ď t. +and +} pβ ´ qβ}1 ď p4 +? +k ` t1qt. +Following the similar argument, we also show that the inequality constrained estimator pβa +is a consistent estimator of the true parameter β. +Theorem 2. +Let +α1 “ +min +}β}1ďR1,}β}2ďR2 +αminrEtexppβTXiqXiXT +i us{2 +and let ρλ satisfy Conditions (A1) – (A6) and Conditions (C1) – (C6) in the supplementary +material hold. Assume α1 ą 3{4µ, and β is in the feasible set. Let λ satisfy +4max +! +}BLpβtq{Bβ}8,α1plogppq{nq1{4) +ď λ ď α1 +6R1 +and n ě logppqmaxp16R4 +1τ 4 +1 {α4 +1,64R4 +1τ 2 +1 {α2 +1q. Then the local minimum of (8) satisfies the +error bounds +} pβa ´ βt}2 ď 6λ +? +k ` 2λ?m +4α1 ´ 3µ +. +and +} pβa ´ βt}1 ď p4 +? +k ` ?mq6λ +? +k ` 2λ?m +4α1 ´ 3µ +. + +10 +Theorems 1 and 2 suggest that when logppq{n Ñ 0, and when λ is suitably chosen, for +example, λ is at least no smaller than Ortlogppq{nu1{4s, both pβ and pβa converge to their +corresponding true values in terms of both l1 and l2 norms, as long as k and m grow slower +than tn{logppqu1{2. These theoretical results suggest that the dimension of βtM, i.e., the +number of parameters involved in the tests, and the number of nonzero entries in βt can grow +at a slower rate of tn{logppqu1{2 under noisy Poisson model. These results also assist us to +find reasonable ranges for λ in practice to obtain consistent estimators. +4.3. Asymptotic linear forms. +We denote rβ as a stationary point of (7), which satisfies +the first order condition that +tBLp rβq{BβT ` Bρλp rβMcq{BβT +McAupβ ´ rβq ě 0, +(12) +for all β P Rp in the feasible set and satisfies CβM “ t. Here A “ p0p´m,m,Ip´m,p´mq is a +matrix that satisfies }A}8 “ }A}1 “ 1. Likewise, we denote rβa as a stationary point of (8), +which satisfies the first order condition that +tBLp rβaq{BβT ` Bρλp rβaMcq{BβT +aMcAupβa ´ rβaq ě 0, +(13) +for all βa P Rp in the feasible set. +To show the asymptotic normality of pβ and pβa, our first step is to establish that the local +minimizers rβ and rβa achieve variable selection consistency. To do this, we follow the prime- +dual construction introduced in Wainwright (2009). We first show that both +min +}β}1ďR1,}β}2ďR2,βPRMYS tLpβq ` ρλpβMcqu, such that CβM “ t +(14) +and +min +}β}1ďR1,}β}2ďR2,βPRMYS tLpβq ` ρλpβMcqu +(15) +have unique local minimizer in the interior of the feasible set. Then we show that all stationary +points of (7) and (8) must have support in M Y S. Since the local minimizers of (7) and (8) +are automatically stationary points of (7) and (8) respectively, the local minimizers of (7) +and (8) must also have support in M Y S. Therefore, the local minimizers of (7) and (8) +are actually the local minimizers of (14) and (15) respectively, so are also unique. In other +words, pβ and pβa are respectively the unique solution of (14) and (15) hence achieve the +variable selection consistency. The details of the above analysis are presented in Theorem +A.1 and Theorem A.2 in the Appendix A in the supplementary material. +In our second step to establish the asymptotic distribution properties of pβ and pβa, we +define +pQpβq “ B2Lpβq +BβBβT , +and define +A2 “ rImˆm,0mˆksTCTpCrImˆm,0mˆkst pQMYS,MYSpβ˚qu´1 +ˆrImˆm,0mˆksTCTq´1CrImˆm,0mˆks, +where β˚ is the point in between pβ and βt and +A˚ +2 “ rImˆm,0mˆksTCTpCrImˆm,0mˆkstQMYS,MYSpβqu´1 +ˆrImˆm,0mˆksTCTq´1CrImˆm,0mˆks, + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +11 +where Qpβq “ EtexppβTXqXXTu is defined in (9). Based on the variable selection consis- +tency established in the first step, we derive the asymptotic linear form of pβMYS and pβaMYS +under null and alternative hypothesis in Theorems 3 and 4, respectively. +Theorem 3. +Assume ρλ satisfies Conditions (A1) – (A6) and Conditions (C1) – +(C7) in the supplementary material hold, λ “ Oprtlogppq{nu1{4s, }C´1 +r Cm´r}2 “ Op1q, +and λ ď α1{p8R1q. Further we assume the boundedness }tQ´1 +pMYSq,MYSpβtqu}8 ď c8, +and }tQpMYSq,MYSpβtqu´1A2Q´1 +pMYSq,MYSpβtq}8 ď c8. In addition assume }hn}2 “ +Ot +a +maxpm ` k ´ r,rq{nu, minp|βj|q ě λpγ ` 5c8q for j P S and n ě c8pm ` kq4logppq. +Then we have +pβMYS ´ βtMYS +“ ´ptQMYS,ăMYSpβtqu´1 ´ tQMYS,MYSpβtqu´1A˚ +2 +ˆtQMYS,MYSpβtqu´1q +"BLpβtq +Bβ +* +MYS +t1 ` opp1qu +`tQMYS,MYSpβtqu´1A˚ +2rtpCCTq´1C,0rˆkuThnst1 ` opp1qu +and pβpMYSqc “ 0. +Theorem 4. +Assume ρλ satisfies Conditions (A1) – (A6) and Conditions (C1) – (C7) +in the supplementary material hold, λ “ Oprtlogppq{nu1{4s, and λ ď α1{p8R1q. Further +we assume }tQpMYSq,MYSpβtqu´1}8 ď c8, minp|βj|q ě λpγ ` 5c8q for j P S and n ě +c8pm ` kq4logppq. Then we have +pβaMYS ´ βtMYS “ ´tQMYS,MYSpβtqu´1 +"BLpβtq +Bβ +* +MYS +t1 ` opp1qu +and pβpMYSqc “ 0. +Theorems 3 and 4 suggest that the asymptotic linear forms of pβMYS and pβaMYS are the +usual product of the inverse of Hessian matrix and the score function. Furthermore, only +the first pm ` kq ˆ pm ` kq block in the Hessian matrix and the first m ` k elements in the +score function contribute to the asymptotic distribution. Therefore, when m`k grows slower +than tn{logppqu1{4 and }hn} Ñ 0, it is easy to see that the asymptotic linear forms converge +in distribution to Gaussian random vectors. It is worth mentioning that the minimal signal +condition minp|βj|q ě λpγ ` 5c8q for j P S is a standard requirement for the optimization +using nonconvex penalty such as SCAD (Fan & Li 2001). This condition is also very weak +because λ Ñ 0, which allows the minimal signal vanishing to zero. +4.4. Asymptotic distribution of the test statistics. +To study the asymptotic behavior of TS +and TW , we first investigate the distribution of their asymptotic form T0 defined by +T0 ” pωn ` ?nhnqTΨ´1pΣ,Q,βtqpωn ` ?nhnq, +where +ωn “ ´?nCrImˆm,0mˆksQ´1 +MYS,MYSpβtq +"BLpβtq +Bβ +* +MYS +. +As shown in Lemma 1, T0 is asymptotically noncentral chi-square distributed with the non- +central parameter approaches nhT +nΨ´1pΣ,Q,βtqhn. + +12 +Lemma 1. +Assume ρλ satisfies Conditions (A1) – (A6) and Conditions (C1) and (D1) +in the supplementary material hold and n ě c8pm ` kq4logppq, then +lim +nÑ8sup +C +|PrpT0 ď xq ´ Prtχ2pr,nhT +nΨ´1pΣ,Q,βtqhnq ď xu| “ 0, +where χ2pr,γq is a non-central chi-square random variable, with non-centrality parameter γ. +Here Condition (D1) provides upper bound of the third moment of each summand in ω +(note that BLpβtq{Bβ is the summation of the derivatives of the negative log-likelihood from +n samples), which is a necessary condition to establish convergence in distribution. See The- +orem 3.1 in Shi et al. (2019) for example. To establish the asymptotic distribution of TW and +TS, in Theorems 5 and 6 respectively, we show that TW and TS are close to T0, hence has +the same testing property asymptotically when r is finite. +Theorem 5. +Assume the conditions in Theorem 4 and Conditions (D1) and (D2) in the +Section B.4.2 in the supplementary material hold, we have TW ´ T0 “ opprq. Therefore, +lim +nÑ8sup +C +|PrpTW ď xq ´ Prtχ2pr,nhT +nΨ´1pΣ,Q,βtqhnq ď xu| “ 0, +where χ2pr,γq is a non-central chi-square random variable, with non-centrality parameter γ. +Theorem 6. +Assume the conditions in Theorem 3, Conditions (D1) and (D2) in the +Section B.4.2 in the supplementary material hold, we have TS ´ T0 “ opprq. Therefore, +lim +nÑ8sup +C +|PrpTS ď xq ´ Prtχ2pr,nhT +nΨ´1pΣ,Q,βtqhnq ď xu| “ 0, +where χ2pr,γq is a non-central chi-square random variable, with non-centrality parameter γ. +Here Condition (D2) in the Section B.4.2 is a regularity condition ensures ΨpΣ,Q,βtq +to be positive definite. Theorems 5 and 6 show that the two test statistics TW and TS indeed +have the same χ2pr,γq distribution as T0 in large samples, hence can be used to perform +the standard chi-square test. A curious question is whether or not a likelihood ratio type of +test can also be constructed. We feel it is hard in this context because it is almost impossible +to obtain a likelihood function in the functional measurement error context. Much work is +needed to overcome this obstacles. +5. Numerical Implementation. +5.1. Computational algorithms. +We compute the estimators pβ and pβa using the popular +ADMM. In what follows, we only detail the algorithm to estimate pβ. The estimator pβa can +be computed in a similar way. For a given λ, we consider +pβ “ argmin}β}1ďR1,}β}2ďR2 tLpβq ` ρλpβMcqu, s.t. CβM “ t +for constants R1,R2. Similar to Shi et al. (2019), this optimization problem is equivalent to +p pβ, pθq “ argmin}β}1ďR1,}β}2ďR2 tLpβq ` ρλpβMcqu, s.t. CβM “ t,βMc “ θ. +By the augmented Lagrangian method, the estimators can be obtained by minimizing +Lpβ,θ,vq “ Lpβq ` ρλpβMcq ` v +T ˆ +CβM ´ t +βMc ´ θ +˙ +` ρ +2 +���� +CβM ´ t +βMc ´ θ +���� +2 +2 +, + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +13 +Algorithm 1 ADMM Algorithm for estimating pβ. +For t “ 0,1,...,tmax, perform: +Step 1. Use the Newton-Raphson algorithm to solve (17) to obtain rβpt`1q. +rβpt`1q “ argminβ +$ +& +%Lpβq ` vptqT +˜ +CβM ´ t +βMc ´ θptq +¸ +` ρ +2 +����� +CβM ´ t +βMc ´ θptq +����� +2 +2 +, +. +-. +(17) +Step 2. Project rβpt`1q to a L1 ball with radius R1 to obtain ˘βpt`1q by the simplex projection method +(Duchi et al. 2008). If || ˘βpt`1q||2 ą R2, we shrink it to get βpt`1q “ ˘βpt`1qR2{|| ˘βpt`1q||2. Otherwise, +βpt`1q “ ˘βpt`1q. +Step 3. Obtain θpt`1q by solving (17), where the penalty term we use is SCAD with a “ 3.7. +θpt`1q “ argminθ +$ +& +%ρλpθq ` ρ +2 +���βpt`1q +Mc +´ θ +��� +2 +2 ` vptqT +¨ +˝Cβpt`1q +M +´ t +βpt`1q +Mc +´ θ +˛ +‚ +, +. +-. +(18) +Step 4. Update the dual variables v by +vpt`1q “ vptq ` ρ +¨ +˝ +Cβpt`1q +M +´ t +βpt`1q +Mc +´ θpt`1q +˛ +‚. +Step 5. If stopping rule }βpt`1q ´ βptq}2 ď δtol or }θpt`1q ´ θptq}2 ď δtol is satisfied, where δtol denotes +the tolerance of error, then terminate the algorithm. +End of the main loop. +with }β}1 ď R1, }β}2 ď R2, where the dual variables v are Lagrange multipliers and ρ ą 0 +is a given penalty parameter. We compute the estimators of pβ,θ,vq through iterations. Let +the sup-script (t) indicate the t-th iteration, we describe the main steps of ADMM methods +in Algorithm 1. +In the implementation, the initial value βp0q can be computed by a penalized Poisson +regression following Jiang & Ma (2021). For the radii R1 and R2, we consider R1 “ +? +2R2 +and R2 “ 1.5}β}p0q +2 . In the implementation, if the algorithm converges to the boundary, we +can increase the corresponding norm R1 or R2 slightly. In contrast, if multiple minimum +problems are encountered, we can decrease R1 and when the estimation procedure leads to a +very large exppβTXq, we can decrease R2, gradually. The tuning parameter λ is selected by +minimizing +BICpλq “ nLp pβq ` cn} pβ}0 +(16) +with respect to λ, where cn is a positive number that may depend on n. In our analysis, we +follow Shi et al. (2019) to adopt cn “ maxtlogn,logplogpnqqlogpu. For simplicity, we set +ρ “ 1. +5.2. Simulation Experiments. +We generate the outcome Yi from the Poisson model +PrpYi “ y | Xiq “ expt´exppβTXiquexppyβTXiq{y!, +where the covariates Xi “ pXi,1,...,Xi,pqT are generated from two distributions: (I) the +multivariate normal distribution with mean zero and covariance matrix Σ. (II) the uni- +form distribution in the interval p´ +? +6{2, +? +6{2q. To generate correlated uniform distribu- +tion, we first draw covariates independently from Up´ +? +6{2, +? +6{2q, and then transform +these covariates by multiplying the Choleski factorization of covariance Σ. We consider two +forms of the covariance matrix: uncorrelated structure Σ “ 0.5Ip and correlated with auto- +regressive AR(1) structure Σ “ p0.5|i´j|`1qpˆp for i,j “ 1,...,p. Furthermore, the noise Ui + +14 +is drawn from the multivariate normal distribution with mean zero and covariance matrix +Ω “ 0.1Σ. The true coefficient β “ pβ1,...,βpqT “ p0.75,´0.75 ` h2,h3,0,...,0,hpqT. +Here hj,j “ 2,3,p are assigned various values to check the empirical powers of the tests. +We set hj “ 0 when j ‰ 2,3 or p. For simplicity, the initial βp0q is set to be a p-dimensional +zero vector. We select parameter λ as described in Section 5.1. The candidate list for λ is +te´2.5,e´2.245,...,e0.5u of length 41. We consider sample size n “ 300,500 and covariate +dimension p “ 50,350,600. The tolerance of error δtol “ 10´4. We repeat each setting 500 +times, and report the size and power of the proposed tests under different hypotheses. We +perform the tests at type I error α “ 0.05 in the following scenarios. +5.2.1. Univariate parameter testing. +We first consider the following three hypotheses on +a single element in β. +H0,1 : β2 “ ´0.75, v.s. Ha,1 : β2 ‰ ´0.75. +H0,2 : β3 “ 0, v.s. Ha,2 : β3 ‰ 0. +H0,3 : βp “ 0, v.s. Ha,3 : βp ‰ 0. +To test a hypothesis set regarding βj, we simulate data with hj “ 0,0.1,0.2,0.4, while set +hk “ 0 for k ‰ j. For example, to test H0,1 and Ha,1, we simulate data with h3 “ 0, hp “ 0, +and h2 “ 0,0.1,0.2,0.4. When h2 “ 0, the null hypothesis H0,1 holds, we study the type I +error of the test. On the other hand, when h2 “ 0.1 to 0.4, the alternative hypothesis is true, +which allows us to examine the power of the test. Tables 1 and 2 summarize the empirical +type I error and powers of the Wald and score tests. It is clear that the empirical type I errors +are controlled at the nominal level 0.05 in all scenarios, indicating that the proposed tests are +consistent. The powers of the Wald and score tests increase gradually when the magnitude +of |hj|’s increases, and have satisfactory powers in general. The Wald and score tests yield +similar performances in all scenarios. This finding is in accordance with theoretical analysis. +5.2.2. Linear hypothesis testing. +We also consider the hypotheses that contain the linear +combinations of two coefficient parameters: +H0,4 : β1 ` β2 “ 0, v.s. Ha,4 : β1 ` β2 ‰ 0. +H0,5 : β3 ` β4 “ 0, v.s. Ha,5 : β3 ` β4 ‰ 0. +H0,6 : β1 ` βp “ 0.75, v.s. Ha,6 : β1 ` βp ‰ 0.75. +H0,7 : β2 ` β3 “ ´0.75, v.s. Ha,7 : β2 ` β3 ‰ ´0.75. +For the first three sets of hypotheses, we still set hj “ 0,0.1,0.2,0.4 if the hypothesis involves +βj for j “ 2,3,p, and set hk “ 0 if the corresponding βk is not involved in the hypotheses. +For the last hypothesis H0,7, we set h2 “ 0, hp “ 0 and vary h3 from 0 to 0.4. Tables 3 and 4 +show that the Wald and score tests control the type I error at nominal level, and their powers +improve when hj increases. +5.2.3. Performance regarding m. +We further investigate how the testing performance +changes as m changes. We consider three sets of hypotheses: +H0,8 : +4ÿ +j“1 +βj “ 0, v.s. Ha,8 : +4ÿ +j“1 +βj ‰ 0. + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +15 +TABLE 1 +The empirical sizes and powers of Wald and score tests for univariate parameter testing with n “ 300. +X „ Normal +X „ Uniform +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +TW +TS +TW +TS +TW +TS +TW +TS +p “ 50 +β2 +H0,1 : β2 “ ´0.75, v.s. Ha,1 : β2 ‰ ´0.75 +-0.75 +0.068 +0.054 +0.056 +0.050 +0.054 +0.044 +0.066 +0.062 +-0.65 +0.352 +0.288 +0.222 +0.176 +0.292 +0.276 +0.232 +0.208 +-0.55 +0.826 +0.778 +0.680 +0.592 +0.736 +0.726 +0.632 +0.598 +-0.35 +1.000 +0.996 +0.996 +0.976 +1.000 +1.000 +0.990 +0.972 +β3 +H0,2 : β3 “ 0, v.s. Ha,2 : β3 ‰ 0 +0.0 +0.056 +0.046 +0.058 +0.038 +0.056 +0.046 +0.068 +0.060 +0.1 +0.302 +0.272 +0.204 +0.172 +0.250 +0.240 +0.214 +0.182 +0.2 +0.752 +0.724 +0.554 +0.524 +0.692 +0.682 +0.530 +0.508 +0.4 +0.996 +0.996 +0.984 +0.960 +1.000 +1.000 +0.976 +0.942 +βp +H0,3 : βp “ 0, v.s. Ha,3 : βp ‰ 0 +0.0 +0.060 +0.044 +0.056 +0.042 +0.062 +0.054 +0.052 +0.050 +0.1 +0.246 +0.222 +0.234 +0.210 +0.276 +0.258 +0.240 +0.226 +0.2 +0.708 +0.672 +0.666 +0.634 +0.714 +0.698 +0.708 +0.682 +0.4 +0.998 +0.998 +0.998 +0.996 +0.998 +0.998 +0.998 +0.994 +p “ 350 +β2 +H0,1 : β2 “ ´0.75, v.s. Ha,1 : β2 ‰ ´0.75 +-0.75 +0.050 +0.036 +0.054 +0.034 +0.064 +0.068 +0.066 +0.056 +-0.65 +0.312 +0.322 +0.260 +0.242 +0.268 +0.266 +0.230 +0.202 +-0.55 +0.750 +0.766 +0.650 +0.644 +0.752 +0.750 +0.612 +0.598 +-0.35 +0.998 +0.998 +0.980 +0.878 +0.998 +0.998 +0.978 +0.892 +β3 +H0,2 : β3 “ 0, v.s. Ha,2 : β3 ‰ 0 +0.0 +0.064 +0.048 +0.066 +0.066 +0.066 +0.058 +0.068 +0.054 +0.1 +0.328 +0.330 +0.224 +0.200 +0.270 +0.262 +0.198 +0.164 +0.2 +0.770 +0.770 +0.590 +0.546 +0.708 +0.706 +0.568 +0.504 +0.4 +1.000 +1.000 +0.950 +0.846 +1.000 +1.000 +0.942 +0.830 +βp +H0,3 : βp “ 0, v.s. Ha,3 : βp ‰ 0 +0.0 +0.072 +0.066 +0.050 +0.046 +0.058 +0.050 +0.066 +0.056 +0.1 +0.346 +0.342 +0.208 +0.198 +0.250 +0.250 +0.220 +0.206 +0.2 +0.736 +0.742 +0.662 +0.646 +0.782 +0.768 +0.654 +0.638 +0.4 +1.000 +1.000 +0.996 +0.994 +1.000 +1.000 +0.994 +0.992 +H0,9 : +8ÿ +j“1 +βj “ 0, v.s. Ha,9 : +8ÿ +j“1 +βj ‰ 0. +H0,10 : +12 +ÿ +j“1 +βj “ 0, v.s. Ha,10 : +12 +ÿ +j“1 +βj ‰ 0, +corresponding to m “ 4,8 and 12. We set h2 “ 0, hp “ 0, and h3 “ 0,0.2,0.4,0.8. The +empirical sizes and powers are displayed in Table 5. These results suggest that under different +m, the empirical sizes remain close to the nominal significance level for both the Wald and +score tests. On the other hand, the empirical power decreases in general when m increases. +For instance, as shown in Table 5, when X follows the multivariate normal distribution with +mean zero and covariance Σ “ 0.5Ip, p “ 350 and h3 “ 0.8, the powers of the Wald test +are 1.000, 0.950 and 0.854 for m “ 4,8 and 12, respectively. This is intuitively sensible, +and suggests that larger sample size is needed to reach a desired power when the hypothesis +concerns more parameters. +5.2.4. Comparison with naive test. +We further compare the performances of our pro- +posed tests with the naive Wald and score tests developed under the noise free framework. + +16 +TABLE 2 +The empirical sizes and powers of Wald and score tests for univariate parameter testing with n “ 500. +X „ Normal +X „ Uniform +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +TW +TS +TW +TS +TW +TS +TW +TS +p “ 50 +β2 +H0,1 : β2 “ ´0.75, v.s. Ha,1 : β2 ‰ ´0.75 +-0.75 +0.066 +0.054 +0.044 +0.040 +0.064 +0.060 +0.060 +0.056 +-0.65 +0.488 +0.450 +0.346 +0.316 +0.442 +0.422 +0.324 +0.304 +-0.55 +0.954 +0.950 +0.864 +0.838 +0.910 +0.906 +0.800 +0.792 +-0.35 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +β3 +H0,2 : β3 “ 0, v.s. Ha,2 : β3 ‰ 0 +0.0 +0.048 +0.046 +0.066 +0.058 +0.052 +0.050 +0.056 +0.056 +0.1 +0.402 +0.382 +0.324 +0.300 +0.390 +0.386 +0.302 +0.296 +0.2 +0.890 +0.888 +0.780 +0.770 +0.892 +0.884 +0.778 +0.766 +0.4 +1.000 +1.000 +1.000 +0.998 +1.000 +1.000 +1.000 +1.000 +βp +H0,3 : βp “ 0, v.s. Ha,3 : βp ‰ 0 +0.0 +0.066 +0.060 +0.064 +0.058 +0.050 +0.048 +0.050 +0.048 +0.1 +0.400 +0.368 +0.350 +0.338 +0.390 +0.380 +0.336 +0.320 +0.2 +0.922 +0.914 +0.896 +0.878 +0.892 +0.890 +0.884 +0.878 +0.4 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +p “ 600 +β2 +H0,1 : β2 “ ´0.75, v.s. Ha,1 : β2 ‰ ´0.75 +-0.75 +0.052 +0.056 +0.046 +0.046 +0.062 +0.056 +0.040 +0.038 +-0.65 +0.458 +0.478 +0.328 +0.330 +0.390 +0.392 +0.396 +0.402 +-0.55 +0.920 +0.930 +0.864 +0.868 +0.926 +0.928 +0.902 +0.902 +-0.35 +0.842 +0.918 +0.988 +0.990 +1.000 +1.000 +1.000 +1.000 +β3 +H0,2 : β3 “ 0, v.s. Ha,2 : β3 ‰ 0 +0.0 +0.076 +0.066 +0.062 +0.058 +0.070 +0.066 +0.066 +0.056 +0.1 +0.454 +0.452 +0.344 +0.342 +0.392 +0.392 +0.454 +0.454 +0.2 +0.904 +0.904 +0.832 +0.822 +0.902 +0.894 +0.870 +0.862 +0.4 +0.974 +0.974 +0.998 +0.980 +1.000 +1.000 +0.998 +0.894 +βp +H0,3 : βp “ 0, v.s. Ha,3 : βp ‰ 0 +0.0 +0.048 +0.046 +0.050 +0.046 +0.060 +0.058 +0.058 +0.054 +0.1 +0.444 +0.444 +0.404 +0.390 +0.414 +0.416 +0.450 +0.448 +0.2 +0.930 +0.928 +0.870 +0.860 +0.912 +0.912 +0.876 +0.874 +0.4 +0.980 +0.980 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +We consider the covariates Xi “ pXi,1,...,Xi,pqT generated from the multivariate normal +distribution with mean zero and covariance matrix 0.7Ip. The noise Ui follows the multivari- +ate normal distribution with mean zero and covariance matrix 0.3Ip. Other settings remain +unchanged. We consider the hypotheses on a single element in β: H0,2, and the linear com- +binations of two coefficient parameters: H0,5 and H0,7 as described previously. We report +the empirical sizes and powers of the Wald and score tests with/without noises for p “ 50 in +Table 6. It is clear that while the proposed tests achieve Type I errors reasonably close to the +nominal level under different null hypotheses, the naive tests lead to precarious performance. +For instance, the Type I errors of Wald and Score tests for H0,5 are as large as 0.474 and +0.554, respectively. These Type I errors are far beyond the significance level. Because they +cannot control the significance level, we do not recommend consider using them in practice. +5.3. Neuroimage application. +We apply our proposed testing procedures to study how +the SUVRs from PET image data affect the MoCa score. We download the preprocessed 18F- +AV-1451 PET image features, and demographic and cognitive assessments from the ADNI +database. The image features include 18F-AV-1451 SUVRs and volumes of the cortical, sub- +cortical regions, brainstem, ventricles and sub-divisions of corpus callosum. Furthermore, the +demographic variables include gender and standardized age (divided by the standard devia- + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +17 +TABLE 3 +The empirical size and power of Wald and score tests for linear hypothesis testing with n “ 300. +X „ Normal +X „ Uniform +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +TW +TS +TW +TS +TW +TS +TW +TS +p “ 50 +β1 ` β2 +H0,4 : β1 ` β2 “ 0, v.s. Ha,4 : β1 ` β2 ‰ 0 +0.0 +0.056 +0.048 +0.040 +0.042 +0.056 +0.050 +0.052 +0.042 +0.1 +0.154 +0.132 +0.174 +0.162 +0.138 +0.136 +0.208 +0.184 +0.2 +0.420 +0.386 +0.574 +0.532 +0.374 +0.354 +0.574 +0.530 +0.4 +0.930 +0.906 +0.990 +0.984 +0.908 +0.902 +0.994 +0.990 +β3 ` β4 +H0,5 : β3 ` β4 “ 0, v.s. Ha,5 : β3 ` β4 ‰ 0. +0.0 +0.058 +0.050 +0.066 +0.056 +0.056 +0.050 +0.068 +0.066 +0.1 +0.130 +0.116 +0.154 +0.138 +0.152 +0.136 +0.190 +0.174 +0.2 +0.450 +0.428 +0.470 +0.450 +0.412 +0.388 +0.450 +0.424 +0.4 +0.930 +0.920 +0.946 +0.930 +0.932 +0.920 +0.958 +0.932 +β1 ` βp +H0,6 : β1 ` βp “ 0.75, v.s. Ha,6 : β1 ` βp ‰ 0.75 +0.75 +0.050 +0.044 +0.060 +0.046 +0.058 +0.048 +0.044 +0.036 +0.85 +0.172 +0.146 +0.138 +0.116 +0.162 +0.154 +0.154 +0.130 +0.95 +0.490 +0.444 +0.408 +0.354 +0.462 +0.436 +0.418 +0.376 +1.15 +0.970 +0.950 +0.936 +0.916 +0.970 +0.958 +0.942 +0.914 +β2 ` β3 +H0,7 : β2 ` β3 “ ´0.75, v.s. Ha,7 : β2 ` β3 ‰ ´0.75 +-0.75 +0.056 +0.050 +0.054 +0.044 +0.060 +0.044 +0.062 +0.062 +-0.65 +0.200 +0.176 +0.182 +0.172 +0.148 +0.138 +0.180 +0.174 +-0.55 +0.484 +0.444 +0.516 +0.486 +0.422 +0.408 +0.468 +0.466 +-0.35 +0.922 +0.910 +0.966 +0.962 +0.920 +0.910 +0.960 +0.962 +p “ 350 +β1 ` β2 +H0,4 : β1 ` β2 “ 0, v.s. Ha,4 : β1 ` β2 ‰ 0 +0.0 +0.062 +0.056 +0.062 +0.056 +0.050 +0.046 +0.048 +0.046 +0.1 +0.164 +0.160 +0.216 +0.202 +0.106 +0.096 +0.206 +0.184 +0.2 +0.472 +0.438 +0.612 +0.572 +0.402 +0.378 +0.536 +0.510 +0.4 +0.940 +0.934 +0.988 +0.988 +0.910 +0.900 +0.982 +0.980 +β3 ` β4 +H0,5 : β3 ` β4 “ 0, v.s. Ha,5 : β3 ` β4 ‰ 0. +0.0 +0.058 +0.046 +0.070 +0.040 +0.038 +0.040 +0.068 +0.048 +0.1 +0.192 +0.188 +0.174 +0.138 +0.126 +0.124 +0.172 +0.136 +0.2 +0.454 +0.442 +0.462 +0.410 +0.392 +0.378 +0.404 +0.356 +0.4 +0.952 +0.952 +0.912 +0.814 +0.944 +0.942 +0.916 +0.828 +β1 ` βp +H0,6 : β1 ` βp “ 0.75, v.s. Ha,6 : β1 ` βp ‰ 0.75 +0.75 +0.046 +0.044 +0.058 +0.038 +0.056 +0.042 +0.060 +0.054 +0.85 +0.148 +0.142 +0.280 +0.292 +0.110 +0.110 +0.240 +0.242 +0.95 +0.466 +0.472 +0.562 +0.566 +0.394 +0.390 +0.496 +0.512 +1.15 +0.942 +0.944 +0.960 +0.940 +0.938 +0.940 +0.954 +0.932 +β2 ` β3 +H0,7 : β2 ` β3 “ ´0.75, v.s. Ha,7 : β2 ` β3 ‰ ´0.75 +-0.75 +0.052 +0.038 +0.052 +0.046 +0.052 +0.038 +0.062 +0.038 +-0.65 +0.154 +0.138 +0.174 +0.138 +0.134 +0.120 +0.142 +0.130 +-0.55 +0.450 +0.420 +0.478 +0.442 +0.400 +0.374 +0.488 +0.456 +-0.35 +0.932 +0.916 +0.968 +0.966 +0.914 +0.912 +0.960 +0.960 +tion) at the image examining time. For each subject, we obtain his/her MoCa score within +14 days of his/her image examining time as the outcome, which ranges from 9 to 30. Fur- +thermore, we remove the covariates with more than 100 missing values. We standardize the +volumes of ROIs by subtracting the means and dividing by the standard deviations. We use +the SUVR from inferior cerebellum as a reference and divide the rest of SUVRs by this ref- +erence as suggested in (Landau et al. 2016). Finally, we have n “ 196 complete samples with +p “ 218 covariates in the analysis. +Since the neuroimage data are longitudinally collected, we estimate the covariance matrix +of U using repeatedly measured image features, while assuming that age and gender are + +18 +TABLE 4 +The empirical size and power of Wald and score tests for linear hypothesis testing with n “ 500. +X „ Normal +X „ Uniform +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +TW +TS +TW +TS +TW +TS +TW +TS +p “ 50 +β1 ` β2 +H0,4 : β1 ` β2 “ 0, v.s. Ha,4 : β1 ` β2 ‰ 0 +0.0 +0.058 +0.052 +0.048 +0.046 +0.062 +0.056 +0.046 +0.044 +0.1 +0.240 +0.220 +0.286 +0.280 +0.202 +0.196 +0.296 +0.274 +0.2 +0.670 +0.656 +0.810 +0.794 +0.586 +0.580 +0.758 +0.754 +0.4 +0.996 +0.996 +1.000 +1.000 +0.988 +0.988 +0.998 +0.998 +β3 ` β4 +H0,5 : β3 ` β4 “ 0, v.s. Ha,5 : β3 ` β4 ‰ 0. +0.0 +0.046 +0.040 +0.064 +0.058 +0.060 +0.048 +0.060 +0.062 +0.1 +0.244 +0.216 +0.254 +0.244 +0.242 +0.238 +0.260 +0.244 +0.2 +0.646 +0.616 +0.696 +0.692 +0.666 +0.648 +0.686 +0.678 +0.4 +0.996 +0.996 +0.998 +0.996 +0.996 +0.998 +0.998 +0.998 +β1 ` βp +H0,6 : β1 ` βp “ 0.75, v.s. Ha,6 : β1 ` βp ‰ 0.75 +0.75 +0.068 +0.062 +0.058 +0.056 +0.050 +0.046 +0.054 +0.048 +0.85 +0.242 +0.222 +0.190 +0.162 +0.226 +0.212 +0.176 +0.160 +0.95 +0.676 +0.640 +0.620 +0.580 +0.646 +0.626 +0.646 +0.614 +1.15 +0.996 +0.994 +0.986 +0.980 +0.996 +0.994 +0.998 +0.990 +β2 ` β3 +H0,7 : β2 ` β3 “ ´0.75, v.s. Ha,7 : β2 ` β3 ‰ ´0.75 +-0.75 +0.060 +0.058 +0.056 +0.044 +0.056 +0.052 +0.050 +0.048 +-0.65 +0.238 +0.234 +0.302 +0.302 +0.214 +0.206 +0.226 +0.218 +-0.55 +0.640 +0.622 +0.730 +0.720 +0.608 +0.598 +0.664 +0.660 +-0.35 +0.992 +0.992 +1.000 +1.000 +0.994 +0.992 +0.998 +0.998 +p “ 600 +β1 ` β2 +H0,4 : β1 ` β2 “ 0, v.s. Ha,4 : β1 ` β2 ‰ 0 +0.0 +0.054 +0.044 +0.056 +0.050 +0.046 +0.042 +0.046 +0.042 +0.1 +0.192 +0.180 +0.286 +0.268 +0.190 +0.182 +0.292 +0.288 +0.2 +0.602 +0.594 +0.790 +0.786 +0.578 +0.558 +0.824 +0.816 +0.4 +0.688 +0.700 +0.998 +1.000 +0.986 +0.984 +1.000 +1.000 +β3 ` β4 +H0,5 : β3 ` β4 “ 0, v.s. Ha,5 : β3 ` β4 ‰ 0. +0.0 +0.072 +0.064 +0.066 +0.060 +0.044 +0.046 +0.060 +0.056 +0.1 +0.222 +0.218 +0.244 +0.238 +0.206 +0.194 +0.264 +0.264 +0.2 +0.654 +0.650 +0.678 +0.670 +0.614 +0.608 +0.712 +0.714 +0.4 +0.974 +0.974 +0.994 +0.980 +0.992 +0.990 +0.994 +0.978 +β1 ` βp +H0,6 : β1 ` βp “ 0.75, v.s. Ha,6 : β1 ` βp ‰ 0.75 +0.75 +0.044 +0.040 +0.056 +0.048 +0.058 +0.056 +0.052 +0.050 +0.85 +0.238 +0.242 +0.316 +0.316 +0.196 +0.200 +0.386 +0.390 +0.95 +0.670 +0.672 +0.716 +0.722 +0.604 +0.616 +0.790 +0.788 +1.15 +0.986 +0.994 +1.000 +1.000 +0.994 +0.994 +0.9996 +0.994 +β2 ` β3 +H0,7 : β2 ` β3 “ ´0.75, v.s. Ha,7 : β2 ` β3 ‰ ´0.75 +-0.75 +0.058 +0.052 +0.052 +0.044 +0.056 +0.052 +0.044 +0.046 +-0.65 +0.222 +0.200 +0.230 +0.208 +0.162 +0.146 +0.264 +0.246 +-0.55 +0.630 +0.608 +0.692 +0.680 +0.544 +0.516 +0.686 +0.678 +-0.35 +0.974 +0.990 +1.000 +1.000 +0.994 +0.992 +1.000 +1.000 +recorded precisely. More specifically, let Ă +Wij denote the observed image features at the jth +examining time. We first perform the regression between Ă +Wij and age of the ith patient at +the jth examining time, and obtain rUij as the residual of the regression. Then we obtain the +estimator for the covariance matrix +rΩ “ +řn +i“1 +řni +j“1p rUij ´ Uiqp rUij ´ UiqT +řn +i“1pni ´ 1q +, +where ni is the number of repeated measurements of Ă +Wij, and Ui “ řni +j“1 rUij{ni. Finally, +because the first two covariates, age and gender, are measured precisely, the first two columns + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +19 +TABLE 5 +The empirical size and power of Wald and score tests under different m. +X „ Normal +X „ Uniform +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +Σ “ 0.5Ip +Σ “ 0.5|i´j|`1 +TW +TS +TW +TS +TW +TS +TW +TS +p “ 50 +ř4 +j“1 βj +H0,8 : ř4 +j“1 βj “ 0, v.s. Ha,8 : ř4 +j“1 βj ‰ 0 +0.0 +0.062 +0.052 +0.046 +0.038 +0.062 +0.060 +0.060 +0.052 +0.2 +0.238 +0.220 +0.434 +0.406 +0.244 +0.228 +0.404 +0.386 +0.4 +0.754 +0.724 +0.914 +0.904 +0.660 +0.646 +0.914 +0.900 +0.8 +1.000 +1.000 +1.000 +1.000 +0.998 +0.998 +1.000 +1.000 +ř8 +j“1 βj +H0,9 : ř8 +j“1 βj “ 0, v.s. Ha,9 : ř8 +j“1 βj ‰ 0 +0.0 +0.054 +0.052 +0.062 +0.058 +0.052 +0.054 +0.062 +0.052 +0.2 +0.162 +0.142 +0.316 +0.308 +0.178 +0.172 +0.296 +0.282 +0.4 +0.410 +0.388 +0.764 +0.732 +0.424 +0.406 +0.764 +0.742 +0.8 +0.966 +0.952 +1.000 +1.000 +0.920 +0.906 +1.000 +1.000 +ř12 +j“1 βj +H0,10 : ř12 +j“1 βj “ 0, v.s. Ha,10 : ř12 +j“1 βj +0.0 +0.046 +0.044 +0.062 +0.062 +0.046 +0.052 +0.052 +0.050 +0.2 +0.116 +0.124 +0.250 +0.240 +0.084 +0.096 +0.210 +0.210 +0.4 +0.288 +0.298 +0.610 +0.604 +0.318 +0.330 +0.642 +0.642 +0.8 +0.854 +0.796 +0.996 +0.994 +0.802 +0.756 +0.994 +0.992 +p “ 350 +ř4 +j“1 βj +H0,8 : ř4 +j“1 βj “ 0, v.s. Ha,8 : ř4 +j“1 βj ‰ 0 +0.0 +0.062 +0.052 +0.062 +0.052 +0.060 +0.054 +0.036 +0.036 +0.2 +0.260 +0.244 +0.420 +0.400 +0.230 +0.222 +0.402 +0.382 +0.4 +0.718 +0.684 +0.918 +0.916 +0.710 +0.672 +0.924 +0.912 +0.8 +1.000 +0.998 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +ř8 +j“1 βj +H0,9 : ř8 +j“1 βj “ 0, v.s. Ha,9 : ř8 +j“1 βj ‰ 0 +0.0 +0.062 +0.060 +0.068 +0.060 +0.066 +0.058 +0.042 +0.040 +0.2 +0.132 +0.130 +0.266 +0.258 +0.180 +0.166 +0.262 +0.238 +0.4 +0.452 +0.420 +0.770 +0.746 +0.380 +0.360 +0.764 +0.752 +0.8 +0.950 +0.936 +1.000 +1.000 +0.936 +0.926 +1.000 +1.000 +ř12 +j“1 βj +H0,10 : ř12 +j“1 βj “ 0, v.s. Ha,10 : ř12 +j“1 βj +0.0 +0.056 +0.056 +0.054 +0.054 +0.052 +0.050 +0.038 +0.038 +0.2 +0.098 +0.106 +0.170 +0.174 +0.100 +0.100 +0.178 +0.172 +0.4 +0.296 +0.296 +0.616 +0.604 +0.276 +0.274 +0.584 +0.574 +0.8 +0.854 +0.792 +0.998 +0.996 +0.838 +0.818 +0.992 +0.990 +TABLE 6 +The empirical size and power of Wald and score tests with/without noise considered. +With noise +Without noise +With noise +Without noise +With noise +Without noise +TW +TS +TW +TS +TW +TS +TW +TS +TW +TS +TW +TS +β3 +H0,2 : β3 “ 0 +H0,5 : β3 ` β4 “ 0 +H0,7 : β2 ` β3 “ ´0.75 +0.0 +0.084 +0.078 +0.774 +0.824 +0.064 +0.066 +0.474 +0.554 +0.056 +0.054 +0.538 +0.604 +0.1 +0.340 +0.316 +0.332 +0.410 +0.106 +0.094 +0.784 +0.838 +0.166 +0.188 +0.234 +0.288 +0.2 +0.692 +0.646 +0.104 +0.092 +0.270 +0.194 +0.930 +0.956 +0.388 +0.378 +0.096 +0.090 +0.4 +0.914 +0.954 +0.690 +0.362 +0.700 +0.540 +0.996 +1.000 +0.882 +0.880 +0.406 +0.196 +and rows of the estimated Ω, denoted by pΩ, are zeros. We set the rest pp ´ 2q ˆ pp ´ 2q sub- +matrix of pΩ to be rΩ. +We test p hypotheses, each of the form +H0 : βj “ 0 +v.s. +Ha : βj ‰ 0, +(19) +for j “ 1,...,p at 0.05 nominal level. To implement the hypothesis testing procedure, in each +test, we first fit a standard penalized Poisson regression model to obtain the initial values of +the coefficient estimators. Then we construct the score test and Wald test statistics based on + +20 +(11) and (10), respectively. The tuning parameter λ is selected by minimizing (16). We obtain +the p-value as the probability of a χ2p1q random variable that is greater than the resulting +score and Wald test statistics. There are 33 and 69 covariate coefficients with significant p- +values at 0.05 nominal level based on the score and Wald tests, respectively. Furthermore, +we plot the boxplot of the resulting p-values in Figure 2. It is clear that the distribution of +the p-values are similar for the score and the Wald test. For each covariate j, we obtain the +score test +Wald test +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distributions p−value +FIG 2. The boxplot of the p-values based on the score and Wald tests. The distributions of the p-values are similar +from both methods. +estimated jth coefficient based on (8) under the corresponding alternative hypothesis, and +plot the estimated coefficients of the SUVRs at the cortical regions on a template brain in +Figure 3. The results show that the SUVRs have negative effects on the cognitive score, +suggesting that the higher the SUVR values, the lower the MoCa score and in turn the worse +the cognitive function, which is consistent with the scientific evidences (Braak & Braak 1991, +Schöll et al. 2016, Baker et al. 2017). Furthermore, the score test is more stringent and gives +less number of significant SUVRs. Among 33 significant predictors from the score test, 27 of +them are also significant in the Wald test. Based on this high agreement between the score and +Wald tests, we believe the difference between the two tests is a small sample phenomenon. +To adjust for the multiple testing, we further performed an analysis to control false dis- +covery rate (FDR) (Benjamini & Hochberg 1995) within 0.05 by treating the p-values as +independent. Since the score test is too stringent, no significant covariate has been identified +at 0.05 FDR by using the score test. Therefore, we only present the results from the Wald test. +We plot the p-values versus 0.05j{218 in Figure 4 in an increasing order, which suggests 36 +covariates are selected as the important predictors. There are 13 cortical SUVRs among the +36 important predictors that are significant. We present their estimated coefficient, p-values +from the Wald test in Table 7. The results show that the majority of the significant cortical +SUVRs are in the temporal lobe, which consists of structures that are vital for declarative or +long-term memory (Smith & Kosslyn 2008). + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +21 +FIG 3. The effects of SUVRs at the cortical regions. The colors represent the values (indicated by the color bars) +of the estimated coefficients of the SUVRs. We only plot the coefficient values corresponding to the significant +brain regions with p-value less than 0.05 from score test (left) and Wald test (right). The white areas are the +non-significant brain regions. The L and R letters in the plot represent the left and right hemispheres. +GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG +GGGGGGGGGGGGGGGGGGGGGGGG +GGGGGGGG +G +GGGGGGGGGG +GG +GGGGGGGGGGGGGGGGGGGGGGGGGG +GG +GGGGG +GGG +GG +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +R +p−value +FIG 4. Sorted p-value versus R “ 0.05j{218,j “ 1,...,218. There are 36 important predictors corresponding +to the p-values (in red) that below the line. +In addition, we perform a 5-fold cross validation and compare the prediction errors among +the four methods: (a) We select the important predictors as those with p-value less than 0.05 +in the test (19) based on the score statistics and then use formula expp pβT +S WSi ´ pβT +S pΩS pβSq to +predict the outcome in the test sample, where WSi is the selected covariates, pβS is estimator +from (6) using selected covariates, pΩS is the subset of pΩ corresponding to the selected co- +variates. (b) We select the important predictors as those with p-value less than 0.05 in the test +(19) based on the Wald statistics and then use formula expp pβT +W WWi ´ pβT +W pΩW pβW q to pre- + +L +R +-0.2647 +0L +R +-0.2626 +022 +Cortical regions +Brain lobes +Estimated coefficient +Wald test p-value +left middle temporal gyrus +Temporal lobe +-0.214 +0.0002 +left inferior parietal cortex +Parietal lobe +-0.214 +0.0003 +left inferior temporal gyrus +Temporal lobe +-0.223 +0.0007 +right inferior parietal cortex +Temporal lobe +-0.211 +0.0007 +left BANKSSTS +Temporal lobe +-0.174 +0.0015 +left fusiform gyrus +Temporal lobe +-0.262 +0.0016 +right middle temporal gyrus +Temporal lobe +-0.229 +0.0024 +left caudal middle frontal gyrus +Frontal lobe +-0.236 +0.0030 +left precuneus cortex +Parietal lobe +-0.215 +0.0034 +left entorhinal cortex +Temporal lobe +-0.217 +0.0036 +right inferior temporal +Temporal lobe +-0.225 +0.0059 +right left entorhinal cortex +Temporal lobe +-0.221 +0.0065 +right BANKSSTS +Temporal lobe +-0.168 +0.0076 +TABLE 7 +The estimated coefficients, p-values from score and Wald tests for the significant SUVRs at 27 cortical regions. +We also include the specific brain lobe that contains each cortical region. BANKSSTS stands for banks of the +superior temporal sulcus. +dict the outcome in the test sample, where WWi is the selected covariates, pβW is estimator +from (6) using selected covariates, pΩW is the subset of pΩ corresponding to the selected co- +variates. (c) We select the important predictors using the standard lasso regression between +the logarithm of the MoCa score and all covariates and then use formula expp pβTWiq to +predict the outcome, where pβ is the estimator from the lasso regression. (d) We select the im- +portant predictors using the penalized Poisson regression between the logarithm of the MoCa +score and all covariates and then use formula expp pβTWiq to predict the outcome, where pβ is +the estimator from the penalized Poisson regression. The penalty parameters in the lasso and +penalized Poisson regression are selected using a sub-routine of 10-folder cross-validation. +Method (d) breaks down because the algorithm does not converge for any selections of the +penalty parameters. Therefore, in Figure 5, we show the distributions of the prediction er- +rors, defined as řn +i“1 |Yi ´ pYi|{|Yi|, only for the methods (a), (b) and (c) after 100 runs of the +5-fold cross-validation. The results shows that Method (a) and (b) have similar performance +and both outperform Method (c) with much smaller prediction errors on average. +Finally, we perform the score and the Wald tests to test whether any SUVRs from any com- +posite regions may have significant association with the MoCa score, where the composite +regions, namely BRAAK12, BRAAK34, BRAAK56, are defined in (Braak & Braak 1991) +and used in Landau et al. (2016) and Schöll et al. (2016). We provide the list of ROIs in each +composite regions in Appendix B.5. Let βSk be the coefficients of the SUVRs from the ROIs +that belong to the composite region k. We test the null hypothesis that βSk “ 0. The results +in Table 8 show that all the tests are significant, suggesting that at least one ROI in each of the +composite region has significant association with the cognitive function. This result partially +agrees with results in (Schöll et al. 2016) that the SUVRs from the composite regions are sig- +nificantly different in healthy subjects and patients with a diagnosis of probable Alzheimer’s +disease. +6. Conclusion and discussion. +We have proposed an amenably penalized noise cor- +rected Poisson model to study the relationship between the cognitive score and high dimen- +sional noisy neuroimage data. Under the sparsity assumption, we established the parameter +convergence rates in both l1 and l2 norms, the variable selection consistency property and +the asymptotic normality of a subvector with possibly infinitely many components. Infer- +ence tools are subsequently developed. The neuroimage application shows that the inference + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +23 +score +Wald +Lasso +0.10 +0.15 +0.20 +0.25 +Prediction errors +FIG 5. The distribution of the prediction errors from 100 runs of the 5-fold cross-validation based on Methods +(a), (b) and (c). Method (d) breaks down because the algorithm does not converge. +Composite regions +TS +score test p-value +TW +Wald test p-value +DF +BRAAK12 +10.10 +0.039 +15.41 +0.0039 +4 +BRAAK34 +42.51 +0.0113 +89.45 +1.774e-09 +24 +BRAAK56 +66.29 +0.0165 +71.44 +0.0055 +44 +TABLE 8 +The score and Wald test results of hypothesis that βSk “ 0. TS and TW are the score and Wald test statistics, DF +is the degree of freedom in the asymptotic distribution of the score and Wald statistics, which equals the number +of the ROIs in the composite regions. +tools generate scientifically meaningful results, which have potential to be used to study the +cognitive function and cognitive changes for neurodegenerative diseases. Further research +along this line is ongoing in our group. The neuroimage dataset and computational code are +available at Jiang et al. (2021). +Thanks to an anonymous referee, we would like to point out one important extension. +Instead of a constant matrix Ω, we can further allow Ω to depend on both the covariate +X and the response Y , hence ΩpY,Xq, and assume EpU|Y,Xq “ 0. This would include +heteroscedastic measurement error and to allow dependent relation between W and Y given +X. All the estimation and inference results will still hold and the regularity conditions and +proofs in the Suppement also do not need to be further modified to accomodate this extension. +Establishing similar results in generalized linear models beyond Poisson or general re- +gression models with non-Gaussian noise turns out to be surprisingly difficult due to various +technical obstacles. The main difficult lies in being unable to construct a loss function that +is positive-definite at the true parameter value. In the case when an estimating equation is +available, although one may be tempted to treat the l2 norm square of the estimating equation +as a loss function, we find other technical issues arise partially because the Hessian of the +loss function may involve the response, hence some of the techniques used here cannot be + +24 +directly applied. Likewise, extending the Poisson model to allow overdispersion also turns +out challenging, regardless if we use a negative binomial model, or incorporate random ef- +fects, or use extra observed covariates. All these will lead to models different from Poisson. +The biggest hurdle of considering general regression model and/or non-Gaussian noise is to +rigorously establish that the loss function is locally convex. More investigation and dedicated +effort are needed in this aspect. +The assumption that the covariance of the measurement is known is widely adopted in the +low and dimensional noisy data literature (Stefanski 1989, Cook & Stefanski 1995, Loh & +Wainwright 2012, Sørensen et al. 2015), because the parameter estimation in the noisy model +with unknown noise covariance is a challenging, especially in high dimensional setting where +the covariance is a high dimensional unknown parameter to be estimated. Thresholding tech- +niques as those proposed in Bickel & Levina (2008), Cai & Liu (2011), Fan et al. (2011) +can be used for the covariance estimation, but the theoretical properties of the resulting esti- +mators are involved, requiring careful treatment of the additional error from the covariance +estimator. In a relatively simple situation when the error variance can be estimated through +estimating a parameter γ via solving fγpγq “ 0, then writing Lpβq as Lpβ,γq, we can acco- +modate the additional parameter by concatenating β with γ and carrying out the subsequent +analysis. For example, in this case the result in Theorem 4 will be updated to +ˆ pβaMYS ´ βtMYS +pγ ´ γ +˙ +“ ´ +" +QMYS,MYSpβt,γq B2Lpβ,γq{BβMYSBγT +Bfγpγq{BβT +MYS +Bfγpγq{BγT +*´1 +ˆ +«! +BLpβt,γq +Bβ +) +MYS +fγpγq +ff +t1 ` opp1qu. +Letting M ” QMYS,MYSpβtq´tB2Lpβt,γq{BβMYSBγTutBfγpγq{BγTu´1tBfγpγq{BβT +MYSu, +then this leads to +pβaMYS ´ βtMYS “ ´M´1 +„"BLpβt,γq +Bβ +* +MYS +` tB2Lpβt,γq{BβMYSBγTutBfγpγq{BγTu´1 +ˆfγpγqst1 ` opp1qu. +We further conduct simulations to evaluate the proposed adjustment in Section B.6 of the +supplementary document. The results suggest that the proposed adjustment controls type I +error rate when Ω contain small number of unknown parameters. Estimation and testing +when Ω has a large number of unknown parameters are challenging problems and deserve +much more extensive investigation. +REFERENCES +Baker, S. L., Lockhart, S. N., Price, J. C., He, M., Huesman, R. H., Schonhaut, D., Faria, J., Rabinovici, G. & +Jagust, W. J. 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(2017), ‘The alzheimer’s disease neuroimaging initiative 3: Continued innovation +for clinical trial improvement’, Alzheimer’s & Dementia 13(5), 561–571. +Zhang, C.-H., Zhang, T. et al. (2012), ‘A general theory of concave regularization for high-dimensional sparse +estimation problems’, Statistical Science 27(4), 576–593. +Zhang, X. & Cheng, G. (2017), ‘Simultaneous inference for high-dimensional linear models’, Journal of the +American Statistical Association 112(518), 757–768. + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +27 +Supplementary Materials to “On High dimensional Poisson models +with measurement error: hypothesis testing for nonlinear +nonconvex optimization” +Appendix A. +We define an auxiliary function qλptq “ λ|t| ´ ρλptq to facilitate the theo- +retical derivation, where qλptq ´ µ{2t2 is concave and everywhere differentiable as shown in +Lemma B.4 in the supplementary material. +A.1. Conditions for the estimation consistency. +Define αminpMq and αmaxpMq as the +minimal and maximal eigenvalues of the matrix M, respectively. Further, we define the sub- +exponential norm and sub-Gaussian norm as +}X}ψ1 “ sup +kě1 +1{kEp|X|kq1{k, +and +}X}ψ2 “ sup +kě1 +1{ +? +kEp|X|kq1{k. +For notational convenience, let +ApβTWiq :“ exppβTWi ´ βTΩβ{2q, +gpWi,β,v,wq :“ vTtpWi ´ Ωβqb2 ´ Ωuw, +and +g1pWi,β,v,wq :“ vTtpWi ´ Ωβqb2uw. +DEFINITION 1. +Loh & Wainwright (2012) (Lower-RE condition). The matrix Γ satisfies +a lower restricted eigenvalue condition with curvature α1 ą 0 and tolerance τpn,pq ą 0 if +βTΓβ ě α1}β}2 +2 ´ τpn,pq}β}2 +1,@β P Rp. +DEFINITION 2. +Loh & Wainwright (2012) (Upper-RE condition). The matrix Γ satisfies +a upper restricted eigenvalue condition with smoothness a2 ą 0 and tolerance τpn,pq ą 0 if +βTΓβ ď a2}β}2 +2 ` τpn,pq}β}2 +1,@β P Rp. +We first state the regularity conditions as follows: +(C1) (a) supi“1,...,n,}v}2ď1 |WT +i v| ď MW +a +}v}0 for a positive constant MW . }Ω}2 “ Op1q. +(b) +D1 ď αminrEtexppβTXqXXTus ď αmaxrEtexppβTXqXXTus ď D2, +DW1 ď αminrEtexpp2βTW ´ βTΩβqpW ´ Ωβqb2us + +28 +ď αmaxrEtexpp2βTW ´ βTΩβqpW ´ Ωβqb2us ď DW2, +αmaxrEtexppβTW ´ βTΩβ{2qpW ´ Ωβqb2us ď DW3, +and Etexpp2βTXqu “ Op1q for any β with }β}2 ď 2R2. +(c) Ep}Wi´Ωβ}2 +2q ď DΩ, for any β with }β}2 ď 2R2. Here D1,D2,DW1,DW2,DW3,DΩ +are positive constants. +(d) }C}2 “ Op1q and }pCCTq´1}2 “ Op1q. +(e) The L2 norm of the true parameter β is bounded, that is }β}2 ď b0 for some +0 ă b0 ă 8. +(C2) For j “ 1,...,p, define +Kj :“ }Uij}ψ2 “ p2Ωjjq1{2 sup +kě1 +k´1{2 Γ1{ktpk ` 1q{2u +π1{p2kq +, +where Γ is the Gamma function, then there exist constants m0,M0 so that +m0 ă K2 +j +nÿ +i“1 +Y 2 +i {n ă M0, +uniformly for all j almost surely. +(C3) Define +KY pXiq “ sup +kě1 +k´1Er|Yi ´ exppβT +t Xiq|k|Xis1{k. +There exist constants m1,m2,M1,M2 so that +m1 ă +nÿ +i“1 +X2 +ijKY pXiq2{n ă M1, +max +i +|Xij|KY pXiq{ +a +logn ă M2, +uniformly for all j “ 1,...,p almost surely. +(C4) Sample size n and dimension of covariates p satisfy +logpnq +a +logppq{n ď C +for an absolute constant C. +(C5) For ej, j “ 1,...,p, define +Kwijpβq “ sup +kě1 +k´1{2Er|pWi ´ ΩβqTej ´ EtpWi ´ ΩβqTej|βTWi,Xiu|k|βTWi,Xis1{k, +we assume EtKwijpβtq4u ă Q0. Then there exist constants m3,M3,Q1 so that +(i) +m3 ă +nÿ +i“1 +Kwijpβtq2 expp2βT +t Wi ´ βT +t Ωβtq{n ă M3 +and + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +29 +(ii) +| +nÿ +i“1 +EtexppβT +t Wi ´ βT +t Ωβt{2qpWi ´ ΩβtqTej|βT +t Wi,Xiu ´ exppβT +t XiqXT +i ej +a +nlogppq +| ă Q1 +uniformly for all j “ 1,...,p in probability. +(C6) For vectors v,w P Rp, }v}2 ď 1,}w}2 ď 1, for β with }β}2 ď 2R2, +Kgvwipβq :“ sup +ką1 +1{kEp|rgpWi,β,v,wq´EtgpWi,β,v,wq|βTWi,Xius|k|βTWi,Xiq1{k. +We assume EtKgvwipβq4u ă Q01, and ErexptA2pβTWiqK2 +gvwipβqus ă Q02. We also +assume that for all v,w, +m4 ă +nÿ +i“1 +|ApβTWiq|2Kgvwipβq2{n ă M4, +(A.20) +m5 ă max +i +|ApβTWiq|Kgvwipβq{ +a +logn ă M5, +(A.21) +and +n´1{2|sup +v,w +nÿ +i“1 +vTpApβTWiqErtpWi ´ Ωβqb2 ´ Ωu|βTWi,Xis +´EtexppβTXiqXiXT +i uqw|2 ă Q2, +(A.22) +in probability. +(C7) For vectors v,w P Rp, }v}2 ď 1,}w}2 ď 1, for β with }β}2 ď 2R2, +Kg1vwipβq :“ sup +ką1 +1{kEp|rg1pWi,β,v,wq´Etg1pWi,β,v,wq|βTWi,Xius|k|βTWi,Xiq1{k. +We assume EtKg1vwipβq4u ă Q11, and ErexptA4pβTWiqK2 +g1vwipβqus ă Q12. We also +assume that for all v, +m6 ă +nÿ +i“1 +|A2pβTWiq|2Kg1vwipβq2{n ă M6, +m7 ă max +i +|A2pβTWiq|Kg1vwipβq{ +a +logn ă M7, +and +m61 ă +nÿ +i“1 +|A2pβTWiq|2Kg1vwipβq2{n ă M61, +m71 ă max +i +|A2pβTWiq|Kg1vwipβq{ +a +logn ă M71,. +Further +n´1{2|sup +v,w +nÿ +i“1 +vTpA2pβTWiqErtpWi ´ Ωβqb2u|βTWi,Xis +´Etexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uqw| ă Q3, + +30 +in probability and +n´1{2|sup +v,w +nÿ +i“1 +vTpApβTWiqErtpWi ´ Ωβqb2u|βTWi,Xis +´EtexppβTWi ´ βTΩβ{2qpWi ´ Ωβqb2uqw| ă Q31, +in probability. +A.2. Proof of Theorems in Section 4.2. +A.2.1. Proof of Theorem 1. +First denote pv “ pβ ´ qβ, by the Taylor expansion of the first +order derivative +tBLp pβq{BβT ´ BLp qβq{BβTupv “ pvTB2Lpβ˚q{BβBβTpv, +where β˚ is a point on the line connecting qβ and pβ and hence is in the feasible set. Therefore, +by Lemma B.6 we have +tBLp pβq{BβT ´ BLp qβq{BβTupv ě α1}pv}2 +2 ´ τ1 +a +logppq{n}pv}2 +1. +(A.23) +We first show that }pv}2 ď 1. If not, we have +tBLp pβq{BβT ´ BLp qβq{BβTupv ě α1}pv}2 ´ 2τ1 +a +logppq{nR1}pv}1. +Together with (12), we obtain +t´Bρλp pβMcq{BβT +McA ´ BLp qβq{BβTupv ě α1}pv}2 ´ 2τ1 +a +logppq{nR1}pv}1. +(A.24) +Further, +t´Bρλp pβMcq{BβT +McA ´ BLp qβq{BβTupv +ď t} ´ Bρλp pβMcq{BβT +Mc}8}A}8 ` }BLp qβq{BβT}8u}pv}1 +ď pλ ` }BLp qβq{BβT}8q}pv}1 +ď 3λ{2}pv}1. +(A.25) +The second inequality holds because the maximum row sum of A is 1 and }Bρλp pβMcq{BβMc}8 ď +λ by Condition (A1)–(A6) and Lemma B.1. The last equality holds because }BLp qβq{BβT}8 ď +λ{2 by the statement assumption. Now combine (A.25) and (A.24), we have +}pv}2 ď α´1 +1 p2τ1 +a +logppq{nR1 ` 3λ{2q}pv}1 +ď α´1 +1 p2τ1 +a +logppq{nR1 ` 3λ{2q2R1. +By the assumption that λ ď α1{p6R1q and n ě logppqp64τ 2 +1 R4 +1q{α2 +1 in the statement, we +conclude that the right hand side is at most one, which contradict to the hypothesis that +}pv}2 ą 1. Therefore }pv}2 ď 1. + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +31 +Further, since function ρλpβMcq ` µ{2}βMc}2 +2 is convex function of βMc by Condition +(A5), we have +ρλp qβMcq ` µ{2} qβMc}2 +2 ´ ρλp pβMcq ´ µ{2} pβMc}2 +2 +ě tBρλp pβMcq{BβT +Mc ` µ pβT +Mcup qβMc ´ pβMcq +which implies +ρλp qβMcq ´ ρλp pβMcq ` µ{2} pβMc ´ βMc}2 +ě tBρλp pβMcq{BβT +Mcup qβMc ´ pβMcq +“ tBρλp pβMcq{BβT +McuAp qβ ´ pβq. +Combine with (12), we have +Bρλp pβMcq{BβT +McAp qβ ´ pβq ě ´BLp pβq{BβTp qβ ´ pβq, +and hence +ρλp qβMcq ´ ρλp pβMcq ` µ{2} pβMc ´ qβMc}2 ě BLp pβq{BβTp pβ ´ qβq. +Now combine with (A.23), we have +α1}pv}2 +2 ´ τ1 +c +logppq +n +}pv}2 +1 +ď ´BLp qβq{BβTpv ` ρλp qβMcq ´ ρλp pβMcq ` µ{2}Apv}2 +2 +ď ´BLp qβq{BβTpv ` ρλp qβMcq ´ ρλp pβMcq ` µ{2}A}1}A}8}pv}2 +2 +“ ´BLp qβq{BβTpv ` ρλp qβMcq ´ ρλp pβMcq ` µ{2}pv}2 +2. +This implies +pα1 ´ µ{2q}pv}2 +2 +ď τ1 +c +logppq +n +}pv}2 +1 ` }BLp qβq{Bβ}8}pv}1 ` ρλp qβMcq ´ ρλp pβMcq +ď +# +2R1τ1 +c +logppq +n +` }BLp qβq{Bβ}8 ++ +t}pvMc}1 ` }pvM}1u ` ρλp qβMcq ´ ρλp pβMcq. +(A.26) +Note that by the assumption that n ě logppqp16R4 +1τ 4 +1 q{α4 +1, we have +2R1τ1 +"logppq +n +*1{2 +“ 2R1τ1 +"logppq +n +*1{4 "logppq +n +*1{4 +ď α1 +"logppq +n +*1{4 +. +Further by the assumption that 4}BLp qβq{Bβ}8 ď λ and 4α1tlogppq{nu1{4 ď λ in the lemma +statement we obtain +2R1τ1 +c +logppq +n +` }BLp qβq{Bβ}8 ď λ{4 ` λ{4 ď λ{2. + +32 +Combine with (A.26) and Lemma B.1 and subadditivity of ρλ in Condition (A2), we have +pα1 ´ µ{2q}pv}2 +2 +ď ρλp qβMcq ´ ρλp pβMcq ` λ{2 +"ρλppvMcq +λ +` }pvM}1 ` µ +2λ}pvMc}2 +2 +* +ď ρλp qβMcq ´ ρλp pβMcq ` ρλp qβMcq ` ρλp pβMcq +2 +` λ}pvM}1{2 ` µ{4}pv}2 +2. +(A.27) +Further, let br be the vector containing the first r element of vector b, and b´r the vec- +tor without the first r element. By the condition that CβM “ t, we have pβr +M “ C´1 +r pt ´ +Cm´r pβ´r +Mq, and qβr +M “ C´1 +r pt ´ Cm´r qβ´r +Mq. We have +}pvM}1 “ }C´1 +r Cm´rpv´r +M}1 ` }pv´r +M}1 +ď ?r}C´1 +r Cm´rpv´r +M}2 ` +? +m ´ r}pv´r +M}2 +ď p?r}C´1 +r Cm´r}2 ` +? +m ´ rq}pv}2. +(A.28) +Now (A.27) becomes +0 ď +ˆ +α1 ´ 3µ +4 +˙ +}pv}2 +2 ď 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ` λ}pvM}1{2. +(A.29) +We consider two cases, 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ą 0 and 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ď +0. When 3ρλp qβMcq ´ ρλp pβMcq ě 0, by Lemma B.2, we have +0 ď 3ρλp qβMcq ´ ρλp pβMcq ď 3λ}pvMcA}1 ´ λ}pvMcAc}1. +(A.30) +Now from (A.30) we further have +}pvMcAc}1 ď 3}pvMcA}1. +Substitue (A.28) and (A.30) into (A.29), we have +p2α1 ´ 3µ +2 q}pv}2 +2 ď 3λ}pvMcA}1 ´ λ}pvMcAc}1 ` λ}pvM}1 +ď 3λ}pvMcA}1 ` λp?r}C´1 +r Cm´r}2 ` +? +m ´ rq}pv}2 +ď 3λ +? +k}pvMcA}2 ` λp?r}C´1 +r Cm´r}2 ` +? +m ´ rq}pv}2 +ď t3λ +? +k ` λp?r}C´1 +r Cm´r}2 ` +? +m ´ rqu}pv}2. +Hence we have that when 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ą 0 +}pv}2 ď 6λ +? +k ` 2λp?r}C´1 +r Cm´r}2 ` ?m ´ rq +4α1 ´ 3µ +. +(A.31) +When 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ď 0, by (A.29) and (A.28) we have +ˆ +α1 ´ 3µ +4 +˙ +}pv}2 +2 ď λ}pvM}1{2 +ď λp?r}C´1 +r Cm´r}2 ` +? +m ´ rq}pv}2{2, + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +33 +which implies that when 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ă 0, +}pv}2 ď 2λp?r}C´1 +r Cm´r}2 ` ?m ´ rq +4α1 ´ 3µ +. +Together with (A.31), we always have +}pv}2 ď 6λ +? +k ` 2λp?r}C´1 +r Cm´r}2 ` ?m ´ rq +4α1 ´ 3µ +. +Further, the L1 distance is +}pv}1 ď }pvMcA}1 ` }pvMcAc}1 ` }pvM}1 +ď 4}pvMcA}1 ` }pvM}1 +ď 4 +? +k}pvMc}2 ` p?r}C´1 +r Cm´r}2 ` +? +m ´ rq}pv}2 +ď 4 +? +k}pv}2 ` p?r}C´1 +r Cm´r}2 ` +? +m ´ rq}pv}2 +ď p4 +? +k ` ?r}C´1 +r Cm´r}2 ` +? +m ´ rq6λ +? +k ` 2λp?r}C´1 +r Cm´r}2 ` ?m ´ rq +4α1 ´ 3µ +. +This proves the result. +A.2.2. Proof of Theorem 2. +This proof is very similar to the proof of Theorem 1 and is +simpler, hence is omitted. +A.3. Proof of Theorems in Section 4.3. +A.3.1. Supporting Theorem of Theorem 3 and its Proof. +Theorem A.1. +Let α1 “ min}β}1ďR1,}β}2ďR2 αminrEtexppβTXiqXiXT +i us{2. Assume +ρλ is µ-amenable, for µ ă α1, and Conditions (C1) – (C6) hold. Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu +and A1 “ tImˆm,0mˆpp´mqu. Write t1 ” ?r}C´1 +r Cm´r}2 ` ?m ´ r and t ” p6λ +? +k ` +2λt1qp4α1 ´ 3µq´1. Further we state the following two conditions. +(a) The parameters λ,R1,R2 satisfy +4max +! +}BLp qβq{Bβ}8,α1plogppq{nq1{4) +ď λ ď +« +α1 +" +16p4 +? +k ` t1qt{λ +*´1ff1{2 +, +max +" +2} qβ}1,2p4 +? +k ` t1qt +* +ď R1 ď min +´α1 +8λ,rnα2 +1{t64τ 2 +1 logppqus1{4,rnα4 +1{t16τ 4 +1 logppqus1{4¯ +. +with } qβ}1 ‰ p4 +? +k ` t1qt and max +! +2} qβ}2,2t +) +ď R2, with } qβ}2 ‰ t. +(b) Let pβMYS be the minimizer for (14), pβ “ p pβT +MYS,0T +p´m´kqT, pz and µ4 satisfy +BLp pβq +Bβ +´ AT Bqλp pβMcq +B pβMc +` λpATpzq ` AT +1 CTµ4 “ 0. +(A.32) + +34 +There exist δ P r4R1τ1 +a +logppq{n{λ,1s so that }pATpzqpMYSqc}8 ď 1 ´ δ, where we name +pATpzq as extended sub-gradient. In addition, +n ě maxrlogppqτ 2 +1 pm ` kq2{pα1 ´ µq2,4logppqτ 2 +1 tt1 ` p2{δ ` 1{2q +? +ku4{pα1 ´ µq2s, α1 ą µ +and βMc is k sparse. +Under the above conditions and the conditions (a) and (b), (7) has a unique local minimizer +pβ. +Proof: We follow the primal-dual witness construction introduced in Wainwright (2009). +Step i: Following Lemma B.10 in the supplementary material, let pβMYS be the minimizer +for (14). We can easily check that when replacing β by βMYS, X by XMYS, and p by m`k, +all the conditions of Theorem 1 are still satisfied. Here to check the condition regarding α1, +we note that for any β, +min +}β}2ďR2,}β}1ďR1 +αminrEtexppβTXiqXiXT +i us +“ +min +}β}2ďR2,}β}1ďR1 +inf +}v}2“1,vPRp vTEtexppβTXiqXiXT +i uv +ď +min +}β}2ďR2,}β}1ďR1 +inf +}v}2“1,vPRm`kpvT,0TqEtexppβTXiqXiXT +i upvT,0TqT +“ +min +}β}2ďR2,}β}1ďR1 +αminrEtexppβTXiqXi,MYSXT +i,MYSus +“ +min +}βMYS}2ďR2,}βMYS}1ďR1 +αminrEtexppβT +MYSXi,MYSqXi,MYSXT +i,MYSus. +Therefore, Theorem 1 applied to the m ` k dimensional case leads to +} pβMYS ´ qβMYS}1 ď p4 +? +k ` t1qt ď R1{2, +and +} pβMYS ´ qβMYS}2 ď t ď R2{2. +Therefore +} pβMYS}1 ď } pβMYS ´ qβMYS}1 ` } qβ}1 ă R1 +and +} pβMYS}2 ď } pβMYS ´ qβMYS}2 ` } qβ}2 ă R2. +Hence pβMYS and pβ must be in the interior of the feasible region. +Step ii: We show that pβ is a local minimum for (7) by verifying the conditions in Lemma +B.12 in the supplementary material. Because +Lpβq ` ρλpβMcq “ Lpβq ´ qλpβMcq ` λ}βMc}1, +we can write f “ L, g “ qλ, and px˚,v˚,w˚ +1,w˚ +2,µ˚ +1,µ˚ +2,µ˚ +3q “ p pβ,pz,ATpz,pz1,0,0,µ4q, +where z1 P B} pβ}2. Lemma B.4 in the supplementary material ensures the concavity and dif- +ferentiability of gpxq ´ µ{2}x}2 +2. Further, since µ˚ +1 “ µ˚ +2 “ 0, (B.2) and (B.3) are satisfied. + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +35 +(B.4) is satisfied by our construction in (A.32). Therefore, it remains to verify (B.5). We +first show that G˚ Ď RMYS so that (B.5) only needs to be satisfies for the vectors belong to +RMYS. Suppose this does not hold, let ν P G˚ such that supppνq Ę M Y S. This implies +there is an index j P pM Y Sqc such that νj ‰ 0. +Now we define ATz1 such that pATz1qk “ pATpzqk for k ‰ j, and pATz1qj “ signpνjq, +where ak is the kth element in vector a. Clearly z1 P B} pβMc}1 and +λνTATz1 ą λνTATpz +because }pATpzqpMYSqc}8 ă 1. Therefore, +νT +« +BLp pβq +Bβ +´ +# +AT Bqλp pβMcq +B pβMc ++ff +` λνTATz1 ` νTAT +1 CTµ4 +ą νT +« +BLp pβq +Bβ +´ +# +AT Bqλp pβMcq +B pβMc ++ff +` λνTATpz ` νAT +1 CTµ4 “ 0. +Now because ν P G˚, we have +νAT +1 CTµ4 “ 0. +This implies +νT +« +BLp pβq +Bβ +´ +# +AT Bqλp pβMcq +B pβMc ++ff +` λνTATz1 ą 0 +which contradicts with the requirement of G˚ that +sup +vPB} pβMc}1 +νT +« +BLp pβq +Bβ +´ +# +AT Bqλp pβMcq +B pβMc ++ff +` λνTATv “ 0. +Therefore, G˚ Ď RMYS. Now by construction suppp pβq Ă M Y S, using Lemma B.10 in the +supplementary material, we conclude that (B.5) holds with κ “ µ. Hence, all conditions of +Lemma B.12 in the supplementary material are satisfied, so we conclude pβ is an isolated +local minimum of (7). +Now by Lemma B.13, because suppp pβq Ď M Y S and pβ is an interior minimizer, the +support of any stationary point of (7) is a subset of MYS. Hence, we can write any stationary +point in the form of rβ “ t rβT +MYS,0T +p´m´kuT, where rβMYS is a stationary point for (14). +Further, note that (14) is strictly convex by Lemma B.10, and hence rβMYS is unique in +the feasible set and therefore pβMYS and rβ are unique. Hence pβ “ rβ is the unique local +minimum. This proves the result. +A.3.2. Supporting Theorem of Theorem 4 and its Proof. +Theorem A.2. +Let α1 “ min}β}1ďR1,}β}2ďR2 αminrEtexppβTXiqXiXT +i us{2. Assume +ρλ is µ-amenable, for µ ă α1, and Conditions (C1) – (C6) in the supplementary material hold. +Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu and A1 “ tImˆm,0mˆpp´mqu. Further assume that + +36 +(a) The parameters λ,R1,R2 satisfy +4max +! +}BLpβtq{Bβ}8,α1plogppq{nq1{4) +ď λ ď +» +–α1 +# +16p4 +? +k ` ?mq6 +? +k ` ?mq +4α1 ´ 3µ ++´1fi +fl +1{2 +, +max +# +2}βt}1,2p4 +? +k ` ?mqλ6 +? +k ` 2?m +4α1 ´ 3µ ++ +ď R1 +ď min +´α1 +8λ,rnα2 +1{t64τ 2 +1 logppqus1{4,rnα4 +1{t16τ 4 +1 logppqus1{4¯ +. +with +}βt}1 ‰ p4 +? +k ` ?mq6λ +? +k ` 2λ?m +4α1 ´ 3µ +and +max +# +2}βt}2,26λ +? +k ` 2λ?m +4α1 ´ 3µ ++ +ď R2, +with +}βt}2 ‰ 6λ +? +k ` 2λ?m +4α1 ´ 3µ +. +(b) Let pβaMYS be the minimizer for (15), pβa “ p pβT +aMYS,0T +p´m´kqT, pz satisfy +# +BLp pβaq +Bβa ++ +´ +# +AT Bqλp pβaMcq +B pβaMc ++ +` λpATpzq “ 0. +(A.33) +There exist δ P r4R1τ1 +a +logppq{n{λ,1s so that }pATpzqpMYSqc}8 ď 1 ´ δ, where we name +pATpzq as extended sub-gradient. In addition, +n ě maxrlogppqτ 2 +1 pm ` kq2{pα1 ´ µq2, +4logppqτ 2 +1 t?m ` p2{δ ` 1{2q +? +ku4{pα1 ´ µq2s, α1 ą µ +and βMc is k spar se. Under the above conditions and the conditions (a) and (b), (8) has a +unique local minimizer pβa. +Proof: The proof follows the same argument as those lead to Theorem A.1 hence is omitted. +A.3.3. Definition needed to prove Theorems 3 and 4. +Let β˚ is the point on the line +connecting pβ and qβ. We write +pQpβ˚q “ +# +pQpMYSqpMYSqpβ˚q pQpMYSqpMYSqcpβ˚q +pQpMYSqcpMYSqpβ˚q pQpMYSqcpMYSqcpβ˚q ++ +(A.34) +Then by the construction in (A.32), let pβ “ p pβT +MYS,0T +pMYSqcqT, pβMYS is the minimizer for +(14), then we have +pQpβ˚qp pβ ´ qβq ` +» +—– +! +BLp qβq +Bβ +) +MYS ´ +! +AT Bqλp pβMcq +BβMc +) +MYS +! +BLp qβq +Bβ +) +pMYSqc ´ +! +AT Bqλp pβMcq +BβMc +) +pMYSqc +fi +ffifl + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +37 +`λ +"pATpzqMYS +pATpzqpMYSqc +* +` +" +pAT +1 CTµ˚ +3qMYS +0pMYSqc +* +“ 0. +Taking the upper m ` k non-zero component, we get +pβMYS ´ qβMYS “ t pQMYS,MYSpβ˚qu´1 +« +´ +# +BLp qβq +Bβ ++ +MYS +` +# +AT Bqλp pβMcq +BβMc ++ +MYS +´λpATpzqMYS ´ pAT +1 CTµ˚ +3qMYS +ȷ +, +(A.35) +while taking the lower p ´ m ´ k components, this leads to +pATpzqpMYSqc “ λ´1 +» +– +# +AT Bqλp pβMcq +BβMc ++ +pMYSqc +´ +# +BLp qβq +Bβ ++ +pMYSqc +fi +fl +´λ´1 pQpMYSqcpMYSqpβ˚qp pβMYS ´ βMYSq +“ λ´1 +» +– +# +AT Bqλp pβMcq +BβMc ++ +pMYSqc +´ +# +BLp qβq +Bβ ++ +pMYSqc +fi +fl +`λ´1 pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 +ˆ +˜«# +BLp qβq +Bβ ++ +MYS +´ +# +AT Bqλp pβMcq +BβMc ++ +MYS +ff +` λpATpzqMYS ` pAT +1 CTµ˚ +3qMYS +¸ +. +Further by Condition (A4), we have +# +AT Bqλp pβMcq +BβMc ++ +pMYSqc +“ +«# +Bλ|pβj| +Bβj ++ +´ +# +Bρλppβjq +Bβj ++ +,j P pM Y Sqc +ffT +“ 0. +Therefore, we have +pATpzqpMYSqc “ λ´1 +» +–´ +# +BLp qβq +Bβ ++ +pMYSqc +fi +fl ` λ´1 pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 +ˆ +˜«# +BLp qβq +Bβ ++ +MYS +´ +# +AT Bqλp pβMcq +BβMc ++ +MYS +ff +` λpATpzqMYS ` pAT +1 CTµ˚ +3qMYS +¸ +. +By the condition that C pβM “ C qβ “ t, from (A.35), we have +0 “ CrImˆm,0mˆkst pQMYS,MYSpβ˚qu´1 +« +´ +# +BLp qβq +Bβ ++ +MYS +` +# +AT Bqλp pβMcq +BβMc ++ +MYS +´ λpATpzqMYS +ff +`CrImˆm,0mˆkst pQMYS,MYSpβ˚qu´1rImˆm,0mˆksTCTµ˚ +3, +which leads to +µ˚ +3 “ pCrImˆm,0mˆkst pQMYS,MYSpβ˚qu´1rImˆm,0mˆksTCTq´1CrImˆm,0mˆkst pQMYS,MYSpβ˚qu´1 + +38 +ˆ +« +´ +# +BLp qβq +Bβ ++ +MYS +` +# +AT Bqλp pβMcq +B qβMc ++ +MYS +´ λpATpzqMYS +ff +and +pA1CTµ˚ +3qMYS +“ A2t pQMYS,MYSpβ˚qu´1 +« +´ +# +BLp qβq +Bβ ++ +MYS +` +# +AT Bqλp pβMcq +BβMc ++ +MYS +´ λpATpzqMYS +ff +, +where +A2 “ rImˆm,0mˆksTCTpCrImˆm,0mˆkst pQMYS,MYSpβ˚qu´1rImˆm,0mˆksTCTq´1CrImˆm,0mˆks. +Hence, to use Theorem A.1, we must show that }pATpzqpMYSqc}8 ă 1. Define Qpβq “ +EtexppβTXqXXTu, and +A˚ +2 “ rImˆm,0mˆksTCTpCrImˆm,0mˆkstQMYS,MYSpβqu´1rImˆm,0mˆksTCTq´1CrImˆm,0mˆks. +A.3.4. Proof of Theorem 3. +First of all, pβpMYSqc “ 0 by construction. For any unit vec- +tor, recall that +vT B2Lpβq +BβBβT w “ exppβTWi ´ βTΩβ{2qvTtpWi ´ Ωβqb2 ´ Ωuw +Denoting ∇3Lpβq to be the third order gradient of L, we have +vT∇3Lpβqw +“ n´1 +nÿ +i“1 +exppβTWi ´ βTΩβ{2qvTtpWi ´ Ωβqb2 ´ ΩuwtWi ´ ΩβuT +´n´1 +nÿ +i“1 +exppβTWi ´ βTΩβ{2qtvTpWi ´ ΩβqwTΩ ` wTpWi ´ ΩβqvTΩu. +Hence define vectors v,w such that their jth element |vj| ą 0,|wj| ą 0 for j P M Y S, and +|vj| “ |wj| “ 0 for j R MYS, and }v}2 “ }w}2 “ 1. Firstly by Theorem 1 and the condition +that }hn}2 “ Ot +a +maxpm ` k ´ r,rq{nu, we have +} pβ ´ βt}2 “ } pβ ´ qβ}2 ` } qβ ´ βt}2 +ď C1λmaxp?r, +? +m ´ r, +? +kq ` +a +pm ` kq{n +ď C2 +"logppq +n +*1{4 ? +m ` k +for some constants C1,C2 ą 0. Further recall that K ” tv P RMYS : }v}2 ď 1u. +sup +v,wPK +ˇˇˇˇvT +" +pQpβ˚q ´ B2Lpβtq +BβBβT +* +w +ˇˇˇˇ +ď sup +v,wPK +«ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvTtpWi ´ Ωβ˚qb2 ´ ΩuwtWi ´ Ωβ˚uTp pβ ´ βtq +ˇˇˇˇ + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +39 +` +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq +ˇˇˇˇ +ff +ď sup +v,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qb2wpWi ´ Ωβ˚qTp pβ ´ βtq +´vTΩwpWi ´ Ωβ˚qTp pβ ´ βtqu +ˇˇˇˇ +` sup +v,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq +ˇˇˇˇ +ď sup +v,wPK +ˇˇˇˇ +n´1 +2 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qpv ` wqT +? +2 +pWi ´ Ωβ˚qb2 v ` w +? +2 pWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +` sup +v,wPK +ˇˇˇˇ +n´1 +2 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qpv ´ wqT +? +2 +pWi ´ Ωβ˚qb2 v ´ w +? +2 pWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +` sup +v,wPK +ˇˇˇˇ +pv ` wqT +? +2 +Ωv ` w +? +2 +n´1 +2 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +` sup +v,wPK +ˇˇˇˇ +pv ´ wqT +? +2 +Ωv ´ w +? +2 +n´1 +2 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +` sup +v,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq +ˇˇˇˇ +ď sup +vPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2vpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +`sup +vPK +ˇˇˇˇvTΩvn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +` sup +v,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq +ˇˇˇˇ +ď sup +vPK +n´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2v|pWi ´ Ωβ˚qTp pβ ´ βtq| +(A.36) +`sup +vPK +vTΩv +››››n´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qpWi ´ Ωβ˚q +›››› +2 +}p pβ ´ βtq}2 +(A.37) +`2 sup +v,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qtwTΩp pβ ´ βtqu +ˇˇˇˇ. +(A.38) +Now for (A.36), because by Condition (C1) in the supplementary material, we have +|pWi ´ Ωβ˚qTp pβ ´ βtq| +“ |pWi ´ Ωβ˚qTp pβ ´ βtq{} pβ ´ βt}2|} pβ ´ βt}2 +ď MW +? +m ` k} pβ ´ βt}2 ` }Ω}2}β˚}2} pβ ´ βt}2 + +40 +ď 2MW +? +m ` k} pβ ´ βt}2. +The last equality holds because }Ω}2 “ Op1q, }β˚}2 ď }βt}2 ` } pβ ´ βt}2 “ Opp1q ` +Op tlogppq{nu1{4 ? +m ` k “ opp +? +m ` kq. From Corollary B.4 in the supplementary mate- +rial, we have +sup +vPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2v sup +i +|pWi ´ Ωβ˚qTp pβ ´ βq| +´ Etexppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2vusup +i +|pWi ´ Ωβ˚qTp pβ ´ βq| +ˇˇˇˇ +ď 2MW +a +pm ` kq{n +? +m ` k} pβ ´ β}2 +with probability 1´Orexpt´pm`kqus. Further since supvPK Etexppβ˚TWi´β˚TΩβ˚{2qvTpWi´ +Ωβ˚qb2vu “ Op1q due to Condition C1(b), hence we get +sup +vPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2vpWi ´ Ωβ˚qTp pβ ´ βq +ˇˇˇˇ +ď sup +vPK +# +n´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2v ++ +sup +i +|pWi ´ Ωβ˚qTp pβ ´ βq| +ď t2MW +a +pm ` kq{n ` Opp1qu +? +m ` k} pβ ´ β}2 +“ Opp +? +m ` k} pβ ´ β}2q. +For (A.37), we first have +}n´1 +nÿ +i“1 +expp2β˚TWi ´ β˚TΩβ˚qpWi ´ Ωβ˚qb2 +MYS +´Etexpp2β˚TWi ´ β˚TΩβ˚qpWi ´ Ωβ˚qb2 +MYSu}2 +“ Opt +a +pm ` kq{nu +by Corollary B.3 in the supplementary material. Further }Etexpp2β˚TWi´β˚TΩβ˚qpWi´ +Ωβ˚qb2 +MYSu}2 “ Op1q. Hence (A.37) is of order Opp} pβ ´ β}2q. Now for (A.38), because +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚q +ˇˇˇˇ +2 +ď n´1 +nÿ +i“1 +texpp2β˚TWi ´ β˚TΩβ˚qvTpWi ´ Ωβ˚qb2v, +by Corollary B.3 in the supplementary material, we have +n´1 +nÿ +i“1 +expp2β˚TWi ´ β˚TΩβ˚qvTpWi ´ Ωβ˚qb2v +´Etexpp2β˚TWi ´ β˚TΩβ˚qvTpWi ´ Ωβ˚qb2vu “ Opp +a +pm ` kq{nq + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +41 +with probability of the order 1 ´ Otexppm ` kqu. Further because Etexpp2β˚TWi ´ +β˚TΩβ˚qvTpWi ´ Ωβ˚qb2vu “ Op1q, the third term of the last line is of the order +Opp} pβ ´ βt}2q. Therefore, we have +sup +v,wPK +ˇˇˇˇvT +" +pQpβ˚q ´ B2Lpβtq +BβBβT +t +* +w +ˇˇˇˇ “ Opp +? +m ` k} pβ ´ βt}2q +Hence for some positive constants C3, +sup +v,wPK +ˇˇˇˇvT +" +pQpβ˚q ´ B2Lpβtq +BβBβT +* +w +ˇˇˇˇ ď C3 +"logppq +n +*1{4 +pm ` kq +(A.39) +with probability 1 ´ Orexpt´pm ` kqus. Further, by Corollary B.2 in the supplementary +material, we also have +sup +v,wPK +ˇˇˇˇvT +„B2Lpβtq +BβBβT ´ Qpβtq +ȷ +w +ˇˇˇˇ “ sup +v,wPK +ˇˇˇˇvT +MYS +„B2Lpβtq +BβBβT ´ Qpβtq +ȷ +MYS,MYS +wMYS +ˇˇˇˇ +“ Opt +a +pm ` kq{nu +“ op +«"logppq +n +*1{4 +pm ` kq +ff +(A.40) +with probability 1 ´ 2expt´pm ` kqu. Further, recall that K1 ” tv P RpMYSqc : }v}2 “ +1,}v}0 “ 1u. +sup +v1PK1,wPK +ˇˇˇˇvT +1 +" +pQpβ˚q ´ B2Lpβtq +BβBβT +* +w +ˇˇˇˇ +ď +sup +v1PK1,wPK +«ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 tpWi ´ Ωβ˚qb2 ´ ΩuwtWi ´ Ωβ˚uTp pβ ´ βtq +ˇˇˇˇ +` +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvT +1 pWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq +ˇˇˇˇ +ff +ď +sup +v1PK1,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvT +1 pWi ´ Ωβ˚qb2wpWi ´ Ωβ˚qTp pβ ´ βtq +´vT +1 ΩwpWi ´ Ωβ˚qTp pβ ´ βtqu +ˇˇˇˇ +` +sup +v1PK1,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvT +1 pWi ´ Ωβ˚qwTΩ +`wTpWi ´ Ωβ˚qvT +1 Ωup pβ ´ βtq +ˇˇˇˇ +ď +sup +v1PK1,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 pWi ´ Ωβ˚qb2wpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +` sup +v1,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 ΩwpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ + +42 +` +sup +v1PK1,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvT +1 pWi ´ Ωβ˚qwTΩ +`wTpWi ´ Ωβ˚qvT +1 Ωup pβ ´ βtq +ˇˇˇˇ +ď +sup +v1PK1,wPK +n´1 +nÿ +i“1 +ˇˇˇˇexppβ˚TWi ´ β˚TΩβ˚{2qvT +1 pWi ´ Ωβ˚qb2w +ˇˇˇˇ +ˇˇˇˇpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +` +sup +v1PK1,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 ΩwpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +` +sup +v1PK1,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvT +1 pWi ´ Ωβ˚qwTΩ +`wTpWi ´ Ωβ˚qvT +1 Ωup pβ ´ βtq +ˇˇˇˇ +ď sup +v1PK1 +1 +2n´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 pWi ´ Ωβ˚qb2v1 sup +i +ˇˇˇˇpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +(A.41) +` sup +wPK +1 +2n´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qwTpWi ´ Ωβ˚qb2w sup +i +ˇˇˇˇpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +(A.42) +` +sup +v1PK1,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 ΩwpWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +(A.43) +` +sup +v1PK1,wPK +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtvT +1 pWi ´ Ωβ˚qwTΩ +`wTpWi ´ Ωβ˚qvT +1 Ωup pβ ´ βtq +ˇˇˇˇ. +(A.44) +For (A.41), from Corollary B.4, we have +sup +v1PK1 +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 pWi ´ Ωβ˚qb2v1 sup +i +|pWi ´ Ωβ˚qTp pβ ´ βtq| +´ Etexppβ˚TWi ´ β˚TΩβ˚{2qvT +1 pWi ´ Ωβ˚qb2v1usup +i +|pWi ´ Ωβ˚qTp pβ ´ βtq| +ˇˇˇˇ +ď 2MW +a +logppq{n +? +m ` k} pβ ´ βt}2 +with probability 1´Orexpt´logppqus. Further since supv1PK1 Etexppβ˚TWi´β˚TΩβ˚{2qvT +1 pWi´ +Ωβ˚qb2v1u “ Op1q due to Condition C1(b), hence we get +sup +v1PK1 +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 pWi ´ Ωβ˚qb2v1pWi ´ Ωβ˚qTp pβ ´ βtq +ˇˇˇˇ +ď sup +v1PK1 +# +n´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 pWi ´ Ωβ˚qb2v1 ++ +sup +i +|pWi ´ Ωβ˚qTp pβ ´ βtq| +ď t2MW +a +logppq{n ` Opp1qu +? +m ` k} pβ ´ βt}2 + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +43 +“ Opp +? +m ` k} pβ ´ β}2q. +For (A.42), we use the same argument as those lead to the order of (A.36), we have (A.42) is +of order Opp +? +m ` k} pβ ´ βt}2q. For (A.43), we use the same argument as those lead to the +order of (A.37), we have (A.43) is of order Opp} pβ ´ βt}2q. +Now for (A.44), because +ˇˇˇˇn´1 +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qvT +1 pWi ´ Ωβ˚q +ˇˇˇˇ +2 +ď n´1 +nÿ +i“1 +texpp2β˚TWi ´ β˚TΩβ˚qvT +1 pWi ´ Ωβ˚qb2v1, +by Corollary B.3 in the supplementary material, we have +n´1 +nÿ +i“1 +expp2β˚TWi ´ β˚TΩβ˚qvT +1 pWi ´ Ωβ˚qb2v1 +´Etexpp2β˚TWi ´ β˚TΩβ˚qvT +1 pWi ´ Ωβ˚qb2v1u “ Opp +a +logppq{nq +with probability of the order 1 ´ Orexpt´logppqus. Further because Etexpp2β˚TWi ´ +β˚TΩβ˚qvT +1 pWi ´ Ωβ˚qb2v1u “ Op1q, and because of the same argument as those lead to +(A.43), we conclude that (A.44) is of the order Opp} pβ ´ βt}2q. Hence follow (A.36), and by +Conditions (C1), Corollaries B.2–B.5 in the supplementary material, for positive constants +C4 we have +sup +v1PK1,wPK +ˇˇˇˇvT +1 +" +pQpβ˚q ´ B2Lpβq +BβBβT +* +w +ˇˇˇˇ “ Opp +? +m ` k} pβ ´ βt}2q +ď C4 +"logppq +n +*1{4 +pm ` kq +(A.45) +and +sup +v1PK1,wPK +ˇˇˇˇvT +1 +„B2Lpβtq +BβBβT ´ Qpβtq +ȷ +w +ˇˇˇˇ +“ +sup +v1PK1,wPK +ˇˇˇˇvT +1pMYSqc +„B2Lpβtq +BβBβT ´ Qpβtq +ȷ +pMYSqc,MYS +wMYS +ˇˇˇˇ +“ Opr +a +maxtpm ` kq,logppqu{ns +“ op +«"logppq +n +*1{4 +pm ` kq +ff +, +(A.46) +with probabilities to the order of 1 ´ 2expr´maxtpm ` kq,logppqus. +Combine (A.39) and (A.40) with Lemma B.11 in the supplementary material, +}pt pQpMYSq,MYSqpβ˚qu´1 ´ tQpMYSq,MYSpβtqu´1}2 +ď +}tQpMYSq,MYSpβtqu´1}2 +2} pQpMYSq,MYSpβ˚q ´ QpMYSq,MYSpβtq}2 +t1 ´ }tQpMYSq,MYSpβtqu´1}2} pQpMYSq,MYSpβ˚q ´ QpMYSq,MYSpβtq}2u + +44 +“ Opp} pQpMYSq,MYSpβ˚q ´ Q0pMYSq,MYSpβtq}2q +“ C9 +«"logppq +n +*1{4 +pm ` kq +ff +, +(A.47) +and hence +}pt pQpMYSq,MYSqpβ˚qu´1 ´ tQpMYSq,MYSpβtqu´1}8 +“ C9pm ` kq3{2 +"logppq +n +*1{4 +. +(A.48) +Further, +›››› pQpMYSqc,MYSpβ˚qt pQpMYSq,MYSpβ˚qu´1 +# +BLp qβq +Bβ ++ +MYS +›››› +8 +ď +››››r pQpMYSqc,MYSpβ˚q ´ QpMYSqc,MYSpβtqst pQpMYSq,MYSpβ˚qu´1 +# +BLp qβq +Bβ ++ +MYS +›››› +8 +` +››››QpMYSqc,MYSpβtqpt pQpMYSq,MYSpβ˚qu´1 +´tQpMYSq,MYSpβtqu´1q +# +BLp qβq +Bβ ++ +MYS +›››› +8 +` +››››QpMYSqc,MYSpβtqtQpMYSq,MYSpβtqu´1 +# +BLp qβq +Bβ ++ +MYS +›››› +8 +ď +sup +v1PK1,wPK +ˇˇˇˇvT +1pMYSqcr pQpMYSqc,MYSpβ˚q ´ QpMYSqc,MYSpβtqswMYS +ˇˇˇˇ +ˆ}t pQpMYSq,MYSpβ˚qu´1}2 +›››› +# +BLp qβq +Bβ ++ +MYS +›››› +2 +`}QpMYSqc,MYSpβtq}2 +››››t pQpMYSq,MYSpβ˚qu´1 ´ tQpMYSq,MYSpβtqu´1 +›››› +2 +›››› +# +BLp qβq +Bβ ++ +MYS +›››› +2 +` +››››QpMYSqc,MYSpβtqtQ0pMYSq,MYSpβtqu´1 +# +BLp qβq +Bβ ++ +MYS +›››› +8 +ď C10 +«"logppq +n +*1{4 +pm ` kq +ff +? +m ` k +a +logppq{n +(A.49) +`C11 +«"logppq +n +*1{4 +pm ` kq +ff +? +m ` k +a +logppq{n +(A.50) +` +››››QpMYSqc,MYSpβtqtQpMYSq,MYSpβtqu´1 +# +BLp qβq +Bβ ++ +MYS +›››› +8 +. +(A.49) is obtained by using (A.45) and (A.46) and the fact that +}t pQpMYSq,MYSpβ˚qu´1}2 + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +45 +ď }t pQpMYSq,MYSpβ˚qu´1 ´ rQpMYSq,MYSpβtqs´1}2 ` }tQpMYSq,MYSpβtqu´1}2 +ď Opp1q. +(A.50) is obtained by using (A.39), (A.40) and the fact that }QpMYSqc,MYSpβq}2 “ Op1q. +Therefore, now note that n ě c8pm ` kq4logppq and λ “ Ortlogppq{nu1{4s by the statement +assumption, together with Condition (C4), we have +›››› pQpMYSqc,MYSpβ˚qt pQpMYSq,MYSpβ˚qu´1 +# +BLp qβq +Bβ ++ +MYS +›››› +8 +ď +››››QpMYSqc,MYSpβtqtQpMYSq,MYSpβtqu´1 +# +BLp qβq +Bβ ++ +MYS +›››› +8 +` oppλq +ď +››››QpMYSqc,MYSpβqtQpMYSq,MYSpβtqu´1 +›››› +2 +›››› +# +BLp qβq +Bβ ++ +MYS +›››› +2 +` oppλq +“ +? +m ` k +a +logppq{n ` oppλq +“ oppλq. +(A.51) +Therefore +}pATpzqpMYSqc}8 +“ }λ´1 +» +–´ +# +BLp qβq +Bβ ++ +pMYSqc +fi +fl ` λ´1 pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 +ˆ +˜«# +BLp qβq +Bβ ++ +MYS +´ +# +AT Bqλp pβMcq +BβMc ++ +MYS +ff +` λpATpzqMYS ` pAT +1 CTµ˚ +3qpMYSq +¸ +}8 +ď λ´1 +›››› +» +–´ +# +BLp qβq +Bβ ++ +pMYSqc +fi +fl +›››› +8 +`λ´1 +›››› pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 +# +BLp qβq +Bβ ++ +MYS +›››› +8 +`λ´1 +›››› pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 +˜« +´ +# +AT Bqλp pβMcq +BβMc ++ +MYS +ff +` λpATpzqMYS +¸›››› +8 +`λ´1} pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1pAT +1 CTµ˚ +3qMYS}8 +“ λ´1 +›››› pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 +˜« +´ +# +AT Bqλp pβMcq +BβMc ++ +MYS +ff +` λpATpzqMYS +¸›››› +8 +`λ´1 +›››› pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1 +ˆ +«# +AT Bqλp pβMcq +BβMc ++ +MYS +´ λpATpzqMYS +ff›››› +8 +` op1q. +(A.52) + +46 +Now recall (A.35) and +pA1CTµ˚ +3qMYS +(A.53) +“ A2t pQMYS,MYSpβ˚qu´1 +« +´ +# +BLp qβq +Bβ ++ +MYS +` +# +AT Bqλp pβMcq +BβMc ++ +MYS +´ λpATpzqMYS +ff +. +Hence +} pβMYS ´ qβMYS}8 +ď }t pQMYS,MYSpβ˚qu´1 +# +BLp qβq +Bβ ++ +MYS +}8 +` +››››t pQMYS,MYSpβ˚qu´1 +«# +AT Bqλp pβMcq +BβMc ++ +MYS +´ λpATpzqMYS +ff›››› +8 +`}t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1 +ˆ +« +´ +# +BLp qβq +Bβ ++ +MYS +` +# +AT Bqλp pβMcq +BβMc ++ +MYS +´ λpATpzqMYS +ff +}8 +ď oppλq ` λ}t pQMYS,MYSpβ˚qu´1}8 ` λ}t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1}8 +“ oppλq ` 2λc8 ` 2λc8, +(A.54) +where oppλq in the second inequality is obtained by using the same argument as those lead to +(A.51). The other two terms are obtained by using Lemma B.4 such that +›››› +«# +AT Bqλp pβMcq +BβMc ++ +MYS +´ λpATpzqMYS +ff›››› +8 +ď λ. +The last equality holds because +}t pQMYS,MYSpβ˚qu´1}8 +ď }t pQMYS,MYSpβ˚qu´1 ´ tQpMYSq,MYSpβtqu´1}8 ` }tQpMYSq,MYSpβtqu´1}8 +ď 2c8 +by (A.48). And similarly, we have +}t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1}8 +ď }t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1 +´tQMYS,MYSpβtqu´1A2tQMYS,MYSpβtqu´1}8 +`}tQMYS,MYSpβtqu´1A2tQMYS,MYSpβtqu´1}8 +“ 2c8. +Now because minp|βj|q ě λpγ ` 5c8q for j P S, we have +min|pβj| ě λγ + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +47 +and q1 +λppβqj “ λsignppβjq “ λpATpzqj. Hence by Lemma B.3 in the supplementary material, +we have +# +AT Bqλp pβMcq +BβMc ++ +MYS +´ λpATpzqMYS “ 0. +(A.55) +Inserting (A.55) into (A.52), the first two terms of (A.52) are zero hence }pATzqpMYSqc} “ +opp1q ă 1. This implies that pβ has support in MYS. Further, because of (A.54), pβ is unique +in the feasible set. Therefore, using (A.35) together with (A.55) and (A.53), we have +pβMYS ´ qβMYS +“ pt pQMYS,MYSpβ˚qu´1 ´ t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1q +ˆ +« +´ +# +BLp qβq +Bβ ++ +MYS +ff +“ ´ptQMYS,MYSpβqu´1 ´ tQMYS,MYSpβtqu´1A˚ +2tQMYS,MYSpβtqu´1q +ˆ +«# +BLp qβq +Bβ ++ +MYS +ff +t1 ` opp1qu +and pβpMYSqc “ 0. Now because +BLp qβq +Bβ +´ BLpβtq +Bβ +“ B2Lpβ˚q +BβBβT p qβ ´ βtq +“ Qpβqp qβ ´ βtq ` +"B2Lpβtq +BβBβT ´ Qpβtq +* +p qβ ´ βtq ` +"B2Lpβ˚q +BβBβT ´ B2Lpβtq +BβBβT +* +p qβ ´ βtq +“ Qpβtqp qβ ´ βtqt1 ` opp1qu. +The last equality holds because +} +"B2Lpβtq +BβBβT ´ Qpβtq +* +}2 “ opp1q, +by (A.40), +} +# +B2Lp rβq +BβBβT ´ B2Lpβtq +BβBβT ++ +}2 “ opp1q +by (A.39), and }Qpβtq}2 “ Op1q. Therefore, +pβMYS ´ qβMYS “ ´ptQMYS,MYSpβtqu´1 ´ tQMYS,MYSpβtqu´1A˚ +2tQMYS,MYSpβtqu´1q +ˆ +„"BLpβtq +Bβ +` Qpβtqp qβ ´ βtq +* +MYS +ȷ +t1 ` opp1qu. +Further, we have +pβMYS ´ βtMYS + +48 +“ pβMYS ´ qβMYS ´ rpCCTq´1C,0rˆksThn +“ ´ptQMYS,MYSpβtq´1u ´ tQMYS,MYSpβtqu´1A˚ +2tQMYS,MYSpβtqu´1q +„"BLpβtq +Bβ +* +MYS +`QMYS,MYSpβtqrpCCTq´1C,0rˆksThn +ȷ +t1 ` opp1qu ´ rpCCTq´1C,0rˆksThn +“ ´tpQMYS,MYSpβtqu´1 ´ tQMYS,MYSpβtqu´1A˚ +2tQMYS,MYSpβtqu´1q +"BLpβtq +Bβ +* +MYS +t1 ` opp1qu +´tQMYS,MYSpβtqu´1A˚ +2rtpCCTq´1C,0rˆkuThnst1 ` opp1qu. +A.3.5. Proof of Theorem 4. +The proof is very similar to that of Theorem 3 hence is +omitted. +A.4. Proof of Results in Section 4.4. +A.4.1. Proof of Lemma 1. +First note that +Ψ´1{2pΣ,Q,βtqωn +“ ´?nΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtq +"BLpβtq +Bβ +* +MYS +“ Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtq +ˆn´1{2 +nÿ +i“1 +tYiWi ´ exppβT +t Wi ´ βT +t Ωβt{2qpWi ´ ΩβtquMYS. +It is easy to see that +nÿ +i“1 +cov +´ +Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtq +ˆn´1{2tYiWi ´ exppβT +t Wi ´ βT +t Ωβt{2qpWi ´ ΩβtquMYS +¯ +“ Irˆr. +(A.56) +Further, +r1{4 +nÿ +i“1 +E}n´1{2Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtq +ˆtYiWi ´ exppβT +t Wi ´ βT +t Ωβt{2qpWi ´ ΩβtquMYS}3 +2 +ď r1{4 +nÿ +i“1 +E}n´1{2tYiΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqWiMYS +´exppβT +t Wi ´ βT +t Ωβt{2qΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqpWi ´ ΩβtqMYSu}3 +2 +ď r1{4 +nÿ +i“1 +E}n´1{2tYiΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqWiMYS + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +49 +´exppβT +t Wi ´ βT +t Ωβt{2qΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqpWi ´ ΩβtqMYSu}3 +2 +ď r1{4D +nÿ +i“1 +n´3{2p}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqWiMYS}3 +2 +`}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqpΩβtqMYS}3 +2q +“ opp1q, +(A.57) +where D is a positive constant. The second to the last inequality holds by Condition (D1). +Also because +}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqWiMYS}2 +ď }Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtq}2}WiMYS}2 +ď }Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtq}2|WT +i v|{}v}2 +ď Op1qMW +? +m ` k “ Op +? +m ` kq, +where v is a p dimensional vector with }v}0 “ m ` k, the last inequality holds by Condition +(C1) (a). Further since }Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqpΩβtqMYS}2 “ +Op1q, +r1{4D +nÿ +i“1 +n´3{2p}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqWiMYS}3 +2 +`}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqpΩβtqMYS}3 +2q +“ Otr1{4pm ` kq3{2n´1{2u “ opp1q, +by the Condition that n ě c8pm`kq4logppq. Combine (A.56) and (A.57), and using Lemma +B.14 in Section B.4.3, let Z is a standard Gaussian random variable we have +lim +nÑ8sup +C +|PrpΨ´1{2pΣ,Q,βtqωn P Cq ´ PrpZ P Cq| “ 0. +(A.58) +Now we choose +Cx “ tz : }z ´ ?nΨ´1{2hn}2 ď xu, +then +lim +nÑ8|PrpΨ´1{2pΣ,Q,βtqωn P Cxq ´ PrpZ P Cxq| “ 0, +which implies +lim +nÑ8|PrpT0 ď xq ´ Prtχ2pr,nhT +nΨ´1pΣ,Q,βtqhnq ď xu| “ 0, +where χ2pr,γq is a non-central chi-square distribution, with non-centrality parameter γ. + +50 +A.4.2. Proof of Theorem 5. +From Theorem 4, we have +pβaMYS ´ βtMYS “ ´tQMYS,MYSpβtqu´1 +"BLpβtq +Bβ +* +MYS +t1 ` opp1qu +Hence, +?nCp pβaM ´ βtMq “ ´?nCrImˆm,0mˆkstQMYS,MYSpβtqu´1 +"BLpβtq +Bβ +* +MYS +t1 ` opp1qu +“ ωnt1 ` opp1qu. +Further, because CβtM “ t ` hn, we have +?nΨ´1{2ppΣ, pQ, pβaqpC pβaM ´ tq “ Ψ´1{2ppΣ, pQ, pβaqωnt1 ` opp1qu ` ?nΨ´1{2ppΣ, pQ, pβaqhn. +Further because hn “ Ot +a +maxpm ` k ´ r,rq{nu, +npC pβaM ´ tqTΨ´1ppΣ, pQ, pβaqpC pβaM ´ tq +“ pωn ` ?nhnqTΨ´1pΣ,Q,βtqpωn ` ?nhnqt1 ` opp1qu +`pωn ` ?nhnqTtΨ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtqupωn ` ?nhnq +ď pωn ` ?nhnqTΨ´1pΣ,Q,βtqpωn ` ?nhnqt1 ` opp1qu +`}ωn ` ?nhn}2 +2}Ψ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtq}2 +“ T0 ` opprq. +(A.59) +The last equality holds because T0 converge in distribution to a non-central chi-square distri- +bution with degree freedom r as shown in Lemma 1 and hence T0 is of the order Opprq. Also +because +}ωn ` ?nhn}2 +2 ď pωn ` ?nhnqTΨ´1pΣ,Q,βtqpωn ` ?nhnq{αmintΨ´1pΣ,Q,βtqu +ď pωn ` ?nhnqTΨ´1pΣ,Q,βtqpωn ` ?nhnq{cΨ +by Condition (D2), we have +}Ψ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtq}2}ωn ` ?nhn}2 +2 +ď +«"logppq +n +*1{4 +pm ` kq +ff +OppT0q. +Therefore, TW ´ T0 “ opprq. +A.4.3. Proof of Theorem 6. +From Theorem 3, we have +pβMYS ´ βtMYS +“ ´ptQMYS,MYSpβtqu´1 ´ tQMYS,MYSpβtqu´1A˚ +2tQMYS,MYSpβtqu´1q +"BLpβtq +Bβ +* +MYS +t1 ` opp1qu +`tQMYS,MYSpβtqu´1A˚ +2rtpCCTq´1C,0rˆkuThnst1 ` opp1qu + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +51 +and pβpMYSqc “ 0. By Taylor expansion we have +# +BLp pβq +Bβ ++ +MYS +“ +"BLpβtq +Bβ +* +MYS +` +"B2Lpβ˚q +BβBβT +* +MYS +p pβ ´ βtqMYS +“ +"BLpβtq +Bβ +* +MYS +` QMYS,MYSpβtqp pβ ´ βtqMYSt1 ` opp1qu +“ A˚ +2tQMYS,MYSpβtqu´1 +"BLpβtq +Bβ +* +MYS +t1 ` opp1qu +`tA˚ +2rtpCCTq´1C,0rˆkuThnst1 ` opp1qu +where β˚ is a point in between βt and pβ. The second equality holds by (A.39) and (A.40) in +Appendix. Therefore, we have +?nΨppΣ, pQ, pβq´1{2CrImˆm,0mˆkst pQMYS,MYSp pβqu´1A˚ +2tQMYS,MYSpβtqu´1 +"BLpβtq +Bβ +* +MYS +“ ?nΨppΣ, pQ, pβq´1{2CrImˆm,0mˆkstQMYS,MYSpβtqu´1 +"BLpβtq +Bβ +* +MYS +t1 ` opp1qu +“ ΨppΣ, pQ, pβq´1{2ωnt1 ` opp1qu, +The first equality holds by (A.39) and (A.40) in the supplementary material. And similarly +?nΨppΣ, pQ, pβq´1{2CrImˆm,0mˆkst pQMYS,MYSp pβqu´1tA˚ +2rtpCCTq´1C,0rˆkuThns +“ ?nΨppΣ, pQ, pβq´1{2hnt1 ` opp1qu +and hence +TS “ pωn ` ?nhnqTΨppΣ, pQ, pβq´1pωn ` ?nhnqt1 ` opp1qu +Now the same steps in (A.59) lead to TS ´ T0 “ opprq. +Appendix B. +B.1. Some Lemmas on the Penalty Function. +Lemma B.1. +Conditions (A1)–(A4) imply that ρλ is λ-Lipschitz and all sub gra- +dients and derivatives of ρλ are bounded by λ in magnitude. Conditions (A1)–(A5) +imply +λ}βMc}1 ď ρλpβMcq ` µ{2}βMc}2 +2,@βMc P Rp´m +Proof: This lemma is a direct consequence of Lemma 4 in Loh & Wainwright (2015). + +52 +Lemma B.2. +Suppose ρλ satisfies Conditions (A1)–(A5). Let v P Rp´m, let A +be the index set of k largest elements of v in magnitude, and let Ac be the index set +of the remaining p ´ m ´ k elements of v. Suppose ξ ą 0 and satisfies +ξρλpvAq ´ ρλpvAcq ě 0. +Then +ξρλpvAq ´ ρλpvAcq ď λpξ}vA}1 ´ }vAc}1q. +Moreover, if βtMc P Rp´m is k-sparse, then for a vector βMc P Rp´m such that +ξρλpβtMcq ´ ρλpβMcq ą 0 and ξ ě 1, we have +ξρλpβtMcq ´ ρλpβMcq ď λpξ}vA}1 ´ }vAc}1q, +where v “ βMc ´ βtMc, A is the index set of the k largest elements of v in magni- +tude, and Ac is the index set of the remaining p ´ m ´ k elements of v. +Proof: This lemma is a direct consequence of Lemma 5 in Loh & Wainwright (2015). +Lemma B.3. +Suppose ρλ is pµ,γq-amenable, |pβj| ě λγ for j P M Y S, then +q1 +λppβjq “ λsignppβjq. +Proof: Because ρλ is pµ,γq-amenable, ρ1ppβjq “ 0 by Condition (A6) and (A1). Hence +q1 +λppβjq “ Bλ|pβj|{Bpβj “ λsignppβjq. This proves the result. +Lemma B.4. +Consider a µ-amenable regularizer ρλ. Then +(a) |ρ1 +λptq| ď λ for all t ‰ 0. +(b) The function qλptq ´ µ{2t2 is concave and everywhere differentiable, where +qλptq “ λ|t| ´ ρλptq. +Proof: This lemma is a direct consequence of Lemma 5 in Loh & Wainwright (2017). +B.2. Some Lemmas on the Criterion Function. +Lemma B.5. +Assume Conditions (C1) – (C4) hold. There exists a constant c1 ą 0 +so that +pr +« +n´1} +nÿ +i“1 +tYiWi ´ exppβT +t Wi ´ βT +t Ωβt{2qpWi ´ Ωβtqu}8 ą c1 +a +logppq{n +ff +ď 6p´1. +Proof: The lemma is the direct consequence of Corollary 1 in Jiang & Ma (2021). We +omit the proof here. +Lemma B.6. +Assume that Conditions (C1), (C4) and (C6) hold, then for +any β with }β}2 ď R2, and for sufficiently large n and p, with probability 1 ´ + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +53 +Orexpt´?nlogpus, n´1 řn +i“1 exppβTWi ´βTΩβ{2qtpWi ´Ωβqb2 ´Ωu satisfies +the lower and upper-RE conditions with +α1 “ +min +}β}1ďR1,}β}2ďR2 +αminrEtexppβTXiqXiXT +i us{2, +α2 “ +max +}β}1ďR1,}β}2ďR2 +3αmaxrEtexppβTXiqXiXT +i us{2 +and τpn,pq “ τ1 +a +logppq{n for a bounded positive constant τ1. +Proof: The lemma is a direct consequence of Lemma 12 in Jiang & Ma (2021) with +c “ 1. +Corollary B.1. +Assume Conditions (C1)–(C4) to hold, }hn}2 “ Ot +a +maxpm ` k ´ r,rq{nu +and m ` k “ opn1{3q, then there exists a positive constant c10 so that +}BLp qβq +Bβ +}8 ď c10 maxr +a +pm ` kq{n, +a +logppq{ns, +with probability 1 ´ Opp´1q ´ Orexpt´ +a +nlogppqus. +Proof: +}BLp qβq +Bβ +}8 +“ n´1} +nÿ +i“1 +tYiWi ´ expp qβTWi ´ qβTΩ qβ{2qpWi ´ Ω qβqu}8 +ď n´1} +nÿ +i“1 +tYiWi ´ exppβtTWi ´ βtTΩβt{2qpWi ´ Ωβtqu}8 +`n´1} +nÿ +i“1 +expp qβTWi ´ qβTΩ qβ{2qpWi ´ qβTΩqu ´ exppβtTWi ´ βtTΩβt{2qpWi ´ Ωβtqu}8. +Let K0 “ tv P Rp : vMc “ 0,}v}2 “ 1u. Then, +n´1} +nÿ +i“1 +expp qβTWi ´ qβTΩ qβ{2qpWi ´ qβTΩq ´ exppβtTWi ´ βtTΩβt{2qpWi ´ Ωβtq}8 +ď n´1} +nÿ +i“1 +exppβ˚TWi ´ β˚TΩβ˚{2qtpWi ´ Ωβ˚qb2 ´ Ωut qβ ´ βtu}2 +ď sup +vPK +n´1 +nÿ +i“1 +vT exppβ˚TWi ´ β˚TΩβ˚{2qtpWi ´ Ωβ˚qb2 ´ Ωuv} qβ ´ βt}2 +ď 3αmaxrEtexppβTXiqXiXT +i us{2}CTpCCTq´1}2}hn}2 ` τ1m +a +logppq{n}CTpCCTq´1}2}hn}2 +ď c10 maxr +a +pm ` kq{n, +a +logppq{ns + +54 +for some constants c10. The first inequality holds by the Taylor expansion with +β˚ be the point on the line connecting qβ and β. The second inequality holds be- +cause qβ ´ β only has m nonzero elements supported in M, and hence only the +corresponding m ˆ m sub-matrix in n´1 řn +i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtpWi ´ +Ωβ˚qb2 ´ Ωu contributes to the L2 norm. The third inequality holds by Lemma B.6 +and the fact that v only as m non-zero elements. The last equality holds because +}hn}2 “ Ot +a +maxpm ` k ´ r,rq{nu ď Op +a +m ` k{nq, and m +a +m ` k{n Ñ 0 be- +cause m ` k “ opn1{3q. This proves the results. +Lemma B.7. +Assume that Conditions (C1) and (C6) hold. If Xi,Ui P Rp, define +K ” tv P RMYS : }v}2 ď 1u, K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ 1u +Pr +˜ +sup +v,wPK +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą nt +¸ +ď 2exp +ˆ +´min +„ +nt2 +324e2M4 +, +nt +36eM5logpnq +ȷ +` 2pm ` kqlogp9q +˙ +, +Pr +˜ +sup +vPK1,wPK +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą nt +¸ +ď 2exp +ˆ +´min +„ +nt2 +36e2M4 +, +nt +12eM5logpnq +ȷ +` pm ` kqlogp9q ` logppq +˙ +, +and +Pr +˜ +sup +vPK1,wPK1 +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą nt +¸ +ď 2exp +ˆ +´min +„ +nt2 +16e2M4 +, +nt +8eM5logpnq +ȷ +` 2logppq +˙ +. +Proof: By Lemma 1 statement 3 and Lemma 3 statement 3 in Jiang & Ma (2021), +we can see that the square of a conditional sub-Gaussian variable is sub-exponential. +Now because vTpWi ´ βTΩq given Xi and βTWi is normal, and hence vTpWi ´ +βTΩqb2v is conditional sub-exponential. Now by the Cauchy–Schwarz inequal- +ity, and without loss of generality we assume vTpWi ´ βTΩqb2v ě wTpWi ´ +βTΩqb2w, we have +|vTpWi ´ βTΩqb2w| ď vTpWi ´ βTΩqb2v. +Hence, by Lemma 3 statement 3 in Jiang & Ma (2021), we have for some bounded +positive K3pβTWi,Xiq, +Erexpt|vTpWi ´ βTΩqb2w|{K3pβTWi,Xiqu|βTWi,Xis +ď Erexpt|vTpWi ´ βTΩqb2v|{K3pβTWi,Xiqu|βTWi,Xis ď e +Hence, vTpWi ´ βTΩqb2w is also conditional sub-exponential variable. Therefore, +we have that +gpWi,β,v,wq ´ EtgpWi,β,v,wq|βTWi,Xiu + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +55 +is centered sub-exponential. Then using the same argument as those that lead to +Lemma 8 in Jiang & Ma (2021) and Condition (C6), for any unit vectors v,w, we +can show that +Pr +˜ +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą nt +¸ +ď 2exp +ˆ +´min +„ +nt2 +16e2M4 +, +nt +8eM5logpnq +ȷ˙ +. +Now we define B “ tu1,...,uru Ă K to be a 1/3-cover of K, if for every v P K, +there is some ui P B such that }v ´ ui}2 ď 1{3. Define ∆v “ v ´ uj where uj “ +arg minui }v ´ ui}2. We have }∆v}2 ď 1{3. Similarly define uk “ arg minui }w ´ +ui}2 for w P K. By ?, we can construct B with |B| ă 9pm`kq. Now for v1,v2 P K, +define +Φpv1,v2q “ vT +1 +« nÿ +i“1 +ApβTWiqtpWi ´ Ωβqb2 ´ Ωu ´ EtexppβTXiqXiXT +i u +n +ff +v2. +We have +|Φpv,wq| “ |Φp∆v ` uj,∆w ` ukq| +ď max +j,k |Φpuj,ukq| ` max +i +|Φp∆v,uiq| ` max +i +|Φpui,∆wq| ` |Φp∆v,∆wq|. +Since }3∆v}2 ď 1 and suppp3∆vq Ď K, 3∆v P K. It follows that +sup +v,wPK +|Φpv,wq| +ď max +j,k |Φpuj,ukq| ` 1{3sup +vPK +max +i +|Φp3∆v,uiq| ` 1{3 sup +wPK +max +i +|Φpuk,3∆wq| ` 1{9 sup +v,wPK +|Φp3∆v,3∆wq| +ď max +j,k |Φpuj,ukq| ` 2{3tsup +vPK +|Φp3∆v,3∆vq|u1{2tmax +i +|Φpui,uiq|u1{2 ` 1{9 sup +v,wPK +|Φpv,wq| +ď max +j,k |Φpuj,ukq| ` sup +v,wPK +t2{3|Φpv,wq| ` 1{9|Φpv,wq|u. +Hence, supv,wPK |Φpv,wq| ď 9{2maxj,k |Φpuj,ukq|. By a union bound while con- +sidering |B| ă 92pm`kq, we have +Pr +˜ +sup +v,wPK +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą 9{2nt +¸ +ď Pr +˜ +max +j,k | +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą nt +¸ +ď Pr +˜ +max +j,k | +nÿ +i“1 +rApβTWiqgpWi,β,v{}v}2,w{}w}2q ´ v{}v}T +2 EtexppβTXiqXiXT +i uw{}w}2s| ą nt +¸ +ď 2exp +ˆ +´min +„ +nt2 +16e2M4 +, +nt +8eM5logpnq +ȷ +` 2pm ` kqlogp9q +˙ +. + +56 +It also follows that +sup +vPK1,wPK +|Φpv,wq| +ď max +vPK1,j |Φpv,ujq| ` 1{3 +sup +vPK1,wPK +|Φpv,3∆wq| +ď max +vPK1,j |Φpv,ujq| ` 1{3 +sup +vPK1,wPK +t|Φpv,wq|, +so supvPK1,wPK |Φpv,wq| ď 3{2maxvPK1,j |Φpv,ujq|. Because v P K1 is a vector +with a single nonzero entry 1, there are only p ´ m ´ k elements in K1. We thus have +Pr +˜ +sup +vPK1,wPK +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą 3{2nt +¸ +ď Pr +˜ +max +vPK1,uj +| +nÿ +i“1 +rApβTWiqgpWi,β,v,ujq ´ vTEtexppβTXiqXiXT +i uujs| ą nt +¸ +ď 2exp +ˆ +´min +„ +nt2 +16e2M4 +, +nt +8eM5logpnq +ȷ +` pm ` kqlogp9q ` logppq +˙ +. +This proves the result. Further, since K1 contains only unit vectors, we have +Pr +˜ +sup +vPK1,wPK1 +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą nt +¸ +ď 2exp +ˆ +´min +„ +nt2 +16e2M4 +, +nt +8eM5logpnq +ȷ +` 2logppq +˙ +. +Corollary B.2. +Assume that Conditions (C1), (C4) and (C6) hold and m ` k “ +otn{log2pnqu. If Xi,Ui P Rp, define K ” tv P RMYS : }v}2 ď 1u, K1 ” tv P +RpMYSqc : }v}2 “ 1,}v}0 “ 1u, there are a0 ą 0,a1 ą 0,b1 ą 0 such that +Pr +˜ +sup +v,wPK +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq +´vTEtexppβTXiqXiXT +i uws| ą na0 +c +m ` k +n +¸ +ď 2expt´pm ` kqu, +Pr +˜ +sup +vPK1,wPK +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq +´vTEtexppβTXiqXiXT +i uws| ą na1 +c +maxrm ` k,logppqs +n +¸ +ď 2expr´maxtlogppq,pm ` kqus, + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +57 +and +Pr +˜ +sup +vPK1,wPK1 +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq +´vTEtexppβTXiqXiXT +i uws| ą nb1 +c +logppq +n +¸ +ď 2expr´logppqs. +Proof: By Lemma B.7, take t “ a0 +a +pm ` kq{n, we have +Pr +˜ +sup +v,wPK +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą nt +¸ +ď 2exp +˜ +´min +« +a2 +0pm ` kq +324e2M4 +, a0 +a +npm ` kq +36eM5logpnq +ff +` 2pm ` kqlogp9q +¸ +“ 2exp +ˆ +´a2 +0pm ` kq +324e2M4 +` 2pm ` kq +˙ +“ 2expt´pm ` kqu +The second to the last equality holds because m`k “ otn{log2pnqu. The last equality +holds by choosing a0 “ +a +972e2M4. +Further, by the second relation in Lemma B.7, take t “ a1 +a +maxtlogppq,pm ` kqu{n, +we have +Pr +˜ +sup +vPK1,wPK +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą nt +¸ +ď 2exp +ˆ +´min +„a2 +1 maxtlogppq,pm ` kqu +36e2M4 +, +a1 +a +nmaxtlogppq,pm ` kqu +12eM5logpnq +ff +` pm ` kqlogp9q ` logppq +¸ +“ 2exp +" +´min +ˆa2 +1 maxtlogppq,pm ` kqu +36e2M4 +, +a1 maxrlogppq?n{t12eM5 +a +logppqlogpnqu, +a +npm ` kq{t12eM5logpnqus +¯ +`2pm ` kq ` logppq +˙ +ď 2exp +ˆ +´min +ˆa2 +1 maxtlogppq,pm ` kqu +36e2M4 +, +a1 maxrlogppq{p12eM5Cq, +a +npm ` kq{t12eM5logpnqus +¯ + +58 +`2pm ` kq ` logppq +* +ď 2exp +" +´min +„a2 +1 maxtlogppq,pm ` kqu +36e2M4 +, +a1 maxtlogppq{p12eM5Cq,pm ` kq{p12eM5qus ` 3maxtpm ` kq,logppqu +* +ď 2expp´a2 maxtlogppq,pm ` kqu ` 3maxtlogppq,pm ` kquq +“ 2expr´maxtlogppq,pm ` kqus, +where a2 “ minta2 +1{p36e2M4q,a1{p12eM5Cq,a1{p12eM5qu, and we select a1 such +that a2 ě 4. The third equality holds because logpnq ď C +a +n{logppq by Condition +(C4). The fourth equality holds pm ` kq “ otn{log2pnqu. +In addition, by the third relation in Lemma B.7, take t “ b1 +a +logppq{n, we have +Pr +˜ +sup +vPK1,wPK1 +| +nÿ +i“1 +rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT +i uws| ą nt +¸ +ď 2exp +˜ +´min +« +b2 +1logppq +16e2M4 +, b1 +a +nlogppq +8eM5logpnq +ff +` 2logppq +¸ +“ 2exp +" +´min +ˆb2 +1logppq +16e2M4 +,b1logppq?n{t8eM5 +a +logppqlogpnqu +˙ +` 2logppq +˙ +ď 2exp +ˆ +´min +ˆb2 +1logppq +16e2M4 +,b1logppq{p8eM5Cq +˙ +` 2logppq +* +ď 2expp´b12logppq ` 2logppqq +“ 2expr´logppqs, +where b12 “ mintb2 +1{p16e2M4q,b1{p8eM5Cqu, and we select b1 such that b12 ě 3. +The second inequality holds because logpnq ď C +a +n{logppq by Condition (C4). +Lemma B.8. +Assume that Conditions (C1) and (C7) hold. If Xi,Ui P Rp, define +K ” tv P RMYS : }v}2 ď 1u, K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ 1u. Then +Pr +˜ +sup +v,wPK +| +nÿ +i“1 +rA2pβTWiqg1pWi,β,v,wq +´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws ą nt +¯ +ď 2exp +ˆ +´min +„ +nt2 +324e2M6 +, +nt +36eM7logpnq +ȷ +` 2pm ` kqlogp9q +˙ +, +Pr +˜ +sup +vPK1,wPK +| +nÿ +i“1 +rA2pβTWiqg1pWi,β,v,wq + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +59 +´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| ą nt +˙ +ď 2exp +" +´min +„ +nt2 +36e2M6 +, +nt +12eM7logpnq +ȷ +` pm ` kqlogp9q ` logppq +* +, +and +Pr +˜ +sup +vPK1,wPK1 +| +nÿ +i“1 +rA2pβTWiqg1pWi,β,v,wq +´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| ą nt +˙ +ď 2exp +" +´min +„ +nt2 +16e2M6 +, +nt +8eM7logpnq +ȷ +` 2logppq +* +. +Proof: The lemma holds by using the same arguments as those lead to Lemma B.7. +Corollary B.3. +Assume that Conditions (C1), (C4) (C7) hold, m`k “ otn{log2pnqu. +If Xi,Ui P Rp, define K ” tv P RMYS : }v}2 ď 1u, K1 ” tv P RpMYSqc : }v}2 “ +1,}v}0 “ 1u. There are a2 ą 0,a3 ą 0,b3 ą 0 such that +Pr +˜ +sup +v,wPK +| +nÿ +i“1 +rA2pβTWiqg1pWi,β,v,wq +´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| ą na2 +c +pm ` kq +n +¸ +ď 2expt´pm ` kqu, +Pr +˜ +sup +vPK1,wPK +| +nÿ +i“1 +rA2pβTWiqg1pWi,β,v,wq +´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| +ą na3 +c +maxrpm ` kq,logppqs +n +¸ +ď 2expr´maxtlogppq,pm ` kqus. +and +Pr +˜ +sup +vPK1,wPK1 +| +nÿ +i“1 +rA2pβTWiqg1pWi,β,v,wq +´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| +ą nb3 +c +logppq +n +¸ +ď 2expt´logppqu. + +60 +Proof: The corollary follows the same arguments as those lead to Corollary B.2. +Corollary B.4. +Assume that Conditions (C1), (C4) (C7) hold, m`k “ otn{log2pnqu. +If Xi,Ui P Rp, define K ” tv P RMYS : }v}2 ď 1u, K1 ” tv P RpMYSqc : }v}2 “ +1,}v}0 “ 1u. There are a21 ą 0,a31 ą 0,b31 ą 0 such that +Pr +˜ +sup +v,wPK +| +nÿ +i“1 +rApβTWiqg1pWi,β,v,wq +´vTEtexppβTWi ´ βTΩβ{2qpWi ´ Ωβqb2uws| ą na21 +c +pm ` kq +n +¸ +ď 2expt´pm ` kqu, +Pr +˜ +sup +vPK1,wPK +| +nÿ +i“1 +rApβTWiqg1pWi,β,v,wq +´vTEtexppβTWi ´ βTΩβ{2qpWi ´ Ωβqb2uws| +ą na31 +c +maxrpm ` kq,logppqs +n +¸ +ď 2expr´maxtlogppq,pm ` kqus, +and +Pr +˜ +sup +vPK1,wPK1 +| +nÿ +i“1 +rApβTWiqg1pWi,β,v,wq +´vTEtexppβTWi ´ βTΩβ{2qpWi ´ Ωβqb2uws| +ą nb31 +c +logppq +n +¸ +ď 2expt´logppqu. +Proof: The corollary follows the same arguments as those that lead to Corollary B.3. +Lemma B.9. +Assume that Conditions (C1) holds, m ` k “ otn{log2pnqu. If +Xi,Ui P Rp, K ” tv P RMYS : }v}2 “ 1u, K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ +1u. Then there exists constants c2,c3,c4,c5,c6,c7,c8,c9,c10 ą 0 +Pr +˜ +sup +v,wPK +| +nÿ +i“1 +tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| ą nt +¸ +ď 2exp +` +´c2nminpc3t2,c4tq ` 2pm ` kqlogp9q +˘ +, +Pr +˜ +sup +vPK1,wPK +| +nÿ +i“1 +tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| ą nt +¸ + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +61 +ď 2exp +` +´c7nminpc5t2,c6tq ` pm ` kqlogp9q ` logppq +˘ +, +and +Pr +˜ +sup +vPK1,wPK1 +| +nÿ +i“1 +tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| ą nt +¸ +ď 2exp +` +´c10nminpc8t2,c9tq ` 2logppq +˘ +. +Proof: The lemma is a consequence of Lemma 15 in Loh & Wainwright (2012) and +using the same arguments as those lead to Lemma B.7. +Corollary B.5. +Assume that Conditions (C1) holds. If Xi,Ui P Rp, K ” tv P +RMYS : }v}2 ď 1u, K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ 1u. There are constants +a4 ą 0,a5 ą 0,b5 ą 0 such that +Pr +˜ +sup +v,wPK +| +nÿ +i“1 +tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| +ą na4 +c +pm ` kq +n +¸ +ď 2expt´pm ` kqu, +Pr +˜ +sup +vPK1,wPK +| +nÿ +i“1 +tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| +ą na5 +c +maxrpm ` kq,logppqs +n +¸ +ď 2expr´maxtlogppq,pm ` kqus, +and +Pr +˜ +sup +vPK1,wPK1 +| +nÿ +i“1 +tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| +ą nb5 +c +logppq +n +¸ +ď 2expt´logppqu. +Proof: The corollary follows the same arguments as those lead to Corollary B.2. +B.3. Lemmas on Criterion Function and Penalty Function. +Lemma B.10. +Consider a µ-amenable regularizer ρλ, with µ ă α1 and n ą +logppqτ2 +1 pm ` kq2{pα1 ´ µq2, where α1,τ1 are defined in Lemma B.6. Then the func- +tion Lpβq´µ}βMc}2 +2{2 and Lpβq`ρλpβMcq are strictly convex on β P RMYS, and +hence the restricted program (14) is also strictly convex. +Proof: First define a vector v P Rp with the jth element |vj| ą 0 if j P M Y S, and +}vj} “ 0, otherwise. By Lemma B.6, we have for β in the feasible set of program + +62 +(14), +vTB2Lpβq +BβBβT v ě α1}v}2 +2 ´ τ1 +c +logppq +n +}v}2 +1, +and }v}1 ď +? +m ` k}v}2, and hence we have +vTB2Lpβq +BβBβT v ě +# +α1 ´ τ1pm ` kq +c +logppq +n ++ +}v}2 +2. +Therefore, +vT +MYS +"B2Lpβq +BβBβT +* +pM`SqpM`Sq +vMYS ´ µvT +SvS +ě +# +α1 ´ µ ´ τ1pm ` kq +c +logppq +n ++ +}v}2 +2, +where +␣ +B2Lpβq{pBβBβTq +( +pMYSq,pMYSq is the pm ` kq ˆ pm ` kq block of +␣ +B2Lpβq{BβBβT( +corresponding to M Y S. Hence Lpβq ´ µ}βMc}2 +2{2 is strictly +convex on RMYS. +Finally, because +Lpβq ´ qλpβMcq “ pLpβq ´ µ}βMc}2 +2{2q ` tµ}βMc}2 +2{2 ´ qλpβMcqu, +where the second part is convex over RMYS by Lemma B.4, hence Lpβq ´ +qλpβMcq restricted to RMYS is strictly convex. Because Lpβq ` ρλpβMcq “ Lpβq ´ +qλpβMcq ` λ}βMc}1, the strict convexity of Lpβq ` ρλpβMcq over RMYS follows. +This proves the result. +Now as we know C contains r independent columns, without loss of generality, +we write C “ pCr,Cm´rq where Cr is a full rank square matrix. +Lemma B.11. +Let A,B P Rpˆp be invertible. For any matrix norm }}, we have +}A´1 ´ B´1} ď +}A´1}2}A ´ B} +1 ´ }A´1}}A ´ B}. +In particular, if }A´1}}A ´ B} ď 1{2, then }A´1 ´ B´1} “ Op}A´1}2}A ´ B}q. +Proof: This lemma is Lemma 11 in Loh & Wainwright (2017). +Lemma B.12. +Suppose x˚ is feasible for the program +min +x tfpxq ´ gpxMcq ` λ}xMc}1u, such that }x}1 ď R1,}x}2 ď R2, and CxM “ t, +(B.1) +where f P C2, g P C1 and gpxMcq ´ κ{2}xMc}2 +2 is concave and C is an r ˆ m ma- +trix. Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu and A1 “ tImˆm,0mˆpp´mqu. Assume +there are v˚ P B}x˚ +Mc}1, w˚ +1 P B}x˚}1, w˚ +2 P B}x˚}2, µ˚ +1 ě 0, µ˚ +2 ě 0, µ˚ +3 such that +µ˚ +1pR1 ´ }x˚}1q “ 0 +(B.2) +µ˚ +2pR2 ´ }x˚}2q “ 0 +(B.3) + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +63 +Bfpx˚q +Bx˚ +´ ATBgpx˚ +Mcq +Bx˚ +Mc +` λATv˚ ` µ˚ +1w˚ +1 ` µ˚ +2w˚ +2 ` AT +1 CTµ˚ +3 “ 0 +(B.4) +sTB2fpxq +BxBxT s ą κ,@s P G˚, +(B.5) +where +G˚ :“ +# +s P Rp : }s}2 “ 1; +sup +w1PB}x˚}1 +sTw1 ď 0 if }x˚}1 “ R1; +sup +w2PB}x˚}2 +sTw2 ď 0 if }x˚}2 “ R2; +sup +vPB}x˚ +Mc}1 +sT +ˆBfpx˚q +Bx˚ +´ ATBgpx˚ +Mcq +Bx˚ +Mc +˙ +` λsTATv “ 0; +µ˚ +1 +sup +w1PB}x˚}1 +sTw1 “ 0,µ˚ +2 +sup +w2PB}x˚}2 +sTw2 “ 0;CsM “ 0 ++ +. +Then x˚ is an isolated local minimum of the program (B.1). +Proof: The proof of this lemma is similar to the proof of Theorem 3 in Fletcher & +Watson (1980) and that of Lemma 10 in Loh & Wainwright (2017), except that we +allow additional constraints }x}2 ď R2 and CxM “ t. +Suppose x˚ is not an isolated local minimum. Then there is a sequence pxkq, so +that xk Ñ x˚ and +φpxkq ď φpx˚q, +where φpxq “ fpxq ´ gpxMcq ` λ}xMc}1. Let sk :“ pxk ´ x˚q{}xk ´ x˚}2, so pskq +is a set of feasible directions. Since pskq Ă B2p1q, where B2p1q is the ball with radius +1, the set must possess a point of accumulation s P B2p1q, and we can extract a con- +vergence subsequence such that pskq Ñ s. With a slight abuse of notation, we still use +pskq to denote the subsequence. We will show that s P G˚. +First of all by the construction, xk’s are all feasible, and hence Cxk +M “ t. There- +fore, Csk +M “ 0, take the limits on the left and right of the equation implies +CsM “ 0. +(B.6) +Further, because the feasible region is closed, s is also in the feasible direction at +x˚. If }x˚}1 “ R1, by the sub-gradient of convexity function }x˚}1 we have +0 ě }xk}1 ´ }x˚}1 “ }x˚ ` }xk ´ x˚}2sk}1 ´ }x˚}1 ě }xk ´ x˚}2skTw1, +for any w1 P B}x˚}1. When k Ñ 8, this leads to +sup +w1PB}x˚}1 +sTw1 ď 0. +(B.7) +Further (B.2) also implies that if }x˚}1 ‰ R1, then µ˚ +1 “ 0. Since µ˚ +1 ě 0, hence we +have +µ˚ +1 +sup +w1PB}x˚}1 +sTw1 ď 0. +(B.8) + +64 +Similarly, by the sub-gradient of convexity function }x˚}2 we have +0 ě }xk}2 ´ }x˚}2 “ }x˚ ` }xk ´ x˚}2sk}2 ´ }x˚}2 ě }xk ´ x˚}2skTw2, +for any w2 P B}x˚}2. When k Ñ 8, this leads to +sup +w2PB}x˚}2 +sTw2 ď 0. +(B.9) +Further (B.3) also implies that if }x˚}2 ‰ R2, then µ˚ +2 “ 0. Since µ˚ +2 ě 0, hence we +have +µ˚ +2 +sup +w2PB}x˚}2 +sTw2 ď 0. +(B.10) +Further by (B.4), we have +sTBfpx˚q +Bx˚ +´ sTATBgpx˚ +Mcq +Bx˚ +Mc +` sTλATv˚ ` µ˚ +1sTw˚ +1 ` µ˚ +2sTw˚ +2 ` sTAT +1 CTµ˚T +3 “ 0 +which, together with (B.6), implies +sTBfpx˚q +Bx˚ +´ sTATBgpx˚ +Mcq +Bx˚ +Mc +` sTλATv˚ “ ´µ˚ +1sTw˚ +1 ´ µ˚ +2sTw˚ +2 ě 0. +(B.11) +By the definition of sub-gradient, we have +}x˚ +Mc ` }xk ´ x˚}2Ask}1 ´ }x˚ +Mc}1 ě }xk ´ x˚}2skTATv +for all v P B}x˚ +Mc}1 and all k. Further because }x˚ +Mc `}xk ´x˚}2Ask}1 ´}x˚ +Mc}1 “ +}xk +Mc}1 ´ }x˚ +Mc}1, we have +sTATv “ lim +kÑ8skTATv ď lim +kÑ8 +}xk +Mc}1 ´ }x˚ +Mc}1 +}xk ´ x˚}2 +(B.12) +for all v P B}x˚ +Mc}1. Furthermore, +sTBfpx˚q +Bx˚ +´ sTATBgpx˚ +Mcq +Bx˚ +Mc +“ lim +kÑ8skTBfpx˚q +Bx˚ +´ skTATBgpx˚ +Mcq +Bx˚ +Mc +“ lim +kÑ8 +xxk ´ x˚, Bfpx˚q +Bx˚ +´ AT Bgpx˚ +Mcq +Bx˚ +Mc y +}xk ´ x˚}2 +“ lim +kÑ8 +fpxkq ´ gpxk +Mcq ´ fpx˚q ` gpx˚ +Mcq +}xk ´ x˚}2 +, +(B.13) +since xk Ñ x˚ and fpxkq´gpxk +Mcq P C1. Combining (B.12) and (B.13), we conclude +that +sup +vPB}x˚ +Mc}1 +sTBfpx˚q +Bx˚ +´ sTATBgpx˚ +Mcq +Bx˚ +Mc +` λsTATv + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +65 +ď lim +kÑ8 +φpxkq ´ φpx˚q +}xk ´ x ˚ } +ď 0. +Combining with (B.11), we have +0 ď sTBfpx˚q +Bx˚ +´ sTATBgpx˚ +Mcq +Bx˚ +Mc +` λsTATv˚ +ď +sup +vPB}x˚ +Mc}1 +sTBfpx˚q +Bx˚ +´ sTATBgpx˚ +Mcq +Bx˚ +Mc +` λsTATv +ď 0, +hence +sup +vPB}x˚ +Mc}1 +sTBfpx˚q +Bx˚ +´ sTATBgpx˚ +Mcq +Bx˚ +Mc +` λsTATv “ 0 +(B.14) +Now together with (B.4), we have +µ˚ +1sTw˚ +1 ` µ˚ +2sTw˚ +2 “ 0. +Further by (B.8) and (B.10), we have +µ˚ +1 +sup +w1PB}x˚}1 +sTw1 “ µ˚ +2 +sup +w2PB}x˚}2 +sTw2 “ 0. +(B.15) +Combine (B.6), (B.7), (B.9), (B.14) and (B.15), we conclude that s P G˚. +By the convexity of the L1 norm, we have +}xk}1 ´ }x˚}1 “ }x˚ ` pxk ´ x˚q}1 ´ }x˚}1 ě pxk ´ x˚qTw1 “ xkTw1 ´ }x˚}1 +for all w1 P B}x˚}1. Therefore, xkTw˚ +1 ď }xk}1 ď R1. Similarly, by the convexity of +the L2 norm, we have +}xk}2 ´ }x˚}2 “ }x˚ ` pxk ´ x˚q}2 ´ }x˚}2 ě pxk ´ x˚qTw2 “ xkTw2 ´ }x˚}2 +for all w2 P B}x˚}2. Therefore, xkTw˚ +2 ď }xk}2 ď R2. Further, xkTAT +1 CTµ˚ +3 ´ +tTµ˚ +3 “ 0. Hence +φpxkq “ fpxkq ´ gpxk +Mcq ` λ}xk +Mc}1 +ě fpxkq ´ gpxk +Mcq ` λxkT +Mcv˚ ` µ˚ +1pxkTw˚ +1 ´ R1q ` µ˚ +2pxkTw˚ +2 ´ R2q +`xkTAT +1 CTµ˚ +3 ´ tTµ˚ +3 +for all v˚ P B}x˚ +Mc}1, and +φpx˚q “ fpx˚q ´ gpx˚ +Mcq ` λx˚T +Mcv˚ ` µ˚ +1px˚Tw˚ +1 ´ R1q ` µ˚ +2px˚Tw˚ +2 ´ R2q +`x˚TAT +1 CTµ˚ +3 ´ tTµ˚ +3. +The equality holds because µ˚ +1px˚Tw˚ +1 ´ R1q “ 0 and µ˚ +2px˚Tw˚ +2 ´ R2q “ 0. Hence +lim +kÑ8tφpxkq ´ φpx˚qu{}xk ´ x˚}2 +2 +ě lim +kÑ8tfpxkq ´ fpx˚q ´ gpxk +Mcq ` gpx˚ +Mcq ` xλATv˚ ` µ˚ +1w˚ +1 ` µ˚ +2w˚ +2 + +66 +`AT +1 CTµ˚ +3,xk ´ x˚yu{}xk ´ x˚}2 +2 +“ lim +kÑ8tfpxkq ´ fpx˚q ´ gpxk +Mcq ` gpx˚ +Mcqu{}xk ´ x˚}2 +2 +´ +BBfpx˚q +Bx˚ +´ ATBgpx˚ +Mcq +Bx˚ +Mc +,xk ´ x˚ +F +{}xk ´ x˚}2 +2 +“ lim +kÑ8 +" +fpxkq ´ fpx˚q ´ +BBfpx˚q +Bx˚ +,xk ´ x˚ +F* +{}xk ´ x˚}2 +2 +´ +" +gpxk +Mcq ´ gpx˚ +Mcq ´ +B +ATBgpx˚ +Mcq +Bx˚ +Mc +,xk ´ x˚ +F* +{}xk ´ x˚}2 +2. +(B.16) +By the concavity of gpxMcq ´ κ{2}xMc}2 +2, we have +" +gpxk +Mcq ´ gpx˚ +Mcq ´ +B +ATBgpx˚ +Mcq +Bx˚ +Mc +,xk ´ x˚ +F* +ď κ{2}xk +Mc ´ x˚ +Mc}2 +2. +Further note that φpxkq ´ φpx˚q ď 0. Combine with (B.16), we have +lim +kÑ8 +" +fpxkq ´ fpx˚q ´ +BBfpx˚q +Bx˚ +,xk ´ x˚ +F* +{}xk ´ x˚}2 +2 +´κ{2}xk +Mc ´ x˚ +Mc}2 +2{}xk ´ x˚}2 +2 ď 0, +which by Taylor expansion implies +sTB2fpx˚q +BxBxT s “ lim +kÑ8pxk ´ x˚qTB2fpx˚q +BxBxT pxk ´ x˚q{}xk ´ x˚}2 +2 +ď lim +kÑ8tκ}xk +Mc ´ x˚ +Mc}2 +2 ` op}pxk +Mc ´ x˚ +Mcq}2 +2qu{}xk ´ x˚}2 +2 +ď κ, +which contradicts with (B.5). Hence x˚ must be an isolated local minimum. +Corollary B.6. +Suppose x˚ is feasible for the program +min +x tfpxq ´ gpxMcq ` λ}xMc}1u, such that }x}1 ď R1,}x}2 ď R2, +(B.17) +where f P C2, g P C1 and gpxMcq ´ κ{2}xMc}2 +2 is concave and C is an r ˆ m ma- +trix. Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu and A1 “ tImˆm,0mˆpp´mqu. Assume +there are v˚ P B}x˚ +Mc}1, w˚ +1 P B}x˚}1, w˚ +2 P B}x˚}2, µ˚ +1 ě 0, µ˚ +2 ě 0 such that +µ˚ +1pR1 ´ }x˚}1q “ 0 +µ˚ +2pR2 ´ }x˚}2q “ 0 +Bfpx˚q +Bx˚ +´ ATBgpx˚ +Mcq +Bx˚ +Mc +` λATv˚ ` µ˚ +1w˚ +1 ` µ˚ +2w˚ +2 “ 0 +sTB2fpxq +BxBxT s ą κ,@s P G˚, +where +G˚ :“ +# +s P Rp : }s}2 “ 1; +sup +w1PB}x˚}1 +sTw1 ď 0 if }x˚}1 “ R1; +sup +w2PB}x˚}2 +sTw2 ď 0 if }x˚}2 “ R2; + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +67 +sup +vPB}x˚ +Mc}1 +sT +ˆBfpx˚q +Bx˚ +´ ATBgpx˚ +Mcq +Bx˚ +Mc +˙ +` λsTATv “ 0; +µ˚ +1 +sup +w1PB}x˚}1 +sTw1 “ 0,µ˚ +2 +sup +w2PB}x˚}2 +sTw2 “ 0 ++ +. +Then x˚ is an isolated local minimum of the program (B.17). +Proof: The Corollary holds by using the same argument as those lead to Lemma B.12, +while ignoring the equality constraint. +Lemma B.13. +Assume Conditions (C1)–(C6) hold, λ ď α1{p8R1q, R1 ď rnα2 +1{t64τ2 +1 logppqus1{4, +δ P r4R1τ1 +a +logppq{n{λ,1s, n ě 4logppqτ2 +1 t?r}C´1 +r Cm´r}2 ` ?m ´ r ` p2{δ ` +1{2q +? +ku4{pα1 ´µq2, and α1 ą µ. Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu and A1 “ +tImˆm,0mˆpp´mqu. Suppose rβ is a stationary point of program (7) and pβ is the inte- +rior local minimizer of (7) and satisfies suppp pβq Ď M Y S. Then suppp rβq Ď M Y S. +Proof: Let rv :“ rβ ´ pβ, by the Taylor expansion of the first order derivative +tBLp rβq{BβT ´ BLp pβq{BβTurv “ rvTB2Lpβ˚q{BβBβTrv, +where β˚ is a point on the line connecting pβ and rβ and hence in the feasible set. +Therefore, by Lemma B.6 we have +tBLp rβq{BβT ´ BLp pβq{BβTurv ě α1}rv}2 +2 ´ τ1 +a +logppq{n}rv}2 +1. +We first show that }rv}2 ď 1. Suppose that }rv}2 ą 1, we have +tBLp rβq{BβT ´ BLp pβq{BβTurv ě α1}rv}2 ´ 2τ1R1 +a +logppq{n}rv}1. +(B.18) +The first order condition implies +! +BLp rβq{Bβ ` ATBρλp rβMcq{BβMc +)T +p pβ ´ rβq ě 0 +(B.19) +and hence +BLp rβq{BβTrv ď ´ATBρλp rβMcq{BβT +Mcrv. +Combine with (B.18), we have +t´ATBρλp rβMcq{BβMc ´ BLp pβq{BβuTrv ě α1}rv}2 ´ 2τ1R1 +a +logppq{n}rv}1. +(B.20) +Further because pβ is an interior local minimum, we have +BLp pβq{Bβ ` ATBρλp pβMcq{BβMc ` AT +1 CTµ4 “ 0. +for some Lagrange multiplier µ4 ą 0. Note that pAT +1 CTµ4qTrv “ µT +t Cp rβM ´ +pβmMq “ 0. Therefore, plug in (B.20), we have +α1}rv}2 ´ 2τ1R1 +a +logppq{n}rv}1 +ď tATBρλp pβMcq{BβMc ` AT +1 CTµ4 ´ ATBρλp rβMcq{BβMcuTrv + +68 +“ tATBρλp pβMcq{BβMc ´ ATBρλp rβMcq{BβMcuTrv +ď t}ATBρλp pβMcq{BβMc}8 ` }ATBρλp rβMcq{BβMc}8u}rv}1 +ď 2λ}rv}1, +where }ATBρλpβMcq{BβMc}8 ď λ holds by Lemma B.4. Hence we have +}rv}2 ď t2λ ` 2τ1R1 +a +logppq{nu}rv}1{α1 ď 2R1t2λ ` 2τ1R1 +a +logppq{nu{α1. +The right hand side is at most 1 because λ ď α1{p8R1q, and n ě logppq64τ2 +1 R4 +1{α2 +1, +which contradicts with }rv}2 ą 1. Therefore }rv}2 ď 1. +Now note that +Lpβq ´ qλpβMcq “ pLpβq ´ µ}βMc}2 +2{2q ` tµ}βMc}2 +2{2 ´ qλpβMcqu +and tµ}βMc}2 +2{2 ´ qλpβMcqu is convex by Lemma B.4, and hence for any β in the +feasible set, we have +rvTB2tLpβq ´ qλpβMcqu +BβBβT +rv ě rvTB2Lpβq +BβBβT rv ´ µrvTATArv +ě pα1 ´ µq}rv}2 +2 ´ τ1 +c +logppq +n +}rv}2 +1. +(B.21) +Further by (B.19), we have +0 ď +# +BLp rβq +BβT ´ Bqλp rβMcq +BβT +Mc +A ++ +p pβ ´ rβq ` λrzTAp pβ ´ rβq, +where rz P B} rβMc}1. Further, because pβ is an interior local minimum, for pz P +B} pβMc}1, we have +0 “ BLp pβq{Bβ ` ATBρλp pβMcq{BβMc ` AT +1 CTµ4 +“ +# +BLp pβq +Bβ +´ ATBqλp pβMcq +BβMc ++ +` λATpz ` AT +1 CTµ4, +which leads to +# +BLp pβq +BβT ´ Bqλp pβMcq +BβT +Mc +A ++ +rv ` λppzTAqrv “ 0. +Hence +0 ď +# +BLp pβq +BβT ´ Bqλp pβMcq +BβT +Mc +A ++ +rv ´ +# +BLp rβq +BβT ´ Bqλp rβMcq +BβT +Mc +A ++ +rv +`λppzTAqp rβ ´ pβq ´ λprzTAqp rβ ´ pβq +“ +# +BLp pβq +BβT ´ Bqλp pβMcq +BβT +Mc +A ++ +rv ´ +# +BLp rβq +BβT ´ Bqλp rβMcq +BβT +Mc +A ++ +rv +`λppzTAq rβ ´ λ} pβMc}1 ´ λ} rβMc}1 ` λprzTAq pβ, + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +69 +which implies +λ} rβMc}1 ´ λppzTAq rβ +ď +# +BLp pβq +BβT ´ Bqλp pβMcq +BβT +Mc +A ++ +rv ´ +# +BLp rβq +BβT ´ Bqλp rβMcq +BβT +Mc +A ++ +rv ´ λ} pβMc}1 ` λprzTAq pβ +ď +# +BLp pβq +BβT ´ Bqλp pβMcq +BβT +Mc +A ++ +rv ´ +# +BLp rβq +BβT ´ Bqλp rβMcq +BβT +Mc +A ++ +rv +ď τ1 +c +logppq +n +}rv}2 +1 ´ pα1 ´ µq}rv}2 +2. +(B.22) +The second inequality holds because |przTAq pβ| “ |rzT pβMc| ď }rz}8} pβMc}1 ď +} pβMc}1, and last inequality holds by the Taylor expansion and (B.21). +Now we first assume that the following statement holds: If }pATpzqMYS}8 ď 1´δ +and λ ě 4R1τ1 +a +logppq{pnq{δ, then +}rv}1 ď t?r}C´1 +r Cm´r}2 ` +? +m ´ r ` p2{δ ` 1{2q +? +ku}rv}2. +We then will have +λ} rβMc}1 ´ λppzTAq rβ +ď +« +τ1 +c +logppq +n +t?r}C´1 +r Cm´r}2 ` +? +m ´ r ` p2{δ ` 1{2q +? +ku2 ´ pα1 ´ µq +ff +}rv}2 +2. +Now because n ě 4logppqτ2 +1 t?r}C´1 +r Cm´r}2 ` ?m ´ r ` p2{δ ` 1{2q +? +ku4{pα1 ´ +µq2, we have +0 “ λ} rβMc}1 ´ λ} rβMc}1 ď λ} rβMc}1 ´ λppzTAq rβ ď ´pα1 ´ µq{2}rv}2 +2 ď 0. +The first inequality holds by the fact that ppzTAq rβ “ pzT rβMc ď }pz}8} rβMc}1 ď +} rβMc}1. Hence we have +λ} rβMc}1 “ λppzTAq rβ. +Now because }ppzTAqpMYSqc}8 ă 1, we conclude that rβpMYSqc “ 0. Hence, +suppp rβq Ă M Y S. This would prove the claim in the statement. +Thus, we only need to show that if }pATpzqMYS}8 ď 1´δ and λ ě 4R1τ1 +a +logppq{pnq{δ, +then +}rv}1 ď tp?r}C´1 +r Cm´r}2 ` +? +m ´ rq ` p2{δ ` 1{2q +? +ku}rv}2. +First from (B.22), we have +pα1 ´ µq}rv}2 +2 ´ τ1 +c +logppq +n +}rv}2 +1 ď +# +BLp rβq +BβT ´ Bqλp rβMcq +BβT +Mc +A ++ +rv ´ +# +BLp pβq +BβT ´ Bqλp pβMcq +BβT +Mc +A ++ +rv +ď λppzTAq rβ ´ λ} pβMc}1 ´ λ} rβMc}1 ` λprzTAq pβ +“ λprzTAq pβ ´ λ} rβMc}1 ` λppzTAqrv. +(B.23) + +70 +The second inequality holds by (B.22). Now since suppp pβq Ď M Y S, we have +λprzTAq pβ ´ λ} rβMc}1 +ď λ} pβMc}1 ´ λ} rβMc}1 +“ λ} pβS}1 ´ λ} rβS}1 ´ λ} rβpMYSqc}1 +“ λ} pβS}1 ´ λ} rβS}1 ´ pλ} rβpMYSqc}1 ´ λ} pβpMYSqc}1q +“ λ} rβS ` pβS ´ rβS}1 ´ λ} rβS}1 ´ pλ} rβpMYSqc}1 ´ λ} rβpMYSqc ` pβpMYSqc ´ rβpMYSqc}1q +ď λ}rvS}1 ´ λ}rvpMYSqc}1. +(B.24) +In addition, +λppzTAqrv “ λtpATpzqSuTrvS ` λtpATpzqpMYSqcuTrvpMYSqc +ď λ}pATpzqS}8}rvS}1 ` λ}pATpzqpMYSqc}8}rvpMYSqc}1 +ď λ}rvS}1 ` λp1 ´ δq}rvpMYSqc}1. +(B.25) +Combine (B.23), (B.24), and (B.25), we have +´τ1 +c +logppq +n +}rv}2 +1 ď pα1 ´ µq}rv}2 +2 ´ τ1 +c +logppq +n +}rv}2 +1 +ď λ}rvS}1 ´ λ}rvpMYSqc}1 ` λ}rvS}1 ` λp1 ´ δq}rvpMYSqc}1 +“ λ2}rvS}1 ´ λδ}rvpMYSqc}1. +Now because δλ{2 ě 2τ1R1 +a +logppq{n ě τ1 +a +logppq{n}rv}1, the above display im- +plies +´2´1δλ}rvS}1 ď λ2}rvS}1 ´ λδ}rvpMYSqc}1 +which leads to +δ}rvpMYSqc}1 ď p2 ` δ{2q}rvS}1. +Then +}rv}1 “ }rvM}1 ` }rvS}1 ` }rvpMYSqc}1 +ď }rvM}1 ` }rvS}1 ` p4{δ ` 1q}rvS}1 +ď }rvM}1 ` p2{δ ` 1{2q +? +k}rv}2 +ď t?r}C´1 +r Cm´r}2 ` +? +m ´ r ` p2{δ ` 1{2q +? +ku}rv}2. +The last equality holds by using the same argument as those lead to (A.28). This +completes the proof. +Corollary B.7. +Assume Conditions (C1)–(C6) hold, λ ď α1{p8R1q, R1 ď +rnα2 +1{t64τ2 +1 logppqus1{4, δ P r4R1τ1 +a +logppq{n{λ,1s, n ě 4logppqτ2 +1 t?m ` p2{δ ` +1{2q +? +ku4{pα1 ´ µq2, and α1 ą µ. Suppose rβa is a stationary point of program (8) +and pβa is the interior local minimize of (8) and satisfies suppp pβaq Ď M Y S. Then +suppp rβaq Ď M Y S. + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +71 +Proof: The Corollary holds by using the same arguments as those lead to Lemma +B.13, while using the consistency result in Theorem 2. +B.4. Supporting Results related to Test Statistics. +B.4.1. Some Definitions. +We define +Σpβq ” ErtYiWi ´ exppβTWi ´ βTΩβ{2qpWi ´ Ωβqub2s +By using the relation (2)–(4) and the additional relations +Etexpp2βT +t Wi ´ βT +t Ωβtq | Xiu “ expp2βT +t Xiq, +Etexpp2βT +t Wi ´ βT +t ΩβtqpWi ´ Ωβtq | Xiu “ expp2βT +t XiqXi, +Erexpp2βT +t Wi ´ βT +t ΩβtqtpWi ´ Ωβtqb2 ´ Ω{2u | Xis “ expp2βT +t XiqXb2 +i , +we have +Σpβtq “ ErtexppβT +t Xiq ´ expp2βT +t XiqupXiXT +i ` Ωq +´expp2βT +t XiqΩβtXT +i ´ expp2βT +t XiqXiβT +t Ω ` expp2βT +t Wi ´ βT +t ΩβtqpWi ´ Ωβtqb2s +and the sample version +pΣpβq “ n´1 +nÿ +i“1 +exppβTWi ´ βTΩβ{2qtpWi ´ Ωβqb2 ´ Ωu +´expp2βTWi ´ βTΩβqtpWi ´ Ωβqb2 ´ Ω{2u +`exppβTWi ´ βTΩβ{2qΩ ´ expp2βTWi ´ βTΩβqΩ +´expp2βTWi ´ βTΩβqΩβpWi ´ ΩβqT ´ expp2βTWi ´ βTΩβqpWi ´ ΩβqβTΩ +`expp2βTWi ´ βTΩβqpWi ´ Ωβqb2 +“ n´1 +nÿ +i“1 +exppβTWi ´ βTΩβ{2qtpWi ´ Ωβqb2u +´expp2βTWi ´ βTΩβqtpWi ´ Ωβqb2 ` Ω{2u +´expp2βTWi ´ βTΩβqΩβpWi ´ ΩβqT ´ expp2βTWi ´ βTΩβqpWi ´ ΩβqβTΩ +`expp2βTWi ´ βTΩβqpWi ´ Ωβqb2. +Further, let +ΨpΣ,Q,βq “ pCrImˆm,0mˆksQ´1 +MYS,MYSpβqΣMYS,MYSpβqQ´1 +MYS,MYSpβqrImˆm,0mˆksTCTq, +and +T0 “ pωn ` ?nhnqTΨ´1pΣ,Q,βtqpωn ` ?nhnq, +where +ωn “ ´?nCrImˆm,0mˆksQ´1 +MYS,MYSpβtq +"BLpβtq +Bβt +* +MYS + +72 +Further define Wald statistics as +TW “ npC pβaM ´ tqTΨppΣ, pQ, pβaq´1pC pβaM ´ tq; +the score statistics +TS “ n +# +BLp pβq +BβT ++ +MYS +pCrImˆm,0mˆksQ´1 +MYS,MYSp pβqqT +ˆΨ´1ppΣ, pQ, pβqCrImˆm,0mˆksQ´1 +MYS,MYSp pβq +# +BLp pβq +Bβ ++ +MYS +. +Because without knowing the distribution of X, the full likelihood is unknown and +hence we do not discuss the likelihood ratio test here. +B.4.2. Assumptions. +(D1) Assume +max +1ďiďnp}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqWiMYS}3 +2 +`}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqpΩβtqMYS}3 +2q´1 +ˆEr}YiΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqWi ´ exppβT +t Wi ´ βT +t Ωβt{2q +ˆΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 +MYS,MYSpβtqpWi ´ βT +t ΩqMYS}3 +2|Wis “ Op1q +(D2) cΨ ď αmintΨpΣ,Q,βqu ď αmaxtΨpΣ,Q,βqu ď CΨ. +B.4.3. Some Lemmas. +Lemma B.14. +Suppose X1,...,Xn are independent m dimensional random vec- +tor which satisfies, EpXiq “ 0 and řn +i“1 covpXiq “ Im. Let Z be a m-dimensional +standard multivariate normal vector, then +sup +C +|Prp +nÿ +i“1 +Xi P Cq ´ PrpZ P Cq| “ Opm1{2 +nÿ +i“1 +E}Xi}3 +2q. +Proof: The lemma follows Theorem 1.1 in Bentkus (2005). +Lemma B.15. +Assume Conditions (C1) – (C7) hold, then +}Ψ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtq}2 “ +"logppq +n +*1{4 +pm ` kq, +and +}Ψ´1ppΣ, pQ, pβq ´ Ψ´1pΣ,Q,βtq}2 “ +"logppq +n +*1{4 +pm ` kq. +Proof: First +}ΨppΣ, pQ, pβaq ´ ΨpΣ,Q,βtq}2 + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +73 +“ Optmaxp}pQMYS,MYSp pβaq´1 ´ QMYS,MYSpβtq´1}2, +}pΣMYS,MYSp pβaq ´ ΣMYS,MYSpβtq}2u. +(B.26) +Using the same arguments as those lead to (A.47), we have }pQp pβaq´1´Qpβtq´1}2 “ +!logppq +n +)1{4 +pm ` kq. Now recall that K ” tv P RMYS : }v}2 ď 1u, for v P K, by the +Taylor expansion +vTtpΣp pβaq ´ pΣpβtquv +“ n´1 +nÿ +i“1 +exppβT +a Wi ´ βT +a Ωβa{2qvTtpWi ´ Ωβaqb2uvpWi ´ ΩβaqTp pβa ´ βtq +`n´1 +nÿ +i“1 +2exppβT +a Wi ´ βT +a Ωβa{2qvTpWi ´ ΩβaqvTΩp pβa ´ βtq +´n´1 +nÿ +i“1 +2expp2βT +a Wi ´ βT +a ΩβaqvTtpWi ´ Ωβaqb2 ` Ω{2uvpWi ´ ΩβaqTp pβa ´ βtq +´n´1 +nÿ +i“1 +2expp2βT +a Wi ´ βT +a ΩβaqvTpWi ´ ΩβaqvTΩp pβa ´ βtq +´n´1 +nÿ +i“1 +2expp2βT +a Wi ´ βT +a ΩβaqvTΩβapWi ´ ΩβaqTvpWi ´ ΩβaqTp pβa ´ βtq +´n´1 +nÿ +i“1 +2expp2βT +a Wi ´ βT +a ΩβaqtvTpWi ´ ΩβaqvTΩ ´ vTΩβavTΩup pβa ´ βtq +´n´1 +nÿ +i“1 +2expp2βT +a Wi ´ βT +a ΩβaqvTpWi ´ ΩβaqβT +a ΩvpWi ´ ΩβaqTp pβa ´ βtq +´n´1 +nÿ +i“1 +expp2βT +a Wi ´ βT +a ΩβaqtvTpWi ´ ΩβaqvTΩ ´ vTΩβavTΩup pβa ´ βtq +`n´1 +nÿ +i“1 +expp2βT +a Wi ´ βT +a ΩβaqvTtpWi ´ Ωβaqb2uvpWi ´ ΩβaqTp pβa ´ βtq +`n´1 +nÿ +i“1 +2expp2βT +a Wi ´ βT +a ΩβaqvTpWi ´ ΩβaqvTΩp pβa ´ βtq +“ n´1 +nÿ +i“1 +exppβT +a Wi ´ βT +a Ωβa{2qvTtpWi ´ Ωβaqb2uvpWi ´ ΩβaqTp pβa ´ βtq +`n´1 +nÿ +i“1 +2exppβT +a Wi ´ βT +a Ωβa{2qvTpWi ´ ΩβaqvTΩp pβa ´ βtq + +74 +´n´1 +nÿ +i“1 +2expp2βT +a Wi ´ βT +a ΩβaqvTΩv{2pWi ´ ΩβaqTp pβa ´ βtq +´n´1 +nÿ +i“1 +4expp2βT +a Wi ´ βT +a ΩβaqvTΩβapWi ´ ΩβaqTvpWi ´ ΩβaqTp pβa ´ βtq +´n´1 +nÿ +i“1 +2expp2βT +a Wi ´ βT +a ΩβaqtvTpWi ´ ΩβaqvTΩ ´ vTΩβavTΩup pβa ´ βtq, +where βa is a point in between pβa and βt. Using the same arguments as those lead to +(A.39) and (A.40), we have +}pΣp pβaq ´ pΣpβtq}2 “ Op +«"logppq +n +*1{4 +pm ` kq +ff +, +and +}pΣpβq ´ Σpβtq}2 “ op +«"logppq +n +*1{4 +pm ` kq +ff +. +Therefore, +}pΣp pβaq ´ pΣpβtq}2 “ Op +«"logppq +n +*1{4 +pm ` kq +ff +. +Combine with (B.26) and the fact that }pQp pβaq´1 ´ Qpβtq´1}2 “ +!logppq +n +)1{4 +pm ` +kq, by Lemma B.11 we have +}Ψ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtq}2 +“ Opt}ΨppΣ, pQ, pβaq ´ ΨpΣ,Q,βtq}2u +“ +"logppq +n +*1{4 +pm ` kq. +The second relation in the statement holds by using the same arguments as those lead +the above results. This proves the result. +B.5. The composite ROIs. +Based on Braak & Braak (1991), Landau et al. (2016), +Schöll et al. (2016), we define the composite regions as follow, where letter L and R +represent the left and right hemispheres, respectively. +• Braak 1 and 2 composite region (Braak12): L_entorhinal, R_entorhinal +• Braak 3 and 4 composite region (Braak34): +L_parahippocampal, L_fusiform, L_lingual, L_amygdala, R_parahippocampal, +R_fusiform, R_lingual, R_amygdala, L_middletemporal, L_caudantcing, L_rostantcing, +L_postcing, L_isthmuscing, L_insula, L_inferiortemporal, L_temppole, R_middletemporal, +R_caudantcing, R_rostantcing, R_postcing, R_isthmuscing, R_insula, R_inferiortemporal, +R_temppole. + +ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS +75 +• Braak 5 and 6 composite region (Braak56): +L_superior_frontal, L_lateral_orbitofrontal, L_medial_orbitofrontal, +L_frontal_pole, L_caudal_middle_frontal, L_rostral_middle_frontal, L_pars_opercularis, +L_pars_orbitalis, L_pars_triangularis, L_lateraloccipital, L_parietalsupramarginal, +L_parietalinferior, L_superiortemporal, L_parietalsuperior, L_precuneus, L_bankSuperiorTemporalSulcus, +L_tranvtemp, R_superior_frontal, R_lateral_orbitofrontal, R_medial_orbitofrontal, +R_frontal_pole, R_caudal_middle_frontal, R_rostral_middle_frontal, R_pars_opercularis, +R_pars_orbitalis, R_pars_triangularis, R_lateraloccipital, R_parietalsupramarginal, +R_parietalinferior, +R_superiortemporal, R_parietalsuperior, R_precuneus, R_bankSuperiorTemporalSulcus, +R_tranvtemp, L_pericalcarine, L_postcentral, L_cuneus, L_precentral, L_paracentral, +R_pericalcarine, R_postcentral, R_cuneus, R_precentral, R_paracentral +B.6. Simulations when measurement error covariance is unknown. +To evaluate the +proposed adjustment estimator in Section 6, we added the following three simulation +settings where the covariance matrices of the measurement errors contain different +numbers of unknown parameters. +1. Ω is a matrix with p{4 unknown parameters, corresponding to p{4 nonzero diago- +nal entries. +2. Ω is a matrix with p{2 unknown parameters, corresponding to p{2 nonzero entries. +3. Ω is a matrix with 6p´15 unknown parameters. Specifically, Ω “ pσijqpˆp, where +σij “ 0.05p1 ´ |i ´ j|{5q for |i ´ j| ď 5 and σij “ 0 for |i ´ j| ą 5. +In these settings, the number of unknown parameters in Ω increases, while all other +settings are the same as in Section 5.2. We evaluate the empirical sizes and powers of +the Wald test for n “ 300, p “ 50 in Table B.9 and p “ 350 in Table B.10. 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(2017), ‘Simultaneous inference for high-dimensional linear models’, Journal of the +American Statistical Association 112(518), 757–768. + diff --git a/RtAyT4oBgHgl3EQfVPc7/content/tmp_files/load_file.txt b/RtAyT4oBgHgl3EQfVPc7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..582b1a951b904c2b9ae1040db9b55bf469c7ad67 --- /dev/null +++ b/RtAyT4oBgHgl3EQfVPc7/content/tmp_files/load_file.txt @@ -0,0 +1,4337 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf,len=4336 +page_content='Submitted to the Annals of Statistics ON HIGH DIMENSIONAL POISSON MODELS WITH MEASUREMENT ERROR: HYPOTHESIS TESTING FOR NONLINEAR NONCONVEX OPTIMIZATION BY FEI JIANG1, YEQING ZHOU2,*, JIANXUAN LIU3,† AND YANYUAN MA4,‡ 1Department of Epidemiology and Biostatistics, The University of California, San Francisco, fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='jiang@ucsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='edu 2School of Mathematical Sciences, Tongji University, *zhouyeqing@tongji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='cn 3Department of Mathematics, Syracuse University, †jliu193@syr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='edu 4Department of Statistics, Pennsylvania State University, ‡yzm63@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='edu We study estimation and testing in the Poisson regression model with noisy high dimensional covariates, which has wide applications in analyz- ing noisy big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Correcting for the estimation bias due to the covariate noise leads to a non-convex target function to minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Treating the high dimensional issue further leads us to augment an amenable penalty term to the target function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We propose to estimate the regression parameter through minimizing the penalized target function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We derive the L1 and L2 conver- gence rates of the estimator and prove the variable selection consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We further establish the asymptotic normality of any subset of the parameters, where the subset can have infinitely many components as long as its cardi- nality grows sufficiently slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We develop Wald and score tests based on the asymptotic normality of the estimator, which permits testing of linear func- tions of the members if the subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We examine the finite sample performance of the proposed tests by extensive simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Finally, the proposed method is successfully applied to the Alzheimer’s Disease Neuroimaging Initiative study, which motivated this work initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Count data are routinely encountered in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For example, cog- nitive scores in a neuroscience study, the number of deaths in an infectious disease study, and the number of clicks on a particular product on an e-commerce platform, are all count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Because most of the count data are concentrated on a few small discrete values rather than expanded on the entire real line and because the distribution of count variables is often skewed, the familiar linear model becomes less ideal to capture these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In the liter- ature, Poisson regression (McCullagh & Nelder 2019) is arguably the most popular model to describe count outcomes, because it naturally models the skewed distribution for posi- tive outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' On the other hand, together with the count data, a large number of covariates are often collected thanks to the ever advancing capability of modern technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' However, these covariates are often contaminated with errors due to imperfect data acquisition and pro- cessing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ignoring these errors can produce biased results, which can finally lead to misleading statistical inference on the model parameters (Carroll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2006) that explain the association between covariates and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Our goal is to develop rigorous statistical inference procedures to test linear hypotheses in the high dimensional Poisson model with noisy covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Such inference tools will enable explaining the association between the count outcome and the individual covariate or combination of covariate, quantifying the un- MSC2020 subject classifications: Primary 00X00, 00X00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' secondary 00X00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Keywords and phrases: High dimension Inference, Measurement Error, Non-convex optimization, Poisson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='00139v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='ST] 31 Dec 2022 2 certainties of the estimated association, and controlling the false discovery rate when testing scientifically important hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let Y be the count outcome and X be its associated covariate vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In the Poisson model, Y is related to X as prpY | Xq “ e´exppβTXqtexppβTXquY {Y !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='. (1) We study the testing problem in (1) under the situation that the covariate vector X is both high dimensional and contaminated with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' When X is accurately observed, the testing problem has been extensively discussed in the literature (Ning & Liu 2017, Zhang & Cheng 2017, Van de Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2014, Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' However, when X is not accurately observed, it is unclear that any of the existing proposed tests are applicable, and testing in the high dimensional noisy Poisson regression model has not been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The major obstacles in constructing valid hypothesis testing procedures are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 1) The existing lasso- type penalized Poisson estimator (Jiang & Ma 2021) does not enjoy the variable selection consistency when the number of parameters is much larger than the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2) The asymptotic normality of the estimator has not been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We develop Wald and score tests targeting at linear hypothesis on the parameters of interest in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To overcome obstacle 1), we improve the penalized Poisson estimator proposed in Jiang & Ma (2021) by using a class of “amenable” penalty functions first defined in Loh & Wainwright (2015, 2017) in combination with a modified log-likelihood function to construct estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We establish the estimation consistency and variable selection consistency of the resulting estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To bypass obstacle 2), we derive the asymptotic linear form of the estimators, and establish the asymptotic normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The asymptotic normal estimator has a wider range of applications than the lasso type estimator does, because it facilitates subsequent inference procedures such as constructing hypothesis testing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Even after establishing the asymptotic normal properties, it is still challenging to gener- alize Wald and score tests to the high dimensional setting for Poisson regression with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This is because under the amenable penalties (Loh & Wainwright 2015, 2017), the asymptotic normality of the estimators is built on a minimal signal condition, which requires the nonzero elements in β to be at least of order λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here λ is the penalty parameter which goes to zero when sample size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now consider testing the null hypothesis β1 “ 0 versus the alternative β1 “ hn, where β1 is the first element of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The minimal signal condi- tion implies that the test will have no power in testing the local alternative when |hn|2 ăă λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To resolve this issue, we remove the penalties on the subvector of the parameters involved in the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' However, it is still unclear how fast the dimension of the subvector can grow while still ensuring sufficient power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To this end, we derive the convergence property of the estimators, which provides the explicit rate at which the dimension of the subvector is al- lowed to grow with the sample size in order to achieve consistency, asymptotic normality, and sufficient power in testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, to implement the score test, we need to esti- mate the regression parameters under the null hypothesis, which involves optimization under linear equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This type of constrained parameter estimation for noisy Poisson model has not yet been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To fill this gap, we develop a general procedure for pa- rameter estimation under linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The constraints include inequality constraints for the parameter estimation under general Poisson model and an additional equality constraint imposed by the null hypothesis, which leads to great challenge in establishing the convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Incorporating inequality constraints is practically important because it allows to incorporate additional parameter information, which will reduce the estimation variation and in turn the sample size needed to achieve satisfactory estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We briefly summarize our contributions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' First, we develop a new estimation pro- cedure of the Poisson model with amenable regularization for noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Second, we show ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 3 the variable selection consistency and the consistency of the resulting estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We provide explicit convergence rate of the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Third, we derive the asymptotic normality of the estimator for the nonzero parameters and the parameters to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Fourth, we propose the Wald and score test procedures by constructing the corresponding test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Fifth, we derive the asymptotic distributions of the Wald and score test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' These five essential elements combined together finally allow us to perform hypothesis testing for Poisson model with high dimensional noisy covariates, which allows us to answer important questions such as “if the left inferior temporal gyrus has a significant impact on the development of Alzheimer’s dis- ease”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' These estimation and inference tasks are not straightforward to achieve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' they require building up a series of theoretical properties first,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' which involves techniques related to analyz- ing conditional sub-Gaussian distribution tails,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' utilizing and modifying various concentration inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' constructing the prime-dual equivalence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' carefully bounding various quantities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' linking different vector and matrix norms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' and establishing a Lyapunov-type bound (Bentkus 2005) on the probability distribution to derive the asymptotic distribution of proposed test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' All these analyses are performed under the unusual constraints involving both lin- ear equality constraints and parameter restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We also modify the alternating direction method of multipliers (ADMM) algorithm to solve a regularized optimization problem un- der linear constraint in constrained parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Although each individual technique in its basic form has been used in the literatures of mathematical analysis, statistics, combina- torics, operations research and computer science, a seamless combination of all these into a general tool to solve the problem under study is very challenging and difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Count data occur frequently in practice, and it is a rule rather than exception that the co- variates can be contaminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In modern data collection mechanism, covariates are almost always high dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence, estimation and inference in Poisson regression with high dimensional noisy covariates is a general problem with wide applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A direct motiva- tion of this work is the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study, which is a multi-site longitudinal study investigating early detection of Alzheimer’s disease (AD) and tracking disease progression biomarkers (Weiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Recently, the advent of tau-targeted positron emission tomography (PET) tracers such as flortaucipir (18F-AV-1451) has made it possible to investigate the relative (to patient’s body weight) tissue radioactivity concentration of the tracers, quantified as standardized uptake value ratio (SUVR), in rela- tionship to the cognitive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, we aim to study the association between cog- nitive scores and SUVRs from PEG image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We extract Montreal Cognitive Assessment (MoCa) scores (Y ) and SUVRs (X) from the PET image in the ADNI study taken within 14 days of the cognitive tests from 196 subjects in the ADNI phase 3 study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We first perform a linear lasso regression between the logarithm of MoCa score and the 218 covariates including age, gender, SUVRs, and volumes of whole brain ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Figure 1 shows the density of the residuals from the lasso regression, which suggests that the residuals are skewed and hence the linear lasso regression does not provide a satisfactory fit for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This motivates us to consider Poisson regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We utilize the Poisson high dimensional hypothesis testing pro- cedure developed in Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2019) to examine which SUVRs are significantly associated with the MoCa scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For each covariate of interest, we test the hypothesis that the corre- sponding coefficient is greater than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We plot the logarithm of the p-values from the score and Wald tests proposed in Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2019) for the coefficients of the SUVRs at cortical ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Figure 1 shows that if using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05/218 as a cut off for the p-value, both the Wald and score test identify the SUVRs at all cortical ROIs as significant predictors, which contradicts the fact that the cognitive functions are controlled by a subset of brain ROIs (Leisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This unsatisfactory result likely attributes to the fact the Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2019)’s method relies on the assumption that the expectation of the exponential of the distance between outcome and regression function is bounded (Condition (A3) in (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2019)) while neuroimage data 4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 Density −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5 Density Residuals G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G 0 10 20 50 60 70 −147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 −146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5 −146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 −145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5 −145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 score test 30 40 log of pvalues covariate index G G G G G G GG G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G GG G 0 10 20 50 60 70 −120 −100 −80 −60 −40 −20 Wald test 30 40 log of pvalues covariate index FIG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Left: The density of the residuals from lasso regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The lasso regression does not provide a satisfactory fit for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Middle and Right: The logarithm of the p values from the Wald and score tests proposed by Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2019) for testing whether the SUVR from each cortical regions is significant predictor for the cognitive score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The Wald and score tests suggest that the SUVRs at all the cortical regions have significant association with the cognitive score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' are often subject to data acquisition and processing errors, which likely leads to violation of this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This motivating example demonstrates the necessity of developing novel statistical inference procedure to test linear hypothesis in the high dimensional Poisson model with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Section 2 discusses related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In Section 3, we describe our model assumptions and the overall estimation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We further detail the estimation with and without the null constraint, and the construction of the test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The fundamental theoretical developments are provided in Section 4, where we establish convergence rates, the asymptotic normal results, and the properties of the test procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We study the practical implementation and the numerical performances in Section 5, where a detailed algorithm is provided, extensive simulations are carried out, and a ADNI data set is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We conclude the paper in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The main mathematical proofs are provided in an Appendix given in a Supplementary Document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Related Works and Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Nonlinear models with high dimensional noisy data are in general hard problems to work with, partly because existing treatments usually lead to non-convex optimization, which violates standard requirements in the high dimensional data analysis literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Thus, only linear models, which are the simplest in all noisy data problems, have received relatively thorough investigation (Loh & Wainwright 2012, Belloni & Rosenbaum 2016, Datta & Zou 2017, Belloni, Rosenbaum & Tsybakov 2017, Belloni, Chernozhukov & Kaul 2017, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Expanding the research framework to the Pois- son regression context is difficult because the link function in the Poisson model is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Subsequently, it is not easy to construct noise adjusted quantities such as a noise adjusted Hessian matrix like in the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In addition, the Hessian matrix involves heavy tailed random variables due to the exponential link, even if all the covariates are sub-Gaussian in their original scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' These difficulties require additional restrictions on the moments of the covariate distribution as well as on the parameter searching space, which complicates all the subsequent computation and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Indeed, the only works we are aware of in the high dimensional Poisson model with noisy data are Jiang & Ma (2021), Sørensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2015, 2018), Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2019), while only the estimator in Jiang & Ma (2021) has been shown to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' However, because all these methods use lasso-type L1 penalty in the estima- tion, the resulting estimators do not enjoy variable selection consistency and their asymptotic distribution results are not established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 5 There is extensive literature on the linear hypothesis testing under high dimensional noise free setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (Ning & Liu 2017) introduced a decorrelated score function to construct confi- dence regions for low dimensional components in high dimensional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Zhang & Cheng (2017) used the desparsifying lasso estimator (Van de Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2014) to propose a maximal- type statistic allowing the number of parameters that are involved in the test to grow with the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Moreover, Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2019) proposed a partial penalized likelihood ratio test, a score test, and a Wald test for testing the linear hypothesis of the parameters in high dimen- sional generalized linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We introduce some general notation that will be used throughout the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For a matrix M, let }M}max be the matrix maximum norm, }M}8 be the L8 norm and }M}p be the Lp norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let Fpβq be the σ-field generated by Xi,βTWi,i “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, let Fx be the sigma-field generated by Xi,i “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For a general vector a, let }a}8 be the vector sup-norm, }a}p be the vector lp-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let ej be the unit vector with 1 on its jth entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For a vector v “ pv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',vpqT, let supppvq be the set of indices with vi ‰ 0 and }v}0 “ |supppvq|, where |U| stands for the cardinality of the set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For a vector v P Rp and a subset S Ď p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',pq, we use vS P RS to denote the vector obtained by restricting v on the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Following Fletcher & Watson (1980), for an arbitrary norm } ¨ }A and its dual normal } ¨ }D, we define B}x}A as the set pv : }x}A “ vTx,}v}D ď 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Thus, for an arbitrary vector x “ px1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',xpqT, B}x}1 “ tv “ pv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',vpqT : vj “ signpxjq if xj ‰ 0, and |vj| ď 1 if xj “ 0u, and B}x}2 “ tv “ pv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',vpqT : vj “ xj{}x}2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Model, Estimation and Test Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Problem Formulation: High dimensional Poisson model with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let Xi be a p-dimensional covariate, for example the image features, and let Yi be a count random response variable, for example the MoCa score from the ADNI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We model the rela- tionship between Yi and Xi (i “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',n) through a Poisson model prpYi “ y | Xi “ xq “ e´exppβT t xqtexppβT t xquy{y!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='. Here, βt is a p-dimensional sparse parameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We allow the number of nonzero entries in βt to grow with the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We consider Poisson model here because our response is a count, and Poisson model is arguably the most standard model for count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Indeed, Poisson model has been widely used to model the distribution of cog- nitive scores (Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2021, Fallah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2011, Mitnitski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We use eβT t x to model the conditional mean of the Poisson model to ensure the positiveness of the mean, and to allow possible skewness in the distribution (McCullagh & Nelder 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We assume βt to be sparse because it often happens that only a few covariates have effect on the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For example, in the ADNI data, because the cognitive functions are controlled by a subset of brain ROIs (Leisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2016), only a subset of brain features contributes to the cognitive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, we assume the covariate Xi is not precisely observed and instead, a contam- inated version of Xi, denoted Wi, is observed, where Wi “ Xi ` Ui, and Ui is the noise that is independent of both Xi and Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For example, in the ADNI data, Xi can be the true image features, while Wi represent the observed image features which can deviate from the truth due to imperfect data collection and processing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Without loss of generality, assume that EpXiq “ 0, which can always be achieved by centering the observed covariates in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, we assume Ui is a normally distributed random noise vector with mean zero and known covariance matrix Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The normal assumption for Ui is the common assumption at the state of the art in the Poisson measurement error literature and allows to derive analytic form of the loss function, which is the only setting that we can directly ex- 6 amine the convexity of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The known Ω assumption is only for convenience of presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In practice, it is often replaced by an estimated version based on multiple observations, validation data or other standard instruments under both low and high dimen- sional settings (Carroll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2006, Loh & Wainwright 2012), and the corresponding analysis is routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let pXi,Wi,Yi,Uiq be independent and identically distributed (iid) and assume pWi,Yiq,i “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',n are the iid observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In this work, we devise estimation procedures for β and establishing theoretical properties of the estimator, we further aim at performing inference, such as conduct hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Throughout, we allow the covariate dimension to be much higher than the number of observations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' p ąą n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We assume βt is in the fea- sible set: tβ : }β}0 ď k, }β}2 ď b0u, which is practically sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A vector β in the feasible set automatically satisfies }β}1 ď b0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' General Estimation Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If the true covariates Xi can be observed and the di- mension p is fixed, this is a standard regression model and we routinely estimate β by mini- mizing the negative loglikelihood, which is proportional to ´n´1 nÿ i“1 tYiXT i β ´ exppβTXiqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here we use exppβTXiq to model the mean of Yi because it preserves the positiveness of the mean estimate, and it is a standard choice in the generalized linear model (McCullagh & Nelder 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' It is useful to note that for normal noise Ui, we have the relation EtexppβT t Wi ´ βT t Ωβt{2q | Xiu “ exppβT t Xiq, (2) EtexppβT t Wi ´ βT t Ωβt{2qpWi ´ Ωβtq | Xiu “ exppβT t XiqXi, (3) ErexppβT t Wi ´ βT t Ωβt{2qtpWi ´ Ωβtqb2 ´ Ωu | Xis “ exppβT t XiqXb2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (4) Due to the conditional independence of Wi and Yi given Xi, (2) leads to EtYiWT i βt ´ exppβT t Wi ´ βT t Ωβt{2q | Xi,Yiu “ YiXT i βt ´ exppβT t Xiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Consequently, it is a reasonable practice to estimate β by minimizing the loss function Lpβq “ ´n´1 nÿ i“1 tYiWT i β ´ exppβTWi ´ βTΩβ{2qu, (5) which has the same mean as the negative log-likelihood function when Xi is accurately ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' When n ą p, the estimator for β can be obtained by minimizing Lpβq using the standard gradient descent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' However, when n ă p, without addition regularization, optimizing (5) is an ill-posed mathematical problem because it does not have a unique solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To take into account the ultra-high dimension nature of the model, using the fact that β is sparse, we propose to estimate β through solving the following constrained minimization problem min }β}1ďR1,}β}2ďR2 tLpβq ` ρλpβqu (6) at suitable R1,R2, where ρλpβq is a suitable penalty function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For convenience, define the set tβ : }β}1 ď R1,}β}2 ď R2u as the feasible set (Fletcher & Watson 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here R1,R2 can be any constants that are greater than the true }β}1 and }β}2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The condition }β}1 ď R1 is imposed to guarantee that the objective function satisfies the restricted eigen- value condition discussed in Loh & Wainwright (2012) and therefore the objective function is convex in the feasible set, while the condition }β}2 ď R2 is imposed to avoid the explo- sion of the mean function exppβTWi ´ βTΩβ{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In practice, we often set R1,R2 to be ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 7 a constant times the L1, L2 norms of the initial estimators of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here, with a slight abuse of notation, we use the same symbol ρλ to denote both multivariate and univariate penalty functions and let ρλpβq “ ř}β}0 j“1 ρλpβjq, where βj is the jth element of β and }β}0 is the number of nonzero elements in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Estimation under Hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Consider testing the hypothesis that CβtM “ t`hn for some hn P Rr, where C is a r ˆ m matrix with r ď m, βtM is a m-dimensional sub- vector of β with index set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The null hypothesis holds when hn “ 0, while the alternative hypothesis holds when hn ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For example if t “ 0, hn “ 1, C “ p1,0q, M contains the index of the first element in βt, then testing CβtM “ t ` hn is testing the null hypothesis that βt1 “ 0 versus the alterative that βt1 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Similarly, we can test βt1 ´ βt2 “ 0 versus βt1 ´ βt2 ‰ 0 by choosing C “ p1,´1q, t “ 0, hn “ 0 or nonzero, and M “ t1,2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In summary, by varying C, t, hn, and M, we can generate different linear hypotheses to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Without loss of generality, we assume βM contains the first m elements of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, let βc M be the vector containing elements that are not in M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' the last p ´ m components of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let S Ď Mc be the index set of the nonzero elements of βtMc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We assume βtMc to be k sparse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' |S| “ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Note that k is allowed to diverge with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Without loss of generality, we assume the first k elements in βtMc are none zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose we are interested in testing whether CβtM “ t or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Under the null hypothesis that H0 : CβtM “ t, we modify the general estimation strategy slightly and consider the estimator resulting from the equality and inequality constrained minimization: pβ “ argmin}β}1ďR1,}β}2ďR2 tLpβq ` ρλpβMcqu, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' CβM “ t (7) for suitable R1,R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Without assuming the null hypothesis, we consider a similar estimator resulting from the inequality constrained minimization: pβa “ argmin}β}1ďR1,}β}2ďR2 tLpβq ` ρλpβMcqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (8) Note that here, both (7) and (8) are slightly different from the general strategy in (6), in that we do not place the penalty ρλ on the parameters in M, which are to be tested for the linear relation CβtM “ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This special treatment is to avoid the situation that the penalty forces some components in βM to be zero, and therefore the null hypothesis CβtM “ t is affected not only by the data but also by our penalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We define Qpβq ” EtexppβTXqXXTu, (9) define the covariance of the residuals Σpβq ” ErtYiWi ´ exppβTWi ´ βTΩβ{2qpWi ´ Ωβqub2s, and define ΨpΣ,Q,βq ” pCrImˆm,0mˆksQ´1 MYS,MYSpβqΣMYS,MYSpβqQ´1 MYS,MYSpβqrImˆm,0mˆksTCTq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, let pΣpβq and pQpβq be a sample estimator of Σpβq and Qpβq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To test CβM “ t, we introduce two statistics, the Wald statistic TW “ npC pβaM ´ tqTΨppΣ, pQ, pβaq´1pC pβaM ´ tq, (10) and the score statistic TS “ n # BLp pβq BβT + MYS pCrImˆm,0mˆks pQ´1 MYS,MYSp pβqqT 8 ˆΨ´1ppΣ, pQ, pβqCrImˆm,0mˆks pQ´1 MYS,MYSp pβq # BLp pβq Bβ + MYS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (11) As we will show later in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4 that TW and TS are both asymptotically chi-square distributed with r degrees of freedom under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, to control the false discovery rate at level α, we reject the null hypothesis if TW ą χ2 1´αprq when we perform Wald test, or if TS ą χ2 1´αprq when we perform score test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here χ2 1´αprq is the 1 ´ α quantile of the chi-square distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theoretical Properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define qβM ” βtM ´ CTpCCTq´1hn, and let qβ “ p qβT tM,βT tMcqT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Thus, the last p ´ m components of qβ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' qβMc, and the last p´m components of β0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' β0Mc, are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' However, the first m components of qβ and β are different, in that C qβM “ t under both null and alternative, while CβtM “ t under the null alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Under some conditions, we first show that the inequality and equality constrained estimator pβ is a consistent estimator of qβ regardless the null or the alternative holds, and when }hn}2 vanishes, pβ is also consistent as an estimator of the true parameter βt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further- more, we show that pβa is a consistent estimator of βt regardless the null or the alternative holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We then establish the asymptotic linear form of the estimators of a subvector pβ and a subvector of pβa, which are formed by components of βt that are either to be tested or nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Finally, using the asymptotic linear forms, we construct test statistics and prove the convergence properties of these test statistics under both null and alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Before we proceed with the specific results, we first list a set of assump- tions on the univariate penalty function ρλ which are similar to those in Loh & Wainwright (2015) and Loh & Wainwright (2017) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A1) The function ρλptq satisfies ρλp0q “ 0 and is symmetric around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A2) On the nonnegative real line t ě 0, the function ρλptq is nondecreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, ρλptq is subadditive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ρλpt1 ` t2q ď ρλpt1q ` ρλpt2q for all t1,t2 ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A3) For t ą 0, the function ρλptq{t is non-increasing in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A4) The function ρλptq is differentiable at all t ‰ 0 and sub-differentiable at t “ 0, with limtÑ0` ρ1 λptq “ λ, where ρ1ptq denotes the derivative of ρptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Together with the symmet- ric Condition in (A1), this leads to limtÑ0´ ρ1 λptq “ ´λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A5) There exists µ ą 0 so that ρλptq ` µt2{2 is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A6) There exists a γ P p0,`8q such that ρ1 λptq “ 0 for all t ě γλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Conditions (A1)–(A3) are some general requirements as discussed in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Condition (A4) restricts the class of penalties by excluding regularizers that are not differ- entiable at 0, for example, the lasso penalty is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Condition (A5) is known as weak convexity (Vial 1982, Chen & Gu 2014) and is a type of curvature constraint that controls the level of nonconvexity of ρλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Condition (A6) is imposed to allow penalty to be zero if the esti- mator is γλ away from zero, which removes the estimation bias for the nonzero parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We say ρλ is µ-amenable if Conditions (A1)–(A5) hold, and we name ρλ pµ,γq-amenable if Conditions (A1)–(A6) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The pµ,γq-amenable penalty includes the smoothly clipped absolute deviation (SCAD) and the minimax concave penalty (Loh & Wainwright 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We need some additional regularity conditions to support the theoretical development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' These conditions impose upper and lower bounds on various quantities to ensure that the up- per bounds are finite and the lower bounds are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' They also restrict the relation between ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 9 the sample size and parameter number so that logppq{n Ñ 0 in a slow rate of 1{tlogpnqu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To save space, we only provide a discussion of these conditions here, while provide the details in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Specifically, Condition (C1) (a) is a standard assumption used in noisy data problem such as that used in Sentürk & Müller (2005) and is usually satisfied in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Condition (C1) (b) guarantees the boundedness and the invertibility of the Hessian matrix (4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' the second derivative of the noise free log likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Conditions (C2) and (C3) bound the total variability of both the response Y and the noise U marginally and condition- ally on the covariates X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Similar requirement is also assumed in Loh & Wainwright (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Condition (C4) shows that the dimension of the covariate can grow exponentially faster than the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Finally, Jiang & Ma (2021) have discussed the Conditions (C5)–(C7) and provided examples showing that the conditions are usually satisfied in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We first show that the equality and inequality constrained estimator pβ is a consistent estimator of qβ in Theorems 1 and 2, which is the same as the true parameter βt, except that the first m components are adjusted to ensure that H0 holds for qβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define α1 ” min }β}1ďR1,}β}2ďR2 αminrEtexppβTXiqXiXT i us{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume }C´1 r Cm´r}2 “ Op1q, ρλ satisfies Conditions (A1) – (A6) and Conditions (C1) – (C6) in the supplementary material hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume α1 ą 3{4µ, and β is in the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let λ satisfy 4max !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' }BLp qβq{Bβ}8,α1plogppq{nq1{4) ď λ ď α1 6R1 and n ě logppqmaxp16R4 1τ 4 1 {α4 1,64R4 1τ 2 1 {α2 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Write t1 ” ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ´ r and t ” p6λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λt1qp4α1 ´ 3µq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then the local minimum of (7) satisfies the error bounds } pβ ´ qβ}2 ď t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' and } pβ ´ qβ}1 ď p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` t1qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Following the similar argument, we also show that the inequality constrained estimator pβa is a consistent estimator of the true parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let α1 “ min }β}1ďR1,}β}2ďR2 αminrEtexppβTXiqXiXT i us{2 and let ρλ satisfy Conditions (A1) – (A6) and Conditions (C1) – (C6) in the supplementary material hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume α1 ą 3{4µ, and β is in the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let λ satisfy 4max !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' }BLpβtq{Bβ}8,α1plogppq{nq1{4) ď λ ď α1 6R1 and n ě logppqmaxp16R4 1τ 4 1 {α4 1,64R4 1τ 2 1 {α2 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then the local minimum of (8) satisfies the error bounds } pβa ´ βt}2 ď 6λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m 4α1 ´ 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' and } pβa ´ βt}1 ď p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='mq6λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m 4α1 ´ 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 10 Theorems 1 and 2 suggest that when logppq{n Ñ 0, and when λ is suitably chosen, for example, λ is at least no smaller than Ortlogppq{nu1{4s, both pβ and pβa converge to their corresponding true values in terms of both l1 and l2 norms, as long as k and m grow slower than tn{logppqu1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' These theoretical results suggest that the dimension of βtM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=', the number of parameters involved in the tests, and the number of nonzero entries in βt can grow at a slower rate of tn{logppqu1{2 under noisy Poisson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' These results also assist us to find reasonable ranges for λ in practice to obtain consistent estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Asymptotic linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We denote rβ as a stationary point of (7), which satisfies the first order condition that tBLp rβq{BβT ` Bρλp rβMcq{BβT McAupβ ´ rβq ě 0, (12) for all β P Rp in the feasible set and satisfies CβM “ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here A “ p0p´m,m,Ip´m,p´mq is a matrix that satisfies }A}8 “ }A}1 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Likewise, we denote rβa as a stationary point of (8), which satisfies the first order condition that tBLp rβaq{BβT ` Bρλp rβaMcq{BβT aMcAupβa ´ rβaq ě 0, (13) for all βa P Rp in the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To show the asymptotic normality of pβ and pβa, our first step is to establish that the local minimizers rβ and rβa achieve variable selection consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To do this, we follow the prime- dual construction introduced in Wainwright (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We first show that both min }β}1ďR1,}β}2ďR2,βPRMYS tLpβq ` ρλpβMcqu, such that CβM “ t (14) and min }β}1ďR1,}β}2ďR2,βPRMYS tLpβq ` ρλpβMcqu (15) have unique local minimizer in the interior of the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then we show that all stationary points of (7) and (8) must have support in M Y S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Since the local minimizers of (7) and (8) are automatically stationary points of (7) and (8) respectively, the local minimizers of (7) and (8) must also have support in M Y S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, the local minimizers of (7) and (8) are actually the local minimizers of (14) and (15) respectively, so are also unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In other words, pβ and pβa are respectively the unique solution of (14) and (15) hence achieve the variable selection consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The details of the above analysis are presented in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1 and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2 in the Appendix A in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In our second step to establish the asymptotic distribution properties of pβ and pβa, we define pQpβq “ B2Lpβq BβBβT , and define A2 “ rImˆm,0mˆksTCTpCrImˆm,0mˆkst pQMYS,MYSpβ˚qu´1 ˆrImˆm,0mˆksTCTq´1CrImˆm,0mˆks, where β˚ is the point in between pβ and βt and A˚ 2 “ rImˆm,0mˆksTCTpCrImˆm,0mˆkstQMYS,MYSpβqu´1 ˆrImˆm,0mˆksTCTq´1CrImˆm,0mˆks, ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 11 where Qpβq “ EtexppβTXqXXTu is defined in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Based on the variable selection consis- tency established in the first step, we derive the asymptotic linear form of pβMYS and pβaMYS under null and alternative hypothesis in Theorems 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume ρλ satisfies Conditions (A1) – (A6) and Conditions (C1) – (C7) in the supplementary material hold, λ “ Oprtlogppq{nu1{4s, }C´1 r Cm´r}2 “ Op1q, and λ ď α1{p8R1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further we assume the boundedness }tQ´1 pMYSq,MYSpβtqu}8 ď c8, and }tQpMYSq,MYSpβtqu´1A2Q´1 pMYSq,MYSpβtq}8 ď c8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In addition assume }hn}2 “ Ot a maxpm ` k ´ r,rq{nu, minp|βj|q ě λpγ ` 5c8q for j P S and n ě c8pm ` kq4logppq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then we have pβMYS ´ βtMYS “ ´ptQMYS,ăMYSpβtqu´1 ´ tQMYS,MYSpβtqu´1A˚ 2 ˆtQMYS,MYSpβtqu´1q "BLpβtq Bβ MYS t1 ` opp1qu `tQMYS,MYSpβtqu´1A˚ 2rtpCCTq´1C,0rˆkuThnst1 ` opp1qu and pβpMYSqc “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume ρλ satisfies Conditions (A1) – (A6) and Conditions (C1) – (C7) in the supplementary material hold, λ “ Oprtlogppq{nu1{4s, and λ ď α1{p8R1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further we assume }tQpMYSq,MYSpβtqu´1}8 ď c8, minp|βj|q ě λpγ ` 5c8q for j P S and n ě c8pm ` kq4logppq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then we have pβaMYS ´ βtMYS “ ´tQMYS,MYSpβtqu´1 "BLpβtq Bβ MYS t1 ` opp1qu and pβpMYSqc “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorems 3 and 4 suggest that the asymptotic linear forms of pβMYS and pβaMYS are the usual product of the inverse of Hessian matrix and the score function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, only the first pm ` kq ˆ pm ` kq block in the Hessian matrix and the first m ` k elements in the score function contribute to the asymptotic distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, when m`k grows slower than tn{logppqu1{4 and }hn} Ñ 0, it is easy to see that the asymptotic linear forms converge in distribution to Gaussian random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' It is worth mentioning that the minimal signal condition minp|βj|q ě λpγ ` 5c8q for j P S is a standard requirement for the optimization using nonconvex penalty such as SCAD (Fan & Li 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This condition is also very weak because λ Ñ 0, which allows the minimal signal vanishing to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Asymptotic distribution of the test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To study the asymptotic behavior of TS and TW , we first investigate the distribution of their asymptotic form T0 defined by T0 ” pωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqTΨ´1pΣ,Q,βtqpωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnq, where ωn “ ´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nCrImˆm,0mˆksQ´1 MYS,MYSpβtq "BLpβtq Bβ MYS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' As shown in Lemma 1, T0 is asymptotically noncentral chi-square distributed with the non- central parameter approaches nhT nΨ´1pΣ,Q,βtqhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 12 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume ρλ satisfies Conditions (A1) – (A6) and Conditions (C1) and (D1) in the supplementary material hold and n ě c8pm ` kq4logppq, then lim nÑ8sup C |PrpT0 ď xq ´ Prtχ2pr,nhT nΨ´1pΣ,Q,βtqhnq ď xu| “ 0, where χ2pr,γq is a non-central chi-square random variable, with non-centrality parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here Condition (D1) provides upper bound of the third moment of each summand in ω (note that BLpβtq{Bβ is the summation of the derivatives of the negative log-likelihood from n samples), which is a necessary condition to establish convergence in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' See The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1 in Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2019) for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To establish the asymptotic distribution of TW and TS, in Theorems 5 and 6 respectively, we show that TW and TS are close to T0, hence has the same testing property asymptotically when r is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume the conditions in Theorem 4 and Conditions (D1) and (D2) in the Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2 in the supplementary material hold, we have TW ´ T0 “ opprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, lim nÑ8sup C |PrpTW ď xq ´ Prtχ2pr,nhT nΨ´1pΣ,Q,βtqhnq ď xu| “ 0, where χ2pr,γq is a non-central chi-square random variable, with non-centrality parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume the conditions in Theorem 3, Conditions (D1) and (D2) in the Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2 in the supplementary material hold, we have TS ´ T0 “ opprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, lim nÑ8sup C |PrpTS ď xq ´ Prtχ2pr,nhT nΨ´1pΣ,Q,βtqhnq ď xu| “ 0, where χ2pr,γq is a non-central chi-square random variable, with non-centrality parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here Condition (D2) in the Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2 is a regularity condition ensures ΨpΣ,Q,βtq to be positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorems 5 and 6 show that the two test statistics TW and TS indeed have the same χ2pr,γq distribution as T0 in large samples, hence can be used to perform the standard chi-square test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A curious question is whether or not a likelihood ratio type of test can also be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We feel it is hard in this context because it is almost impossible to obtain a likelihood function in the functional measurement error context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Much work is needed to overcome this obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Numerical Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Computational algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We compute the estimators pβ and pβa using the popular ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In what follows, we only detail the algorithm to estimate pβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The estimator pβa can be computed in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For a given λ, we consider pβ “ argmin}β}1ďR1,}β}2ďR2 tLpβq ` ρλpβMcqu, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' CβM “ t for constants R1,R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Similar to Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2019), this optimization problem is equivalent to p pβ, pθq “ argmin}β}1ďR1,}β}2ďR2 tLpβq ` ρλpβMcqu, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' CβM “ t,βMc “ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' By the augmented Lagrangian method, the estimators can be obtained by minimizing Lpβ,θ,vq “ Lpβq ` ρλpβMcq ` v T ˆ CβM ´ t βMc ´ θ ˙ ` ρ 2 ���� CβM ´ t βMc ´ θ ���� 2 2 , ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 13 Algorithm 1 ADMM Algorithm for estimating pβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For t “ 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',tmax, perform: Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Use the Newton-Raphson algorithm to solve (17) to obtain rβpt`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' rβpt`1q “ argminβ $ & %Lpβq ` vptqT ˜ CβM ´ t βMc ´ θptq ¸ ` ρ 2 ����� CβM ´ t βMc ´ θptq ����� 2 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (17) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Project rβpt`1q to a L1 ball with radius R1 to obtain ˘βpt`1q by the simplex projection method (Duchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If || ˘βpt`1q||2 ą R2, we shrink it to get βpt`1q “ ˘βpt`1qR2{|| ˘βpt`1q||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Otherwise, βpt`1q “ ˘βpt`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Obtain θpt`1q by solving (17), where the penalty term we use is SCAD with a “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' θpt`1q “ argminθ $ & %ρλpθq ` ρ 2 ���βpt`1q Mc ´ θ ��� 2 2 ` vptqT ¨ ˝Cβpt`1q M ´ t βpt`1q Mc ´ θ ˛ ‚ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (18) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Update the dual variables v by vpt`1q “ vptq ` ρ ¨ ˝ Cβpt`1q M ´ t βpt`1q Mc ´ θpt`1q ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If stopping rule }βpt`1q ´ βptq}2 ď δtol or }θpt`1q ´ θptq}2 ď δtol is satisfied, where δtol denotes the tolerance of error, then terminate the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' End of the main loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' with }β}1 ď R1, }β}2 ď R2, where the dual variables v are Lagrange multipliers and ρ ą 0 is a given penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We compute the estimators of pβ,θ,vq through iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let the sup-script (t) indicate the t-th iteration, we describe the main steps of ADMM methods in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In the implementation, the initial value βp0q can be computed by a penalized Poisson regression following Jiang & Ma (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For the radii R1 and R2, we consider R1 “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2R2 and R2 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5}β}p0q 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In the implementation, if the algorithm converges to the boundary, we can increase the corresponding norm R1 or R2 slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In contrast, if multiple minimum problems are encountered, we can decrease R1 and when the estimation procedure leads to a very large exppβTXq, we can decrease R2, gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The tuning parameter λ is selected by minimizing BICpλq “ nLp pβq ` cn} pβ}0 (16) with respect to λ, where cn is a positive number that may depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In our analysis, we follow Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2019) to adopt cn “ maxtlogn,logplogpnqqlogpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For simplicity, we set ρ “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Simulation Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We generate the outcome Yi from the Poisson model PrpYi “ y | Xiq “ expt´exppβTXiquexppyβTXiq{y!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=', where the covariates Xi “ pXi,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',Xi,pqT are generated from two distributions: (I) the multivariate normal distribution with mean zero and covariance matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (II) the uni- form distribution in the interval p´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 6{2, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 6{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To generate correlated uniform distribu- tion, we first draw covariates independently from Up´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 6{2, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 6{2q, and then transform these covariates by multiplying the Choleski factorization of covariance Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We consider two forms of the covariance matrix: uncorrelated structure Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5Ip and correlated with auto- regressive AR(1) structure Σ “ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5|i´j|`1qpˆp for i,j “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, the noise Ui 14 is drawn from the multivariate normal distribution with mean zero and covariance matrix Ω “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The true coefficient β “ pβ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',βpqT “ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75,´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75 ` h2,h3,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',0,hpqT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here hj,j “ 2,3,p are assigned various values to check the empirical powers of the tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We set hj “ 0 when j ‰ 2,3 or p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For simplicity, the initial βp0q is set to be a p-dimensional zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We select parameter λ as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The candidate list for λ is te´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5,e´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='245,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5u of length 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We consider sample size n “ 300,500 and covariate dimension p “ 50,350,600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The tolerance of error δtol “ 10´4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We repeat each setting 500 times, and report the size and power of the proposed tests under different hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We perform the tests at type I error α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 in the following scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Univariate parameter testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We first consider the following three hypotheses on a single element in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' H0,1 : β2 “ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,1 : β2 ‰ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' H0,2 : β3 “ 0, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,2 : β3 ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' H0,3 : βp “ 0, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,3 : βp ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To test a hypothesis set regarding βj, we simulate data with hj “ 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4, while set hk “ 0 for k ‰ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For example, to test H0,1 and Ha,1, we simulate data with h3 “ 0, hp “ 0, and h2 “ 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' When h2 “ 0, the null hypothesis H0,1 holds, we study the type I error of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' On the other hand, when h2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4, the alternative hypothesis is true, which allows us to examine the power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Tables 1 and 2 summarize the empirical type I error and powers of the Wald and score tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' It is clear that the empirical type I errors are controlled at the nominal level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 in all scenarios, indicating that the proposed tests are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The powers of the Wald and score tests increase gradually when the magnitude of |hj|’s increases, and have satisfactory powers in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The Wald and score tests yield similar performances in all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This finding is in accordance with theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Linear hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We also consider the hypotheses that contain the linear combinations of two coefficient parameters: H0,4 : β1 ` β2 “ 0, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,4 : β1 ` β2 ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' H0,5 : β3 ` β4 “ 0, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,5 : β3 ` β4 ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' H0,6 : β1 ` βp “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,6 : β1 ` βp ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' H0,7 : β2 ` β3 “ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,7 : β2 ` β3 ‰ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For the first three sets of hypotheses, we still set hj “ 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4 if the hypothesis involves βj for j “ 2,3,p, and set hk “ 0 if the corresponding βk is not involved in the hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For the last hypothesis H0,7, we set h2 “ 0, hp “ 0 and vary h3 from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Tables 3 and 4 show that the Wald and score tests control the type I error at nominal level, and their powers improve when hj increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Performance regarding m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We further investigate how the testing performance changes as m changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We consider three sets of hypotheses: H0,8 : 4ÿ j“1 βj “ 0, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,8 : 4ÿ j“1 βj ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 15 TABLE 1 The empirical sizes and powers of Wald and score tests for univariate parameter testing with n “ 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' X „ Normal X „ Uniform Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5Ip Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5|i´j|`1 Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5Ip Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5|i´j|`1 TW TS TW TS TW TS TW TS p “ 50 β2 H0,1 : β2 “ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,9 : 8ÿ j“1 βj ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' H0,10 : 12 ÿ j“1 βj “ 0, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,10 : 12 ÿ j“1 βj ‰ 0, corresponding to m “ 4,8 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We set h2 “ 0, hp “ 0, and h3 “ 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The empirical sizes and powers are displayed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' These results suggest that under different m, the empirical sizes remain close to the nominal significance level for both the Wald and score tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' On the other hand, the empirical power decreases in general when m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For instance, as shown in Table 5, when X follows the multivariate normal distribution with mean zero and covariance Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5Ip, p “ 350 and h3 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8, the powers of the Wald test are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='950 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='854 for m “ 4,8 and 12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This is intuitively sensible, and suggests that larger sample size is needed to reach a desired power when the hypothesis concerns more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Comparison with naive test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We further compare the performances of our pro- posed tests with the naive Wald and score tests developed under the noise free framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 16 TABLE 2 The empirical sizes and powers of Wald and score tests for univariate parameter testing with n “ 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' X „ Normal X „ Uniform Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5Ip Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5|i´j|`1 Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5Ip Σ “ 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The noise Ui follows the multivari- ate normal distribution with mean zero and covariance matrix 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3Ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Other settings remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We consider the hypotheses on a single element in β: H0,2, and the linear com- binations of two coefficient parameters: H0,5 and H0,7 as described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We report the empirical sizes and powers of the Wald and score tests with/without noises for p “ 50 in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' It is clear that while the proposed tests achieve Type I errors reasonably close to the nominal level under different null hypotheses, the naive tests lead to precarious performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For instance, the Type I errors of Wald and Score tests for H0,5 are as large as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='474 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='554, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' These Type I errors are far beyond the significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Because they cannot control the significance level, we do not recommend consider using them in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Neuroimage application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We apply our proposed 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Fur- thermore, we remove the covariates with more than 100 missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We standardize the volumes of ROIs by subtracting the means and dividing by the standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We use the SUVR from inferior cerebellum as a reference and divide the rest of SUVRs by this ref- erence as suggested in (Landau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Finally, we have n “ 196 complete samples with p “ 218 covariates in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Since the neuroimage data are longitudinally collected, we estimate the covariance matrix of U using repeatedly measured image features, while assuming that age and gender are 18 TABLE 4 The empirical size and power of Wald and score tests for linear hypothesis testing with n “ 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' X „ Normal X „ Uniform Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5Ip Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5|i´j|`1 Σ “ 0.' metadata={'source': 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´ Uiqp rUij ´ UiqT řn i“1pni ´ 1q , where ni is the number of repeated measurements of Ă Wij, and Ui “ řni j“1 rUij{ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Finally, because the first two covariates, age and gender, are measured precisely, the first two columns ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 19 TABLE 5 The empirical size and power of Wald and score tests under different m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' X „ Normal X „ Uniform Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5Ip Σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5|i´j|`1 Σ “ 0.' metadata={'source': 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are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We set the rest pp ´ 2q ˆ pp ´ 2q sub- matrix of pΩ to be rΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We test p hypotheses, each of the form H0 : βj “ 0 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha : βj ‰ 0, (19) for j “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',p at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To implement the hypothesis testing procedure, in each test, we first fit a standard penalized Poisson regression model to obtain the initial values of the coefficient estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then we construct the score test and Wald test statistics based on 20 (11) and (10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The tuning parameter λ is selected by minimizing (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We obtain the p-value as the probability of a χ2p1q random variable that is greater than the resulting score and Wald test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' There are 33 and 69 covariate coefficients with significant p- values at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 nominal level based on the score and Wald tests, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, we plot the boxplot of the resulting p-values in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' It is clear that the distribution of the p-values are similar for the score and the Wald test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For each covariate j, we obtain the score test Wald test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 Distributions p−value FIG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The boxplot of the p-values based on the score and Wald tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The distributions of the p-values are similar from both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' estimated jth coefficient based on (8) under the corresponding alternative hypothesis, and plot the estimated coefficients of the SUVRs at the cortical regions on a template brain in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The results show that the SUVRs have negative effects on the cognitive score, suggesting that the higher the SUVR values, the lower the MoCa score and in turn the worse the cognitive function, which is consistent with the scientific evidences (Braak & Braak 1991, Schöll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2016, Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, the score test is more stringent and gives less number of significant SUVRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Among 33 significant predictors from the score test, 27 of them are also significant in the Wald test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Based on this high agreement between the score and Wald tests, we believe the difference between the two tests is a small sample phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To adjust for the multiple testing, we further performed an analysis to control false dis- covery rate (FDR) (Benjamini & Hochberg 1995) within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 by treating the p-values as independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Since the score test is too stringent, no significant covariate has been identified at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 FDR by using the score test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, we only present the results from the Wald test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We plot the p-values versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05j{218 in Figure 4 in an increasing order, which suggests 36 covariates are selected as the important predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' There are 13 cortical SUVRs among the 36 important predictors that are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We present their estimated coefficient, p-values from the Wald test in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The results show that the majority of the significant cortical SUVRs are in the temporal lobe, which consists of structures that are vital for declarative or long-term memory (Smith & Kosslyn 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 21 FIG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The effects of SUVRs at the cortical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The colors represent the values (indicated by the color bars) of the estimated coefficients of the SUVRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We only plot the coefficient values corresponding to the significant brain regions with p-value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 from score test (left) and Wald test (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The white areas are the non-significant brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The L and R letters in the plot represent the left and right hemispheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGG G GGGGGGGGGG GG GGGGGGGGGGGGGGGGGGGGGGGGGG GG GGGGG GGG GG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0 R p−value FIG 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Sorted p-value versus R “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05j{218,j “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' There are 36 important predictors corresponding to the p-values (in red) that below the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In addition, we perform a 5-fold cross validation and compare the prediction errors among the four methods: (a) We select the important predictors as those with p-value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 in the test (19) based on the score statistics and then use formula expp pβT S WSi ´ pβT S pΩS pβSq to predict the outcome in the test sample, where WSi is the selected covariates, pβS is estimator from (6) using selected covariates, pΩS is the subset of pΩ corresponding to the selected co- variates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (b) We select the important predictors as those with p-value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 in the test (19) based on the Wald statistics and then use formula expp pβT W WWi ´ pβT W pΩW pβW q to pre- L R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2647 0L R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2626 022 Cortical regions Brain lobes Estimated coefficient Wald test p-value left middle temporal gyrus Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0002 left inferior parietal cortex Parietal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0003 left inferior temporal gyrus Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0007 right inferior parietal cortex Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0007 left BANKSSTS Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0015 left fusiform gyrus Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0016 right middle temporal gyrus Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0024 left caudal middle frontal gyrus Frontal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0030 left precuneus cortex Parietal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0034 left entorhinal cortex Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0036 right inferior temporal Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0059 right left entorhinal cortex Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0065 right BANKSSTS Temporal lobe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0076 TABLE 7 The estimated coefficients, p-values from score and Wald tests for the significant SUVRs at 27 cortical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We also include the specific brain lobe that contains each cortical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' BANKSSTS stands for banks of the superior temporal sulcus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' dict the outcome in the test sample, where WWi is the selected covariates, pβW is estimator from (6) using selected covariates, pΩW is the subset of pΩ corresponding to the selected co- variates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (c) We select the important predictors using the standard lasso regression between the logarithm of the MoCa score and all covariates and then use formula expp pβTWiq to predict the outcome, where pβ is the estimator from the lasso regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (d) We select the im- portant predictors using the penalized Poisson regression between the logarithm of the MoCa score and all covariates and then use formula expp pβTWiq to predict the outcome, where pβ is the estimator from the penalized Poisson regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The penalty parameters in the lasso and penalized Poisson regression are selected using a sub-routine of 10-folder cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Method (d) breaks down because the algorithm does not converge for any selections of the penalty parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, in Figure 5, we show the distributions of the prediction er- rors, defined as řn i“1 |Yi ´ pYi|{|Yi|, only for the methods (a), (b) and (c) after 100 runs of the 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The results shows that Method (a) and (b) have similar performance and both outperform Method (c) with much smaller prediction errors on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Finally, we perform the score and the Wald tests to test whether any SUVRs from any com- posite regions may have significant association with the MoCa score, where the composite regions, namely BRAAK12, BRAAK34, BRAAK56, are defined in (Braak & Braak 1991) and used in Landau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2016) and Schöll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We provide the list of ROIs in each composite regions in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let βSk be the coefficients of the SUVRs from the ROIs that belong to the composite region k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We test the null hypothesis that βSk “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The results in Table 8 show that all the tests are significant, suggesting that at least one ROI in each of the composite region has significant association with the cognitive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This result partially agrees with results in (Schöll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2016) that the SUVRs from the composite regions are sig- nificantly different in healthy subjects and patients with a diagnosis of probable Alzheimer’s disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Conclusion and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We have proposed an amenably penalized noise cor- rected Poisson model to study the relationship between the cognitive score and high dimen- sional noisy neuroimage data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Under the sparsity assumption, we established the parameter convergence rates in both l1 and l2 norms, the variable selection consistency property and the asymptotic normality of a subvector with possibly infinitely many components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Infer- ence tools are subsequently developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The neuroimage application shows that the inference ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 23 score Wald Lasso 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='25 Prediction errors FIG 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The distribution of the prediction errors from 100 runs of the 5-fold cross-validation based on Methods (a), (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Method (d) breaks down because the algorithm does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Composite regions TS score test p-value TW Wald test p-value DF BRAAK12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='039 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0039 4 BRAAK34 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0113 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='774e-09 24 BRAAK56 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0165 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0055 44 TABLE 8 The score and Wald test results of hypothesis that βSk “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' TS and TW are the score and Wald test statistics, DF is the degree of freedom in the asymptotic distribution of the score and Wald statistics, which equals the number of the ROIs in the composite regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' tools generate scientifically meaningful results, which have potential to be used to study the cognitive function and cognitive changes for neurodegenerative diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further research along this line is ongoing in our group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The neuroimage dataset and computational code are available at Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Thanks to an anonymous referee, we would like to point out one important extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Instead of a constant matrix Ω, we can further allow Ω to depend on both the covariate X and the response Y , hence ΩpY,Xq, and assume EpU|Y,Xq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This would include heteroscedastic measurement error and to allow dependent relation between W and Y given X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' All the estimation and inference results will still hold and the regularity conditions and proofs in the Suppement also do not need to be further modified to accomodate this extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Establishing similar results in generalized linear models beyond Poisson or general re- gression models with non-Gaussian noise turns out to be surprisingly difficult due to various technical obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The main difficult lies in being unable to construct a loss function that is positive-definite at the true parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In the case when an estimating equation is available, although one may be tempted to treat the l2 norm square of the estimating equation as a loss function, we find other technical issues arise partially because the Hessian of the loss function may involve the response, hence some of the techniques used here cannot be 24 directly applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Likewise, extending the Poisson model to allow overdispersion also turns out challenging, regardless if we use a negative binomial model, or incorporate random ef- fects, or use extra observed covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' All these will lead to models different from Poisson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The biggest hurdle of considering general regression model and/or non-Gaussian noise is to rigorously establish that the loss function is locally convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' More investigation and dedicated effort are needed in this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The assumption that the covariance of the measurement is known is widely adopted in the low and dimensional noisy data literature (Stefanski 1989, Cook & Stefanski 1995, Loh & Wainwright 2012, Sørensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2015), because the parameter estimation in the noisy model with unknown noise covariance is a challenging, especially in high dimensional setting where the covariance is a high dimensional unknown parameter to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Thresholding tech- niques as those proposed in Bickel & Levina (2008), Cai & Liu (2011), Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2011) can be used for the covariance estimation, but the theoretical properties of the resulting esti- mators are involved, requiring careful treatment of the additional error from the covariance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In a relatively simple situation when the error variance can be estimated through estimating a parameter γ via solving fγpγq “ 0, then writing Lpβq as Lpβ,γq, we can acco- modate the additional parameter by concatenating β with γ and carrying out the subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For example, in this case the result in Theorem 4 will be updated to ˆ pβaMYS ´ βtMYS pγ ´ γ ˙ “ ´ " QMYS,MYSpβt,γq B2Lpβ,γq{BβMYSBγT Bfγpγq{BβT MYS Bfγpγq{BγT ´1 ˆ «!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' BLpβt,γq Bβ ) MYS fγpγq ff t1 ` opp1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Letting M ” QMYS,MYSpβtq´tB2Lpβt,γq{BβMYSBγTutBfγpγq{BγTu´1tBfγpγq{BβT MYSu, then this leads to pβaMYS ´ βtMYS “ ´M´1 „"BLpβt,γq Bβ MYS ` tB2Lpβt,γq{BβMYSBγTutBfγpγq{BγTu´1 ˆfγpγqst1 ` opp1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We further conduct simulations to evaluate the proposed adjustment in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6 of the supplementary document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The results suggest that the proposed adjustment controls type I error rate when Ω contain small number of unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Estimation and testing when Ω has a large number of unknown parameters are challenging problems and deserve much more extensive investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' REFERENCES Baker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=', Lockhart, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2017), ‘The alzheimer’s disease neuroimaging initiative 3: Continued innovation for clinical trial improvement’, Alzheimer’s & Dementia 13(5), 561–571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=', Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2012), ‘A general theory of concave regularization for high-dimensional sparse estimation problems’, Statistical Science 27(4), 576–593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' & Cheng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2017), ‘Simultaneous inference for high-dimensional linear models’, Journal of the American Statistical Association 112(518), 757–768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 27 Supplementary Materials to “On High dimensional Poisson models with measurement error: hypothesis testing for nonlinear nonconvex optimization” Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We define an auxiliary function qλptq “ λ|t| ´ ρλptq to facilitate the theo- retical derivation, where qλptq ´ µ{2t2 is concave and everywhere differentiable as shown in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Conditions for the estimation consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define αminpMq and αmaxpMq as the minimal and maximal eigenvalues of the matrix M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, we define the sub- exponential norm and sub-Gaussian norm as }X}ψ1 “ sup kě1 1{kEp|X|kq1{k, and }X}ψ2 “ sup kě1 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' kEp|X|kq1{k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For notational convenience, let ApβTWiq :“ exppβTWi ´ βTΩβ{2q, gpWi,β,v,wq :“ vTtpWi ´ Ωβqb2 ´ Ωuw, and g1pWi,β,v,wq :“ vTtpWi ´ Ωβqb2uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' DEFINITION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Loh & Wainwright (2012) (Lower-RE condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The matrix Γ satisfies a lower restricted eigenvalue condition with curvature α1 ą 0 and tolerance τpn,pq ą 0 if βTΓβ ě α1}β}2 2 ´ τpn,pq}β}2 1,@β P Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Loh & Wainwright (2012) (Upper-RE condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The matrix Γ satisfies a upper restricted eigenvalue condition with smoothness a2 ą 0 and tolerance τpn,pq ą 0 if βTΓβ ď a2}β}2 2 ` τpn,pq}β}2 1,@β P Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We first state the regularity conditions as follows: (C1) (a) supi“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',n,}v}2ď1 |WT i v| ď MW a }v}0 for a positive constant MW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' }Ω}2 “ Op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (b) D1 ď αminrEtexppβTXqXXTus ď αmaxrEtexppβTXqXXTus ď D2, DW1 ď αminrEtexpp2βTW ´ βTΩβqpW ´ Ωβqb2us 28 ď αmaxrEtexpp2βTW ´ βTΩβqpW ´ Ωβqb2us ď DW2, αmaxrEtexppβTW ´ βTΩβ{2qpW ´ Ωβqb2us ď DW3, and Etexpp2βTXqu “ Op1q for any β with }β}2 ď 2R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (c) Ep}Wi´Ωβ}2 2q ď DΩ, for any β with }β}2 ď 2R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here D1,D2,DW1,DW2,DW3,DΩ are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (d) }C}2 “ Op1q and }pCCTq´1}2 “ Op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (e) The L2 norm of the true parameter β is bounded, that is }β}2 ď b0 for some 0 ă b0 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (C2) For j “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',p, define Kj :“ }Uij}ψ2 “ p2Ωjjq1{2 sup kě1 k´1{2 Γ1{ktpk ` 1q{2u π1{p2kq , where Γ is the Gamma function, then there exist constants m0,M0 so that m0 ă K2 j nÿ i“1 Y 2 i {n ă M0, uniformly for all j almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (C3) Define KY pXiq “ sup kě1 k´1Er|Yi ´ exppβT t Xiq|k|Xis1{k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' There exist constants m1,m2,M1,M2 so that m1 ă nÿ i“1 X2 ijKY pXiq2{n ă M1, max i |Xij|KY pXiq{ a logn ă M2, uniformly for all j “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',p almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (C4) Sample size n and dimension of covariates p satisfy logpnq a logppq{n ď C for an absolute constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (C5) For ej, j “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',p, define Kwijpβq “ sup kě1 k´1{2Er|pWi ´ ΩβqTej ´ EtpWi ´ ΩβqTej|βTWi,Xiu|k|βTWi,Xis1{k, we assume EtKwijpβtq4u ă Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then there exist constants m3,M3,Q1 so that (i) m3 ă nÿ i“1 Kwijpβtq2 expp2βT t Wi ´ βT t Ωβtq{n ă M3 and ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 29 (ii) | nÿ i“1 EtexppβT t Wi ´ βT t Ωβt{2qpWi ´ ΩβtqTej|βT t Wi,Xiu ´ exppβT t XiqXT i ej a nlogppq | ă Q1 uniformly for all j “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',p in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (C6) For vectors v,w P Rp, }v}2 ď 1,}w}2 ď 1, for β with }β}2 ď 2R2, Kgvwipβq :“ sup ką1 1{kEp|rgpWi,β,v,wq´EtgpWi,β,v,wq|βTWi,Xius|k|βTWi,Xiq1{k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We assume EtKgvwipβq4u ă Q01, and ErexptA2pβTWiqK2 gvwipβqus ă Q02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We also assume that for all v,w, m4 ă nÿ i“1 |ApβTWiq|2Kgvwipβq2{n ă M4, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='20) m5 ă max i |ApβTWiq|Kgvwipβq{ a logn ă M5, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='21) and n´1{2|sup v,w nÿ i“1 vTpApβTWiqErtpWi ´ Ωβqb2 ´ Ωu|βTWi,Xis ´EtexppβTXiqXiXT i uqw|2 ă Q2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='22) in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (C7) For vectors v,w P Rp, }v}2 ď 1,}w}2 ď 1, for β with }β}2 ď 2R2, Kg1vwipβq :“ sup ką1 1{kEp|rg1pWi,β,v,wq´Etg1pWi,β,v,wq|βTWi,Xius|k|βTWi,Xiq1{k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We assume EtKg1vwipβq4u ă Q11, and ErexptA4pβTWiqK2 g1vwipβqus ă Q12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We also assume that for all v, m6 ă nÿ i“1 |A2pβTWiq|2Kg1vwipβq2{n ă M6, m7 ă max i |A2pβTWiq|Kg1vwipβq{ a logn ă M7, and m61 ă nÿ i“1 |A2pβTWiq|2Kg1vwipβq2{n ă M61, m71 ă max i |A2pβTWiq|Kg1vwipβq{ a logn ă M71,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further n´1{2|sup v,w nÿ i“1 vTpA2pβTWiqErtpWi ´ Ωβqb2u|βTWi,Xis ´Etexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uqw| ă Q3, 30 in probability and n´1{2|sup v,w nÿ i“1 vTpApβTWiqErtpWi ´ Ωβqb2u|βTWi,Xis ´EtexppβTWi ´ βTΩβ{2qpWi ´ Ωβqb2uqw| ă Q31, in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Theorems in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' First denote pv “ pβ ´ qβ, by the Taylor expansion of the first order derivative tBLp pβq{BβT ´ BLp qβq{BβTupv “ pvTB2Lpβ˚q{BβBβTpv, where β˚ is a point on the line connecting qβ and pβ and hence is in the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6 we have tBLp pβq{BβT ´ BLp qβq{BβTupv ě α1}pv}2 2 ´ τ1 a logppq{n}pv}2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='23) We first show that }pv}2 ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If not, we have tBLp pβq{BβT ´ BLp qβq{BβTupv ě α1}pv}2 ´ 2τ1 a logppq{nR1}pv}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Together with (12), we obtain t´Bρλp pβMcq{BβT McA ´ BLp qβq{BβTupv ě α1}pv}2 ´ 2τ1 a logppq{nR1}pv}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='24) Further, t´Bρλp pβMcq{BβT McA ´ BLp qβq{BβTupv ď t} ´ Bρλp pβMcq{BβT Mc}8}A}8 ` }BLp qβq{BβT}8u}pv}1 ď pλ ` }BLp qβq{BβT}8q}pv}1 ď 3λ{2}pv}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='25) The second inequality holds because the maximum row sum of A is 1 and }Bρλp pβMcq{BβMc}8 ď λ by Condition (A1)–(A6) and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The last equality holds because }BLp qβq{BβT}8 ď λ{2 by the statement assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now combine (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='25) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='24), we have }pv}2 ď α´1 1 p2τ1 a logppq{nR1 ` 3λ{2q}pv}1 ď α´1 1 p2τ1 a logppq{nR1 ` 3λ{2q2R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' By the assumption that λ ď α1{p6R1q and n ě logppqp64τ 2 1 R4 1q{α2 1 in the statement, we conclude that the right hand side is at most one, which contradict to the hypothesis that }pv}2 ą 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore }pv}2 ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 31 Further, since function ρλpβMcq ` µ{2}βMc}2 2 is convex function of βMc by Condition (A5), we have ρλp qβMcq ` µ{2} qβMc}2 2 ´ ρλp pβMcq ´ µ{2} pβMc}2 2 ě tBρλp pβMcq{BβT Mc ` µ pβT Mcup qβMc ´ pβMcq which implies ρλp qβMcq ´ ρλp pβMcq ` µ{2} pβMc ´ βMc}2 ě tBρλp pβMcq{BβT Mcup qβMc ´ pβMcq “ tBρλp pβMcq{BβT McuAp qβ ´ pβq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Combine with (12), we have Bρλp pβMcq{BβT McAp qβ ´ pβq ě ´BLp pβq{BβTp qβ ´ pβq, and hence ρλp qβMcq ´ ρλp pβMcq ` µ{2} pβMc ´ qβMc}2 ě BLp pβq{BβTp pβ ´ qβq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now combine with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='23), we have α1}pv}2 2 ´ τ1 c logppq n }pv}2 1 ď ´BLp qβq{BβTpv ` ρλp qβMcq ´ ρλp pβMcq ` µ{2}Apv}2 2 ď ´BLp qβq{BβTpv ` ρλp qβMcq ´ ρλp pβMcq ` µ{2}A}1}A}8}pv}2 2 “ ´BLp qβq{BβTpv ` ρλp qβMcq ´ ρλp pβMcq ` µ{2}pv}2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This implies pα1 ´ µ{2q}pv}2 2 ď τ1 c logppq n }pv}2 1 ` }BLp qβq{Bβ}8}pv}1 ` ρλp qβMcq ´ ρλp pβMcq ď # 2R1τ1 c logppq n ` }BLp qβq{Bβ}8 + t}pvMc}1 ` }pvM}1u ` ρλp qβMcq ´ ρλp pβMcq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='26) Note that by the assumption that n ě logppqp16R4 1τ 4 1 q{α4 1, we have 2R1τ1 "logppq n 1{2 “ 2R1τ1 "logppq n 1{4 "logppq n 1{4 ď α1 "logppq n 1{4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further by the assumption that 4}BLp qβq{Bβ}8 ď λ and 4α1tlogppq{nu1{4 ď λ in the lemma statement we obtain 2R1τ1 c logppq n ` }BLp qβq{Bβ}8 ď λ{4 ` λ{4 ď λ{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 32 Combine with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='26) and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1 and subadditivity of ρλ in Condition (A2), we have pα1 ´ µ{2q}pv}2 2 ď ρλp qβMcq ´ ρλp pβMcq ` λ{2 "ρλppvMcq λ ` }pvM}1 ` µ 2λ}pvMc}2 2 ď ρλp qβMcq ´ ρλp pβMcq ` ρλp qβMcq ` ρλp pβMcq 2 ` λ}pvM}1{2 ` µ{4}pv}2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='27) Further, let br be the vector containing the first r element of vector b, and b´r the vec- tor without the first r element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' By the condition that CβM “ t, we have pβr M “ C´1 r pt ´ Cm´r pβ´r Mq, and qβr M “ C´1 r pt ´ Cm´r qβ´r Mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We have }pvM}1 “ }C´1 r Cm´rpv´r M}1 ` }pv´r M}1 ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´rpv´r M}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ r}pv´r M}2 ď p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ rq}pv}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='28) Now (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='27) becomes 0 ď ˆ α1 ´ 3µ 4 ˙ }pv}2 2 ď 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ` λ}pvM}1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='29) We consider two cases, 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ą 0 and 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' When 3ρλp qβMcq ´ ρλp pβMcq ě 0, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2, we have 0 ď 3ρλp qβMcq ´ ρλp pβMcq ď 3λ}pvMcA}1 ´ λ}pvMcAc}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='30) Now from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='30) we further have }pvMcAc}1 ď 3}pvMcA}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Substitue (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='28) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='30) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='29), we have p2α1 ´ 3µ 2 q}pv}2 2 ď 3λ}pvMcA}1 ´ λ}pvMcAc}1 ` λ}pvM}1 ď 3λ}pvMcA}1 ` λp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ rq}pv}2 ď 3λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k}pvMcA}2 ` λp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ rq}pv}2 ď t3λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` λp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ rqu}pv}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence we have that when 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ą 0 }pv}2 ď 6λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ´ rq 4α1 ´ 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='31) When 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ď 0, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='29) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='28) we have ˆ α1 ´ 3µ 4 ˙ }pv}2 2 ď λ}pvM}1{2 ď λp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ rq}pv}2{2, ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 33 which implies that when 3{2ρλp qβMcq ´ 1{2ρλp pβMcq ă 0, }pv}2 ď 2λp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ´ rq 4α1 ´ 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Together with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='31), we always have }pv}2 ď 6λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ´ rq 4α1 ´ 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, the L1 distance is }pv}1 ď }pvMcA}1 ` }pvMcAc}1 ` }pvM}1 ď 4}pvMcA}1 ` }pvM}1 ď 4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k}pvMc}2 ` p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ rq}pv}2 ď 4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k}pv}2 ` p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ rq}pv}2 ď p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ rq6λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ´ rq 4α1 ´ 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This proof is very similar to the proof of Theorem 1 and is simpler, hence is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Theorems in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Supporting Theorem of Theorem 3 and its Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let α1 “ min}β}1ďR1,}β}2ďR2 αminrEtexppβTXiqXiXT i us{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume ρλ is µ-amenable, for µ ă α1, and Conditions (C1) – (C6) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu and A1 “ tImˆm,0mˆpp´mqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Write t1 ” ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ´ r and t ” p6λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λt1qp4α1 ´ 3µq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further we state the following two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (a) The parameters λ,R1,R2 satisfy 4max !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' }BLp qβq{Bβ}8,α1plogppq{nq1{4) ď λ ď « α1 " 16p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` t1qt{λ ´1ff1{2 , max " 2} qβ}1,2p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` t1qt ď R1 ď min ´α1 8λ,rnα2 1{t64τ 2 1 logppqus1{4,rnα4 1{t16τ 4 1 logppqus1{4¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' with } qβ}1 ‰ p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` t1qt and max !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2} qβ}2,2t ) ď R2, with } qβ}2 ‰ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (b) Let pβMYS be the minimizer for (14), pβ “ p pβT MYS,0T p´m´kqT, pz and µ4 satisfy BLp pβq Bβ ´ AT Bqλp pβMcq B pβMc ` λpATpzq ` AT 1 CTµ4 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='32) 34 There exist δ P r4R1τ1 a logppq{n{λ,1s so that }pATpzqpMYSqc}8 ď 1 ´ δ, where we name pATpzq as extended sub-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In addition, n ě maxrlogppqτ 2 1 pm ` kq2{pα1 ´ µq2,4logppqτ 2 1 tt1 ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ku4{pα1 ´ µq2s, α1 ą µ and βMc is k sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Under the above conditions and the conditions (a) and (b), (7) has a unique local minimizer pβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: We follow the primal-dual witness construction introduced in Wainwright (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Step i: Following Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='10 in the supplementary material, let pβMYS be the minimizer for (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We can easily check that when replacing β by βMYS, X by XMYS, and p by m`k, all the conditions of Theorem 1 are still satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Here to check the condition regarding α1, we note that for any β, min }β}2ďR2,}β}1ďR1 αminrEtexppβTXiqXiXT i us “ min }β}2ďR2,}β}1ďR1 inf }v}2“1,vPRp vTEtexppβTXiqXiXT i uv ď min }β}2ďR2,}β}1ďR1 inf }v}2“1,vPRm`kpvT,0TqEtexppβTXiqXiXT i upvT,0TqT “ min }β}2ďR2,}β}1ďR1 αminrEtexppβTXiqXi,MYSXT i,MYSus “ min }βMYS}2ďR2,}βMYS}1ďR1 αminrEtexppβT MYSXi,MYSqXi,MYSXT i,MYSus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, Theorem 1 applied to the m ` k dimensional case leads to } pβMYS ´ qβMYS}1 ď p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` t1qt ď R1{2, and } pβMYS ´ qβMYS}2 ď t ď R2{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore } pβMYS}1 ď } pβMYS ´ qβMYS}1 ` } qβ}1 ă R1 and } pβMYS}2 ď } pβMYS ´ qβMYS}2 ` } qβ}2 ă R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence pβMYS and pβ must be in the interior of the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Step ii: We show that pβ is a local minimum for (7) by verifying the conditions in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='12 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Because Lpβq ` ρλpβMcq “ Lpβq ´ qλpβMcq ` λ}βMc}1, we can write f “ L, g “ qλ, and px˚,v˚,w˚ 1,w˚ 2,µ˚ 1,µ˚ 2,µ˚ 3q “ p pβ,pz,ATpz,pz1,0,0,µ4q, where z1 P B} pβ}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4 in the supplementary material ensures the concavity and dif- ferentiability of gpxq ´ µ{2}x}2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, since µ˚ 1 “ µ˚ 2 “ 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 35 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4) is satisfied by our construction in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, it remains to verify (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We first show that G˚ Ď RMYS so that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5) only needs to be satisfies for the vectors belong to RMYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose this does not hold, let ν P G˚ such that supppνq Ę M Y S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This implies there is an index j P pM Y Sqc such that νj ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now we define ATz1 such that pATz1qk “ pATpzqk for k ‰ j, and pATz1qj “ signpνjq, where ak is the kth element in vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Clearly z1 P B} pβMc}1 and λνTATz1 ą λνTATpz because }pATpzqpMYSqc}8 ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, νT « BLp pβq Bβ ´ # AT Bqλp pβMcq B pβMc +ff ` λνTATz1 ` νTAT 1 CTµ4 ą νT « BLp pβq Bβ ´ # AT Bqλp pβMcq B pβMc +ff ` λνTATpz ` νAT 1 CTµ4 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now because ν P G˚, we have νAT 1 CTµ4 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This implies νT « BLp pβq Bβ ´ # AT Bqλp pβMcq B pβMc +ff ` λνTATz1 ą 0 which contradicts with the requirement of G˚ that sup vPB} pβMc}1 νT « BLp pβq Bβ ´ # AT Bqλp pβMcq B pβMc +ff ` λνTATv “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, G˚ Ď RMYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now by construction suppp pβq Ă M Y S, using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='10 in the supplementary material, we conclude that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5) holds with κ “ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence, all conditions of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='12 in the supplementary material are satisfied, so we conclude pβ is an isolated local minimum of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='13, because suppp pβq Ď M Y S and pβ is an interior minimizer, the support of any stationary point of (7) is a subset of MYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence, we can write any stationary point in the form of rβ “ t rβT MYS,0T p´m´kuT, where rβMYS is a stationary point for (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, note that (14) is strictly convex by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='10, and hence rβMYS is unique in the feasible set and therefore pβMYS and rβ are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence pβ “ rβ is the unique local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Supporting Theorem of Theorem 4 and its Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let α1 “ min}β}1ďR1,}β}2ďR2 αminrEtexppβTXiqXiXT i us{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume ρλ is µ-amenable, for µ ă α1, and Conditions (C1) – (C6) in the supplementary material hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu and A1 “ tImˆm,0mˆpp´mqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further assume that 36 (a) The parameters λ,R1,R2 satisfy 4max !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' }BLpβtq{Bβ}8,α1plogppq{nq1{4) ď λ ď » –α1 # 16p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='mq6 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='mq 4α1 ´ 3µ +´1fi fl 1{2 , max # 2}βt}1,2p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='mqλ6 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m 4α1 ´ 3µ + ď R1 ď min ´α1 8λ,rnα2 1{t64τ 2 1 logppqus1{4,rnα4 1{t16τ 4 1 logppqus1{4¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' with }βt}1 ‰ p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='mq6λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m 4α1 ´ 3µ and max # 2}βt}2,26λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m 4α1 ´ 3µ + ď R2, with }βt}2 ‰ 6λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k ` 2λ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m 4α1 ´ 3µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (b) Let pβaMYS be the minimizer for (15), pβa “ p pβT aMYS,0T p´m´kqT, pz satisfy # BLp pβaq Bβa + ´ # AT Bqλp pβaMcq B pβaMc + ` λpATpzq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='33) There exist δ P r4R1τ1 a logppq{n{λ,1s so that }pATpzqpMYSqc}8 ď 1 ´ δ, where we name pATpzq as extended sub-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In addition, n ě maxrlogppqτ 2 1 pm ` kq2{pα1 ´ µq2, 4logppqτ 2 1 t?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ku4{pα1 ´ µq2s, α1 ą µ and βMc is k spar se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Under the above conditions and the conditions (a) and (b), (8) has a unique local minimizer pβa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The proof follows the same argument as those lead to Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1 hence is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Definition needed to prove Theorems 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let β˚ is the point on the line connecting pβ and qβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We write pQpβ˚q “ # pQpMYSqpMYSqpβ˚q pQpMYSqpMYSqcpβ˚q pQpMYSqcpMYSqpβ˚q pQpMYSqcpMYSqcpβ˚q + (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='34) Then by the construction in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='32), let pβ “ p pβT MYS,0T pMYSqcqT, pβMYS is the minimizer for (14), then we have pQpβ˚qp pβ ´ qβq ` » —– !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' BLp qβq Bβ ) MYS ´ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' AT Bqλp pβMcq BβMc ) MYS !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' BLp qβq Bβ ) pMYSqc ´ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' AT Bqλp pβMcq BβMc ) pMYSqc fi ffifl ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 37 `λ "pATpzqMYS pATpzqpMYSqc ` " pAT 1 CTµ˚ 3qMYS 0pMYSqc “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Taking the upper m ` k non-zero component, we get pβMYS ´ qβMYS “ t pQMYS,MYSpβ˚qu´1 « ´ # BLp qβq Bβ + MYS ` # AT Bqλp pβMcq BβMc + MYS ´λpATpzqMYS ´ pAT 1 CTµ˚ 3qMYS ȷ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='35) while taking the lower p ´ m ´ k components, this leads to pATpzqpMYSqc “ λ´1 » – # AT Bqλp pβMcq BβMc + pMYSqc ´ # BLp qβq Bβ + pMYSqc fi fl ´λ´1 pQpMYSqcpMYSqpβ˚qp pβMYS ´ βMYSq “ λ´1 » – # AT Bqλp pβMcq BβMc + pMYSqc ´ # BLp qβq Bβ + pMYSqc fi fl `λ´1 pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 ˆ ˜«# BLp qβq Bβ + MYS ´ # AT Bqλp pβMcq BβMc + MYS ff ` λpATpzqMYS ` pAT 1 CTµ˚ 3qMYS ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further by Condition (A4), we have # AT Bqλp pβMcq BβMc + pMYSqc “ «# Bλ|pβj| Bβj + ´ # Bρλppβjq Bβj + ,j P pM Y Sqc ffT “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, we have pATpzqpMYSqc “ λ´1 » –´ # BLp qβq Bβ + pMYSqc fi fl ` λ´1 pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 ˆ ˜«# BLp qβq Bβ + MYS ´ # AT Bqλp pβMcq BβMc + MYS ff ` λpATpzqMYS ` pAT 1 CTµ˚ 3qMYS ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' By the condition that C pβM “ C qβ “ t, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='35),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' we have 0 “ CrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆkst pQMYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1 « ´ # BLp qβq Bβ + MYS ` # AT Bqλp pβMcq BβMc + MYS ´ λpATpzqMYS ff `CrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆkst pQMYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1rImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksTCTµ˚ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' which leads to µ˚ 3 “ pCrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆkst pQMYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1rImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksTCTq´1CrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆkst pQMYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1 38 ˆ « ´ # BLp qβq Bβ + MYS ` # AT Bqλp pβMcq B qβMc + MYS ´ λpATpzqMYS ff and pA1CTµ˚ 3qMYS “ A2t pQMYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1 « ´ # BLp qβq Bβ + MYS ` # AT Bqλp pβMcq BβMc + MYS ´ λpATpzqMYS ff ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' where A2 “ rImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksTCTpCrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆkst pQMYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1rImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksTCTq´1CrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence, to use Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1, we must show that }pATpzqpMYSqc}8 ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define Qpβq “ EtexppβTXqXXTu, and A˚ 2 “ rImˆm,0mˆksTCTpCrImˆm,0mˆkstQMYS,MYSpβqu´1rImˆm,0mˆksTCTq´1CrImˆm,0mˆks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' First of all, pβpMYSqc “ 0 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For any unit vec- tor, recall that vT B2Lpβq BβBβT w “ exppβTWi ´ βTΩβ{2qvTtpWi ´ Ωβqb2 ´ Ωuw Denoting ∇3Lpβq to be the third order gradient of L, we have vT∇3Lpβqw “ n´1 nÿ i“1 exppβTWi ´ βTΩβ{2qvTtpWi ´ Ωβqb2 ´ ΩuwtWi ´ ΩβuT ´n´1 nÿ i“1 exppβTWi ´ βTΩβ{2qtvTpWi ´ ΩβqwTΩ ` wTpWi ´ ΩβqvTΩu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence define vectors v,w such that their jth element |vj| ą 0,|wj| ą 0 for j P M Y S, and |vj| “ |wj| “ 0 for j R MYS, and }v}2 “ }w}2 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Firstly by Theorem 1 and the condition that }hn}2 “ Ot a maxpm ` k ´ r,rq{nu, we have } pβ ´ βt}2 “ } pβ ´ qβ}2 ` } qβ ´ βt}2 ď C1λmaxp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ r, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' kq ` a pm ` kq{n ď C2 "logppq n 1{4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k for some constants C1,C2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further recall that K ” tv P RMYS : }v}2 ď 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇvT " pQpβ˚q ´ B2Lpβtq BβBβT w ˇˇˇˇ ď sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK «ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvTtpWi ´ Ωβ˚qb2 ´ ΩuwtWi ´ Ωβ˚uTp pβ ´ βtq ˇˇˇˇ ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 39 ` ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq ˇˇˇˇ ff ď sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qb2wpWi ´ Ωβ˚qTp pβ ´ βtq ´vTΩwpWi ´ Ωβ˚qTp pβ ´ βtqu ˇˇˇˇ ` sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq ˇˇˇˇ ď sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇ n´1 2 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qpv ` wqT ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2 pWi ´ Ωβ˚qb2 v ` w ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2 pWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ ` sup v,wPK ˇˇˇˇ n´1 2 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qpv ´ wqT ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2 pWi ´ Ωβ˚qb2 v ´ w ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2 pWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ ` sup v,wPK ˇˇˇˇ pv ` wqT ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2 Ωv ` w ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2 n´1 2 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ ` sup v,wPK ˇˇˇˇ pv ´ wqT ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2 Ωv ´ w ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2 n´1 2 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ ` sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq ˇˇˇˇ ď sup vPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2vpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ `sup vPK ˇˇˇˇvTΩvn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ ` sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvTpWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq ˇˇˇˇ ď sup vPK n´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2v|pWi ´ Ωβ˚qTp pβ ´ βtq| (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='36) `sup vPK vTΩv ››››n´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qpWi ´ Ωβ˚q ›››› 2 }p pβ ´ βtq}2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='37) `2 sup v,wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qtwTΩp pβ ´ βtqu ˇˇˇˇ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='38) Now for (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='36), because by Condition (C1) in the supplementary material, we have |pWi ´ Ωβ˚qTp pβ ´ βtq| “ |pWi ´ Ωβ˚qTp pβ ´ βtq{} pβ ´ βt}2|} pβ ´ βt}2 ď MW ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ βt}2 ` }Ω}2}β˚}2} pβ ´ βt}2 40 ď 2MW ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ βt}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The last equality holds because }Ω}2 “ Op1q, }β˚}2 ď }βt}2 ` } pβ ´ βt}2 “ Opp1q ` Op tlogppq{nu1{4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k “ opp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' From Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4 in the supplementary mate- rial, we have sup vPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2v sup i |pWi ´ Ωβ˚qTp pβ ´ βq| ´ Etexppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2vusup i |pWi ´ Ωβ˚qTp pβ ´ βq| ˇˇˇˇ ď 2MW a pm ` kq{n ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ β}2 with probability 1´Orexpt´pm`kqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further since supvPK Etexppβ˚TWi´β˚TΩβ˚{2qvTpWi´ Ωβ˚qb2vu “ Op1q due to Condition C1(b), hence we get sup vPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2vpWi ´ Ωβ˚qTp pβ ´ βq ˇˇˇˇ ď sup vPK # n´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚qb2v + sup i |pWi ´ Ωβ˚qTp pβ ´ βq| ď t2MW a pm ` kq{n ` Opp1qu ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ β}2 “ Opp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ β}2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='37), we first have }n´1 nÿ i“1 expp2β˚TWi ´ β˚TΩβ˚qpWi ´ Ωβ˚qb2 MYS ´Etexpp2β˚TWi ´ β˚TΩβ˚qpWi ´ Ωβ˚qb2 MYSu}2 “ Opt a pm ` kq{nu by Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further }Etexpp2β˚TWi´β˚TΩβ˚qpWi´ Ωβ˚qb2 MYSu}2 “ Op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='37) is of order Opp} pβ ´ β}2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now for (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='38), because ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvTpWi ´ Ωβ˚q ˇˇˇˇ 2 ď n´1 nÿ i“1 texpp2β˚TWi ´ β˚TΩβ˚qvTpWi ´ Ωβ˚qb2v, by Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3 in the supplementary material, we have n´1 nÿ i“1 expp2β˚TWi ´ β˚TΩβ˚qvTpWi ´ Ωβ˚qb2v ´Etexpp2β˚TWi ´ β˚TΩβ˚qvTpWi ´ Ωβ˚qb2vu “ Opp a pm ` kq{nq ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 41 with probability of the order 1 ´ Otexppm ` kqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further because Etexpp2β˚TWi ´ β˚TΩβ˚qvTpWi ´ Ωβ˚qb2vu “ Op1q, the third term of the last line is of the order Opp} pβ ´ βt}2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, we have sup v,wPK ˇˇˇˇvT " pQpβ˚q ´ B2Lpβtq BβBβT t w ˇˇˇˇ “ Opp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ βt}2q Hence for some positive constants C3, sup v,wPK ˇˇˇˇvT " pQpβ˚q ´ B2Lpβtq BβBβT w ˇˇˇˇ ď C3 "logppq n 1{4 pm ` kq (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='39) with probability 1 ´ Orexpt´pm ` kqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, by Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2 in the supplementary material, we also have sup v,wPK ˇˇˇˇvT „B2Lpβtq BβBβT ´ Qpβtq ȷ w ˇˇˇˇ “ sup v,wPK ˇˇˇˇvT MYS „B2Lpβtq BβBβT ´ Qpβtq ȷ MYS,MYS wMYS ˇˇˇˇ “ Opt a pm ` kq{nu “ op «"logppq n 1{4 pm ` kq ff (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='40) with probability 1 ´ 2expt´pm ` kqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, recall that K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇvT 1 " pQpβ˚q ´ B2Lpβtq BβBβT w ˇˇˇˇ ď sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK «ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 tpWi ´ Ωβ˚qb2 ´ ΩuwtWi ´ Ωβ˚uTp pβ ´ βtq ˇˇˇˇ ` ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvT 1 pWi ´ Ωβ˚qwTΩ ` wTpWi ´ Ωβ˚qvTΩup pβ ´ βtq ˇˇˇˇ ff ď sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvT 1 pWi ´ Ωβ˚qb2wpWi ´ Ωβ˚qTp pβ ´ βtq ´vT 1 ΩwpWi ´ Ωβ˚qTp pβ ´ βtqu ˇˇˇˇ ` sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvT 1 pWi ´ Ωβ˚qwTΩ `wTpWi ´ Ωβ˚qvT 1 Ωup pβ ´ βtq ˇˇˇˇ ď sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 pWi ´ Ωβ˚qb2wpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ ` sup v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 ΩwpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ 42 ` sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvT 1 pWi ´ Ωβ˚qwTΩ `wTpWi ´ Ωβ˚qvT 1 Ωup pβ ´ βtq ˇˇˇˇ ď sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK n´1 nÿ i“1 ˇˇˇˇexppβ˚TWi ´ β˚TΩβ˚{2qvT 1 pWi ´ Ωβ˚qb2w ˇˇˇˇ ˇˇˇˇpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ ` sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 ΩwpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ ` sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvT 1 pWi ´ Ωβ˚qwTΩ `wTpWi ´ Ωβ˚qvT 1 Ωup pβ ´ βtq ˇˇˇˇ ď sup v1PK1 1 2n´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 pWi ´ Ωβ˚qb2v1 sup i ˇˇˇˇpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='41) ` sup wPK 1 2n´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qwTpWi ´ Ωβ˚qb2w sup i ˇˇˇˇpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='42) ` sup v1PK1,wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 ΩwpWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='43) ` sup v1PK1,wPK ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtvT 1 pWi ´ Ωβ˚qwTΩ `wTpWi ´ Ωβ˚qvT 1 Ωup pβ ´ βtq ˇˇˇˇ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='44) For (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='41), from Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4, we have sup v1PK1 ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 pWi ´ Ωβ˚qb2v1 sup i |pWi ´ Ωβ˚qTp pβ ´ βtq| ´ Etexppβ˚TWi ´ β˚TΩβ˚{2qvT 1 pWi ´ Ωβ˚qb2v1usup i |pWi ´ Ωβ˚qTp pβ ´ βtq| ˇˇˇˇ ď 2MW a logppq{n ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ βt}2 with probability 1´Orexpt´logppqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further since supv1PK1 Etexppβ˚TWi´β˚TΩβ˚{2qvT 1 pWi´ Ωβ˚qb2v1u “ Op1q due to Condition C1(b), hence we get sup v1PK1 ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 pWi ´ Ωβ˚qb2v1pWi ´ Ωβ˚qTp pβ ´ βtq ˇˇˇˇ ď sup v1PK1 # n´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 pWi ´ Ωβ˚qb2v1 + sup i |pWi ´ Ωβ˚qTp pβ ´ βtq| ď t2MW a logppq{n ` Opp1qu ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ βt}2 ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 43 “ Opp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ β}2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='42), we use the same argument as those lead to the order of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='36), we have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='42) is of order Opp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ βt}2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='43), we use the same argument as those lead to the order of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='37), we have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='43) is of order Opp} pβ ´ βt}2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now for (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='44), because ˇˇˇˇn´1 nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qvT 1 pWi ´ Ωβ˚q ˇˇˇˇ 2 ď n´1 nÿ i“1 texpp2β˚TWi ´ β˚TΩβ˚qvT 1 pWi ´ Ωβ˚qb2v1, by Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3 in the supplementary material, we have n´1 nÿ i“1 expp2β˚TWi ´ β˚TΩβ˚qvT 1 pWi ´ Ωβ˚qb2v1 ´Etexpp2β˚TWi ´ β˚TΩβ˚qvT 1 pWi ´ Ωβ˚qb2v1u “ Opp a logppq{nq with probability of the order 1 ´ Orexpt´logppqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further because Etexpp2β˚TWi ´ β˚TΩβ˚qvT 1 pWi ´ Ωβ˚qb2v1u “ Op1q, and because of the same argument as those lead to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='43), we conclude that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='44) is of the order Opp} pβ ´ βt}2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence follow (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='36), and by Conditions (C1), Corollaries B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2–B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5 in the supplementary material, for positive constants C4 we have sup v1PK1,wPK ˇˇˇˇvT 1 " pQpβ˚q ´ B2Lpβq BβBβT w ˇˇˇˇ “ Opp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k} pβ ´ βt}2q ď C4 "logppq n 1{4 pm ` kq (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='45) and sup v1PK1,wPK ˇˇˇˇvT 1 „B2Lpβtq BβBβT ´ Qpβtq ȷ w ˇˇˇˇ “ sup v1PK1,wPK ˇˇˇˇvT 1pMYSqc „B2Lpβtq BβBβT ´ Qpβtq ȷ pMYSqc,MYS wMYS ˇˇˇˇ “ Opr a maxtpm ` kq,logppqu{ns “ op «"logppq n 1{4 pm ` kq ff , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='46) with probabilities to the order of 1 ´ 2expr´maxtpm ` kq,logppqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Combine (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='39) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='40) with Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='11 in the supplementary material, }pt pQpMYSq,MYSqpβ˚qu´1 ´ tQpMYSq,MYSpβtqu´1}2 ď }tQpMYSq,MYSpβtqu´1}2 2} pQpMYSq,MYSpβ˚q ´ QpMYSq,MYSpβtq}2 t1 ´ }tQpMYSq,MYSpβtqu´1}2} pQpMYSq,MYSpβ˚q ´ QpMYSq,MYSpβtq}2u 44 “ Opp} pQpMYSq,MYSpβ˚q ´ Q0pMYSq,MYSpβtq}2q “ C9 «"logppq n 1{4 pm ` kq ff , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='47) and hence }pt pQpMYSq,MYSqpβ˚qu´1 ´ tQpMYSq,MYSpβtqu´1}8 “ C9pm ` kq3{2 "logppq n 1{4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='48) Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ›››› pQpMYSqc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qt pQpMYSq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1 # BLp qβq Bβ + MYS ›››› 8 ď ››››r pQpMYSqc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚q ´ QpMYSqc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqst pQpMYSq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1 # BLp qβq Bβ + MYS ›››› 8 ` ››››QpMYSqc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqpt pQpMYSq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1 ´tQpMYSq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqu´1q # BLp qβq Bβ + MYS ›››› 8 ` ››››QpMYSqc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqtQpMYSq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqu´1 # BLp qβq Bβ + MYS ›››› 8 ď sup v1PK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK ˇˇˇˇvT 1pMYSqcr pQpMYSqc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚q ´ QpMYSqc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqswMYS ˇˇˇˇ ˆ}t pQpMYSq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1}2 ›››› # BLp qβq Bβ + MYS ›››› 2 `}QpMYSqc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtq}2 ››››t pQpMYSq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1 ´ tQpMYSq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqu´1 ›››› 2 ›››› # BLp qβq Bβ + MYS ›››› 2 ` ››››QpMYSqc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqtQ0pMYSq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqu´1 # BLp qβq Bβ + MYS ›››› 8 ď C10 «"logppq n 1{4 pm ` kq ff ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k a logppq{n (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='49) `C11 «"logppq n 1{4 pm ` kq ff ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k a logppq{n (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='50) ` ››››QpMYSqc,MYSpβtqtQpMYSq,MYSpβtqu´1 # BLp qβq Bβ + MYS ›››› 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='49) is obtained by using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='45) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='46) and the fact that }t pQpMYSq,MYSpβ˚qu´1}2 ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 45 ď }t pQpMYSq,MYSpβ˚qu´1 ´ rQpMYSq,MYSpβtqs´1}2 ` }tQpMYSq,MYSpβtqu´1}2 ď Opp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='50) is obtained by using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='39), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='40) and the fact that }QpMYSqc,MYSpβq}2 “ Op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, now note that n ě c8pm ` kq4logppq and λ “ Ortlogppq{nu1{4s by the statement assumption, together with Condition (C4), we have ›››› pQpMYSqc,MYSpβ˚qt pQpMYSq,MYSpβ˚qu´1 # BLp qβq Bβ + MYS ›››› 8 ď ››››QpMYSqc,MYSpβtqtQpMYSq,MYSpβtqu´1 # BLp qβq Bβ + MYS ›››› 8 ` oppλq ď ››››QpMYSqc,MYSpβqtQpMYSq,MYSpβtqu´1 ›››› 2 ›››› # BLp qβq Bβ + MYS ›››› 2 ` oppλq “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k a logppq{n ` oppλq “ oppλq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='51) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Therefore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='}pATpzqpMYSqc}8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='“ }λ´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='–´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='BLp qβq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Bβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pMYSqc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='fi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='fl ` λ´1 pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='˜«# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='BLp qβq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Bβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='AT Bqλp pβMcq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='BβMc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='ff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='` λpATpzqMYS ` pAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1 CTµ˚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3qpMYSq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='¸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='}8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='ď λ´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='›››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='–´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='BLp qβq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Bβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pMYSqc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='fi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='fl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='›››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='`λ´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='›››› pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='BLp qβq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Bβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='›››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='`λ´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='›››› pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='˜« ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='AT Bqλp pβMcq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='BβMc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='ff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='` λpATpzqMYS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='¸›››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='`λ´1} pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1pAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1 CTµ˚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3qMYS}8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='“ λ´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='›››› pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='˜« ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='AT Bqλp pβMcq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='BβMc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='ff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='` λpATpzqMYS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='¸›››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='`λ´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='›››› pQpMYSqcpMYSqpβ˚qt pQpMYSqpMYSqpβ˚qu´1A2t pQMYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβ˚qu´1 ˆ «# AT Bqλp pβMcq BβMc + MYS ´ λpATpzqMYS ff›››› 8 ` op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='52) 46 Now recall (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='35) and pA1CTµ˚ 3qMYS (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='53) “ A2t pQMYS,MYSpβ˚qu´1 « ´ # BLp qβq Bβ + MYS ` # AT Bqλp pβMcq BβMc + MYS ´ λpATpzqMYS ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence } pβMYS ´ qβMYS}8 ď }t pQMYS,MYSpβ˚qu´1 # BLp qβq Bβ + MYS }8 ` ››››t pQMYS,MYSpβ˚qu´1 «# AT Bqλp pβMcq BβMc + MYS ´ λpATpzqMYS ff›››› 8 `}t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1 ˆ « ´ # BLp qβq Bβ + MYS ` # AT Bqλp pβMcq BβMc + MYS ´ λpATpzqMYS ff }8 ď oppλq ` λ}t pQMYS,MYSpβ˚qu´1}8 ` λ}t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1}8 “ oppλq ` 2λc8 ` 2λc8, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='54) where oppλq in the second inequality is obtained by using the same argument as those lead to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The other two terms are obtained by using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4 such that ›››› «# AT Bqλp pβMcq BβMc + MYS ´ λpATpzqMYS ff›››› 8 ď λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The last equality holds because }t pQMYS,MYSpβ˚qu´1}8 ď }t pQMYS,MYSpβ˚qu´1 ´ tQpMYSq,MYSpβtqu´1}8 ` }tQpMYSq,MYSpβtqu´1}8 ď 2c8 by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' And similarly, we have }t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1}8 ď }t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1 ´tQMYS,MYSpβtqu´1A2tQMYS,MYSpβtqu´1}8 `}tQMYS,MYSpβtqu´1A2tQMYS,MYSpβtqu´1}8 “ 2c8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now because minp|βj|q ě λpγ ` 5c8q for j P S, we have min|pβj| ě λγ ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 47 and q1 λppβqj “ λsignppβjq “ λpATpzqj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3 in the supplementary material, we have # AT Bqλp pβMcq BβMc + MYS ´ λpATpzqMYS “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='55) Inserting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='55) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='52), the first two terms of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='52) are zero hence }pATzqpMYSqc} “ opp1q ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This implies that pβ has support in MYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, because of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='54), pβ is unique in the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='35) together with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='55) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='53), we have pβMYS ´ qβMYS “ pt pQMYS,MYSpβ˚qu´1 ´ t pQMYS,MYSpβ˚qu´1A2t pQMYS,MYSpβ˚qu´1q ˆ « ´ # BLp qβq Bβ + MYS ff “ ´ptQMYS,MYSpβqu´1 ´ tQMYS,MYSpβtqu´1A˚ 2tQMYS,MYSpβtqu´1q ˆ «# BLp qβq Bβ + MYS ff t1 ` opp1qu and pβpMYSqc “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now because BLp qβq Bβ ´ BLpβtq Bβ “ B2Lpβ˚q BβBβT p qβ ´ βtq “ Qpβqp qβ ´ βtq ` "B2Lpβtq BβBβT ´ Qpβtq p qβ ´ βtq ` "B2Lpβ˚q BβBβT ´ B2Lpβtq BβBβT p qβ ´ βtq “ Qpβtqp qβ ´ βtqt1 ` opp1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The last equality holds because } "B2Lpβtq BβBβT ´ Qpβtq }2 “ opp1q, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='40), } # B2Lp rβq BβBβT ´ B2Lpβtq BβBβT + }2 “ opp1q by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='39), and }Qpβtq}2 “ Op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, pβMYS ´ qβMYS “ ´ptQMYS,MYSpβtqu´1 ´ tQMYS,MYSpβtqu´1A˚ 2tQMYS,MYSpβtqu´1q ˆ „"BLpβtq Bβ ` Qpβtqp qβ ´ βtq MYS ȷ t1 ` opp1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, we have pβMYS ´ βtMYS 48 “ pβMYS ´ qβMYS ´ rpCCTq´1C,0rˆksThn “ ´ptQMYS,MYSpβtq´1u ´ tQMYS,MYSpβtqu´1A˚ 2tQMYS,MYSpβtqu´1q „"BLpβtq Bβ MYS `QMYS,MYSpβtqrpCCTq´1C,0rˆksThn ȷ t1 ` opp1qu ´ rpCCTq´1C,0rˆksThn “ ´tpQMYS,MYSpβtqu´1 ´ tQMYS,MYSpβtqu´1A˚ 2tQMYS,MYSpβtqu´1q "BLpβtq Bβ MYS t1 ` opp1qu ´tQMYS,MYSpβtqu´1A˚ 2rtpCCTq´1C,0rˆkuThnst1 ` opp1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The proof is very similar to that of Theorem 3 hence is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' First note that Ψ´1{2pΣ,Q,βtqωn “ ´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtq "BLpβtq Bβ MYS “ Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtq ˆn´1{2 nÿ i“1 tYiWi ´ exppβT t Wi ´ βT t Ωβt{2qpWi ´ ΩβtquMYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' It is easy to see that nÿ i“1 cov ´ Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtq ˆn´1{2tYiWi ´ exppβT t Wi ´ βT t Ωβt{2qpWi ´ ΩβtquMYS ¯ “ Irˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='56) Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' r1{4 nÿ i“1 E}n´1{2Ψ´1{2pΣ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='βtqCrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksQ´1 MYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtq ˆtYiWi ´ exppβT t Wi ´ βT t Ωβt{2qpWi ´ ΩβtquMYS}3 2 ď r1{4 nÿ i“1 E}n´1{2tYiΨ´1{2pΣ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='βtqCrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksQ´1 MYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqWiMYS ´exppβT t Wi ´ βT t Ωβt{2qΨ´1{2pΣ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='βtqCrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksQ´1 MYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqpWi ´ ΩβtqMYSu}3 2 ď r1{4 nÿ i“1 E}n´1{2tYiΨ´1{2pΣ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='βtqCrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksQ´1 MYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqWiMYS ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 49 ´exppβT t Wi ´ βT t Ωβt{2qΨ´1{2pΣ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='βtqCrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksQ´1 MYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqpWi ´ ΩβtqMYSu}3 2 ď r1{4D nÿ i“1 n´3{2p}Ψ´1{2pΣ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='βtqCrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksQ´1 MYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqWiMYS}3 2 `}Ψ´1{2pΣ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='βtqCrImˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='0mˆksQ´1 MYS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='MYSpβtqpΩβtqMYS}3 2q “ opp1q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='57) where D is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The second to the last inequality holds by Condition (D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Also because }Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtqWiMYS}2 ď }Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtq}2}WiMYS}2 ď }Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtq}2|WT i v|{}v}2 ď Op1qMW ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k “ Op ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` kq, where v is a p dimensional vector with }v}0 “ m ` k, the last inequality holds by Condition (C1) (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further since }Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtqpΩβtqMYS}2 “ Op1q, r1{4D nÿ i“1 n´3{2p}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtqWiMYS}3 2 `}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtqpΩβtqMYS}3 2q “ Otr1{4pm ` kq3{2n´1{2u “ opp1q, by the Condition that n ě c8pm`kq4logppq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Combine (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='56) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='57), and using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='14 in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3, let Z is a standard Gaussian random variable we have lim nÑ8sup C |PrpΨ´1{2pΣ,Q,βtqωn P Cq ´ PrpZ P Cq| “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='58) Now we choose Cx “ tz : }z ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nΨ´1{2hn}2 ď xu, then lim nÑ8|PrpΨ´1{2pΣ,Q,βtqωn P Cxq ´ PrpZ P Cxq| “ 0, which implies lim nÑ8|PrpT0 ď xq ´ Prtχ2pr,nhT nΨ´1pΣ,Q,βtqhnq ď xu| “ 0, where χ2pr,γq is a non-central chi-square distribution, with non-centrality parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 50 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' From Theorem 4, we have pβaMYS ´ βtMYS “ ´tQMYS,MYSpβtqu´1 "BLpβtq Bβ MYS t1 ` opp1qu Hence, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nCp pβaM ´ βtMq “ ´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nCrImˆm,0mˆkstQMYS,MYSpβtqu´1 "BLpβtq Bβ MYS t1 ` opp1qu “ ωnt1 ` opp1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, because CβtM “ t ` hn, we have ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nΨ´1{2ppΣ, pQ, pβaqpC pβaM ´ tq “ Ψ´1{2ppΣ, pQ, pβaqωnt1 ` opp1qu ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nΨ´1{2ppΣ, pQ, pβaqhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further because hn “ Ot a maxpm ` k ´ r,rq{nu, npC pβaM ´ tqTΨ´1ppΣ, pQ, pβaqpC pβaM ´ tq “ pωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqTΨ´1pΣ,Q,βtqpωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqt1 ` opp1qu `pωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqTtΨ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtqupωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnq ď pωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqTΨ´1pΣ,Q,βtqpωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqt1 ` opp1qu `}ωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhn}2 2}Ψ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtq}2 “ T0 ` opprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='59) The last equality holds because T0 converge in distribution to a non-central chi-square distri- bution with degree freedom r as shown in Lemma 1 and hence T0 is of the order Opprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Also because }ωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhn}2 2 ď pωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqTΨ´1pΣ,Q,βtqpωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnq{αmintΨ´1pΣ,Q,βtqu ď pωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqTΨ´1pΣ,Q,βtqpωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnq{cΨ by Condition (D2), we have }Ψ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtq}2}ωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhn}2 2 ď «"logppq n 1{4 pm ` kq ff OppT0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, TW ´ T0 “ opprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' From Theorem 3, we have pβMYS ´ βtMYS “ ´ptQMYS,MYSpβtqu´1 ´ tQMYS,MYSpβtqu´1A˚ 2tQMYS,MYSpβtqu´1q "BLpβtq Bβ MYS t1 ` opp1qu `tQMYS,MYSpβtqu´1A˚ 2rtpCCTq´1C,0rˆkuThnst1 ` opp1qu ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 51 and pβpMYSqc “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' By Taylor expansion we have # BLp pβq Bβ + MYS “ "BLpβtq Bβ MYS ` "B2Lpβ˚q BβBβT MYS p pβ ´ βtqMYS “ "BLpβtq Bβ MYS ` QMYS,MYSpβtqp pβ ´ βtqMYSt1 ` opp1qu “ A˚ 2tQMYS,MYSpβtqu´1 "BLpβtq Bβ MYS t1 ` opp1qu `tA˚ 2rtpCCTq´1C,0rˆkuThnst1 ` opp1qu where β˚ is a point in between βt and pβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The second equality holds by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='39) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='40) in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, we have ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nΨppΣ, pQ, pβq´1{2CrImˆm,0mˆkst pQMYS,MYSp pβqu´1A˚ 2tQMYS,MYSpβtqu´1 "BLpβtq Bβ MYS “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nΨppΣ, pQ, pβq´1{2CrImˆm,0mˆkstQMYS,MYSpβtqu´1 "BLpβtq Bβ MYS t1 ` opp1qu “ ΨppΣ, pQ, pβq´1{2ωnt1 ` opp1qu, The first equality holds by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='39) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='40) in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' And similarly ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nΨppΣ, pQ, pβq´1{2CrImˆm,0mˆkst pQMYS,MYSp pβqu´1tA˚ 2rtpCCTq´1C,0rˆkuThns “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nΨppΣ, pQ, pβq´1{2hnt1 ` opp1qu and hence TS “ pωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqTΨppΣ, pQ, pβq´1pωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqt1 ` opp1qu Now the same steps in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='59) lead to TS ´ T0 “ opprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Some Lemmas on the Penalty Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Conditions (A1)–(A4) imply that ρλ is λ-Lipschitz and all sub gra- dients and derivatives of ρλ are bounded by λ in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Conditions (A1)–(A5) imply λ}βMc}1 ď ρλpβMcq ` µ{2}βMc}2 2,@βMc P Rp´m Proof: This lemma is a direct consequence of Lemma 4 in Loh & Wainwright (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 52 Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose ρλ satisfies Conditions (A1)–(A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let v P Rp´m, let A be the index set of k largest elements of v in magnitude, and let Ac be the index set of the remaining p ´ m ´ k elements of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose ξ ą 0 and satisfies ξρλpvAq ´ ρλpvAcq ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then ξρλpvAq ´ ρλpvAcq ď λpξ}vA}1 ´ }vAc}1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Moreover, if βtMc P Rp´m is k-sparse, then for a vector βMc P Rp´m such that ξρλpβtMcq ´ ρλpβMcq ą 0 and ξ ě 1, we have ξρλpβtMcq ´ ρλpβMcq ď λpξ}vA}1 ´ }vAc}1q, where v “ βMc ´ βtMc, A is the index set of the k largest elements of v in magni- tude, and Ac is the index set of the remaining p ´ m ´ k elements of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: This lemma is a direct consequence of Lemma 5 in Loh & Wainwright (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose ρλ is pµ,γq-amenable, |pβj| ě λγ for j P M Y S, then q1 λppβjq “ λsignppβjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: Because ρλ is pµ,γq-amenable, ρ1ppβjq “ 0 by Condition (A6) and (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence q1 λppβjq “ Bλ|pβj|{Bpβj “ λsignppβjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Consider a µ-amenable regularizer ρλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then (a) |ρ1 λptq| ď λ for all t ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (b) The function qλptq ´ µ{2t2 is concave and everywhere differentiable, where qλptq “ λ|t| ´ ρλptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: This lemma is a direct consequence of Lemma 5 in Loh & Wainwright (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Some Lemmas on the Criterion Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume Conditions (C1) – (C4) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' There exists a constant c1 ą 0 so that pr « n´1} nÿ i“1 tYiWi ´ exppβT t Wi ´ βT t Ωβt{2qpWi ´ Ωβtqu}8 ą c1 a logppq{n ff ď 6p´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The lemma is the direct consequence of Corollary 1 in Jiang & Ma (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We omit the proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume that Conditions (C1), (C4) and (C6) hold, then for any β with }β}2 ď R2, and for sufficiently large n and p, with probability 1 ´ ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 53 Orexpt´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nlogpus, n´1 řn i“1 exppβTWi ´βTΩβ{2qtpWi ´Ωβqb2 ´Ωu satisfies the lower and upper-RE conditions with α1 “ min }β}1ďR1,}β}2ďR2 αminrEtexppβTXiqXiXT i us{2, α2 “ max }β}1ďR1,}β}2ďR2 3αmaxrEtexppβTXiqXiXT i us{2 and τpn,pq “ τ1 a logppq{n for a bounded positive constant τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The lemma is a direct consequence of Lemma 12 in Jiang & Ma (2021) with c “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume Conditions (C1)–(C4) to hold, }hn}2 “ Ot a maxpm ` k ´ r,rq{nu and m ` k “ opn1{3q, then there exists a positive constant c10 so that }BLp qβq Bβ }8 ď c10 maxr a pm ` kq{n, a logppq{ns, with probability 1 ´ Opp´1q ´ Orexpt´ a nlogppqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: }BLp qβq Bβ }8 “ n´1} nÿ i“1 tYiWi ´ expp qβTWi ´ qβTΩ qβ{2qpWi ´ Ω qβqu}8 ď n´1} nÿ i“1 tYiWi ´ exppβtTWi ´ βtTΩβt{2qpWi ´ Ωβtqu}8 `n´1} nÿ i“1 expp qβTWi ´ qβTΩ qβ{2qpWi ´ qβTΩqu ´ exppβtTWi ´ βtTΩβt{2qpWi ´ Ωβtqu}8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let K0 “ tv P Rp : vMc “ 0,}v}2 “ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then, n´1} nÿ i“1 expp qβTWi ´ qβTΩ qβ{2qpWi ´ qβTΩq ´ exppβtTWi ´ βtTΩβt{2qpWi ´ Ωβtq}8 ď n´1} nÿ i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtpWi ´ Ωβ˚qb2 ´ Ωut qβ ´ βtu}2 ď sup vPK n´1 nÿ i“1 vT exppβ˚TWi ´ β˚TΩβ˚{2qtpWi ´ Ωβ˚qb2 ´ Ωuv} qβ ´ βt}2 ď 3αmaxrEtexppβTXiqXiXT i us{2}CTpCCTq´1}2}hn}2 ` τ1m a logppq{n}CTpCCTq´1}2}hn}2 ď c10 maxr a pm ` kq{n, a logppq{ns 54 for some constants c10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The first inequality holds by the Taylor expansion with β˚ be the point on the line connecting qβ and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The second inequality holds be- cause qβ ´ β only has m nonzero elements supported in M, and hence only the corresponding m ˆ m sub-matrix in n´1 řn i“1 exppβ˚TWi ´ β˚TΩβ˚{2qtpWi ´ Ωβ˚qb2 ´ Ωu contributes to the L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The third inequality holds by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6 and the fact that v only as m non-zero elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The last equality holds because }hn}2 “ Ot a maxpm ` k ´ r,rq{nu ď Op a m ` k{nq, and m a m ` k{n Ñ 0 be- cause m ` k “ opn1{3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This proves the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume that Conditions (C1) and (C6) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Ui P Rp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' define K ” tv P RMYS : }v}2 ď 1u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' K1 ” tv P RpMYSqc : }v}2 “ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='}v}0 “ 1u Pr ˜ sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK | nÿ i“1 rApβTWiqgpWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wq ´ vTEtexppβTXiqXiXT i uws| ą nt ¸ ď 2exp ˆ ´min „ nt2 324e2M4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' nt 36eM5logpnq ȷ ` 2pm ` kqlogp9q ˙ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Pr ˜ sup vPK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK | nÿ i“1 rApβTWiqgpWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wq ´ vTEtexppβTXiqXiXT i uws| ą nt ¸ ď 2exp ˆ ´min „ nt2 36e2M4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' nt 12eM5logpnq ȷ ` pm ` kqlogp9q ` logppq ˙ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' and Pr ˜ sup vPK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK1 | nÿ i“1 rApβTWiqgpWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wq ´ vTEtexppβTXiqXiXT i uws| ą nt ¸ ď 2exp ˆ ´min „ nt2 16e2M4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' nt 8eM5logpnq ȷ ` 2logppq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: By Lemma 1 statement 3 and Lemma 3 statement 3 in Jiang & Ma (2021), we can see that the square of a conditional sub-Gaussian variable is sub-exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now because vTpWi ´ βTΩq given Xi and βTWi is normal, and hence vTpWi ´ βTΩqb2v is conditional sub-exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now by the Cauchy–Schwarz inequal- ity, and without loss of generality we assume vTpWi ´ βTΩqb2v ě wTpWi ´ βTΩqb2w, we have |vTpWi ´ βTΩqb2w| ď vTpWi ´ βTΩqb2v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence, by Lemma 3 statement 3 in Jiang & Ma (2021), we have for some bounded positive K3pβTWi,Xiq, Erexpt|vTpWi ´ βTΩqb2w|{K3pβTWi,Xiqu|βTWi,Xis ď Erexpt|vTpWi ´ βTΩqb2v|{K3pβTWi,Xiqu|βTWi,Xis ď e Hence, vTpWi ´ βTΩqb2w is also conditional sub-exponential variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, we have that gpWi,β,v,wq ´ EtgpWi,β,v,wq|βTWi,Xiu ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 55 is centered sub-exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then using the same argument as those that lead to Lemma 8 in Jiang & Ma (2021) and Condition (C6), for any unit vectors v,w, we can show that Pr ˜ | nÿ i“1 rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT i uws| ą nt ¸ ď 2exp ˆ ´min „ nt2 16e2M4 , nt 8eM5logpnq ȷ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now we define B “ tu1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',uru Ă K to be a 1/3-cover of K, if for every v P K, there is some ui P B such that }v ´ ui}2 ď 1{3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define ∆v “ v ´ uj where uj “ arg minui }v ´ ui}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We have }∆v}2 ď 1{3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Similarly define uk “ arg minui }w ´ ui}2 for w P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' By ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=', we can construct B with |B| ă 9pm`kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now for v1,v2 P K, define Φpv1,v2q “ vT 1 « nÿ i“1 ApβTWiqtpWi ´ Ωβqb2 ´ Ωu ´ EtexppβTXiqXiXT i u n ff v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We have |Φpv,wq| “ |Φp∆v ` uj,∆w ` ukq| ď max j,k |Φpuj,ukq| ` max i |Φp∆v,uiq| ` max i |Φpui,∆wq| ` |Φp∆v,∆wq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Since }3∆v}2 ď 1 and suppp3∆vq Ď K, 3∆v P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' It follows that sup v,wPK |Φpv,wq| ď max j,k |Φpuj,ukq| ` 1{3sup vPK max i |Φp3∆v,uiq| ` 1{3 sup wPK max i |Φpuk,3∆wq| ` 1{9 sup v,wPK |Φp3∆v,3∆wq| ď max j,k |Φpuj,ukq| ` 2{3tsup vPK |Φp3∆v,3∆vq|u1{2tmax i |Φpui,uiq|u1{2 ` 1{9 sup v,wPK |Φpv,wq| ď max j,k |Φpuj,ukq| ` sup v,wPK t2{3|Φpv,wq| ` 1{9|Φpv,wq|u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence, supv,wPK |Φpv,wq| ď 9{2maxj,k |Φpuj,ukq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' By a union bound while con- sidering |B| ă 92pm`kq, we have Pr ˜ sup v,wPK | nÿ i“1 rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT i uws| ą 9{2nt ¸ ď Pr ˜ max j,k | nÿ i“1 rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT i uws| ą nt ¸ ď Pr ˜ max j,k | nÿ i“1 rApβTWiqgpWi,β,v{}v}2,w{}w}2q ´ v{}v}T 2 EtexppβTXiqXiXT i uw{}w}2s| ą nt ¸ ď 2exp ˆ ´min „ nt2 16e2M4 , nt 8eM5logpnq ȷ ` 2pm ` kqlogp9q ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 56 It also follows that sup vPK1,wPK |Φpv,wq| ď max vPK1,j |Φpv,ujq| ` 1{3 sup vPK1,wPK |Φpv,3∆wq| ď max vPK1,j |Φpv,ujq| ` 1{3 sup vPK1,wPK t|Φpv,wq|, so supvPK1,wPK |Φpv,wq| ď 3{2maxvPK1,j |Φpv,ujq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Because v P K1 is a vector with a single nonzero entry 1, there are only p ´ m ´ k elements in K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We thus have Pr ˜ sup vPK1,wPK | nÿ i“1 rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT i uws| ą 3{2nt ¸ ď Pr ˜ max vPK1,uj | nÿ i“1 rApβTWiqgpWi,β,v,ujq ´ vTEtexppβTXiqXiXT i uujs| ą nt ¸ ď 2exp ˆ ´min „ nt2 16e2M4 , nt 8eM5logpnq ȷ ` pm ` kqlogp9q ` logppq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, since K1 contains only unit vectors, we have Pr ˜ sup vPK1,wPK1 | nÿ i“1 rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT i uws| ą nt ¸ ď 2exp ˆ ´min „ nt2 16e2M4 , nt 8eM5logpnq ȷ ` 2logppq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume that Conditions (C1), (C4) and (C6) hold and m ` k “ otn{log2pnqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Ui P Rp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' define K ” tv P RMYS : }v}2 ď 1u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' K1 ” tv P RpMYSqc : }v}2 “ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='}v}0 “ 1u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' there are a0 ą 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a1 ą 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='b1 ą 0 such that Pr ˜ sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK | nÿ i“1 rApβTWiqgpWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wq ´vTEtexppβTXiqXiXT i uws| ą na0 c m ` k n ¸ ď 2expt´pm ` kqu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Pr ˜ sup vPK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK | nÿ i“1 rApβTWiqgpWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wq ´vTEtexppβTXiqXiXT i uws| ą na1 c maxrm ` k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='logppqs n ¸ ď 2expr´maxtlogppq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pm ` kqus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 57 and Pr ˜ sup vPK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK1 | nÿ i“1 rApβTWiqgpWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wq ´vTEtexppβTXiqXiXT i uws| ą nb1 c logppq n ¸ ď 2expr´logppqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7, take t “ a0 a pm ` kq{n, we have Pr ˜ sup v,wPK | nÿ i“1 rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT i uws| ą nt ¸ ď 2exp ˜ ´min « a2 0pm ` kq 324e2M4 , a0 a npm ` kq 36eM5logpnq ff ` 2pm ` kqlogp9q ¸ “ 2exp ˆ ´a2 0pm ` kq 324e2M4 ` 2pm ` kq ˙ “ 2expt´pm ` kqu The second to the last equality holds because m`k “ otn{log2pnqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The last equality holds by choosing a0 “ a 972e2M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, by the second relation in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7, take t “ a1 a maxtlogppq,pm ` kqu{n, we have Pr ˜ sup vPK1,wPK | nÿ i“1 rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT i uws| ą nt ¸ ď 2exp ˆ ´min „a2 1 maxtlogppq,pm ` kqu 36e2M4 , a1 a nmaxtlogppq,pm ` kqu 12eM5logpnq ff ` pm ` kqlogp9q ` logppq ¸ “ 2exp " ´min ˆa2 1 maxtlogppq,pm ` kqu 36e2M4 , a1 maxrlogppq?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='n{t12eM5 a logppqlogpnqu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' a npm ` kq{t12eM5logpnqus ¯ `2pm ` kq ` logppq ˙ ď 2exp ˆ ´min ˆa2 1 maxtlogppq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pm ` kqu 36e2M4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' a1 maxrlogppq{p12eM5Cq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' a npm ` kq{t12eM5logpnqus ¯ 58 `2pm ` kq ` logppq ď 2exp " ´min „a2 1 maxtlogppq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pm ` kqu 36e2M4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' a1 maxtlogppq{p12eM5Cq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pm ` kq{p12eM5qus ` 3maxtpm ` kq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='logppqu ď 2expp´a2 maxtlogppq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pm ` kqu ` 3maxtlogppq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pm ` kquq “ 2expr´maxtlogppq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pm ` kqus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' where a2 “ minta2 1{p36e2M4q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a1{p12eM5Cq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a1{p12eM5qu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' and we select a1 such that a2 ě 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The third equality holds because logpnq ď C a n{logppq by Condition (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The fourth equality holds pm ` kq “ otn{log2pnqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In addition, by the third relation in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7, take t “ b1 a logppq{n, we have Pr ˜ sup vPK1,wPK1 | nÿ i“1 rApβTWiqgpWi,β,v,wq ´ vTEtexppβTXiqXiXT i uws| ą nt ¸ ď 2exp ˜ ´min « b2 1logppq 16e2M4 , b1 a nlogppq 8eM5logpnq ff ` 2logppq ¸ “ 2exp " ´min ˆb2 1logppq 16e2M4 ,b1logppq?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='n{t8eM5 a logppqlogpnqu ˙ ` 2logppq ˙ ď 2exp ˆ ´min ˆb2 1logppq 16e2M4 ,b1logppq{p8eM5Cq ˙ ` 2logppq ď 2expp´b12logppq ` 2logppqq “ 2expr´logppqs, where b12 “ mintb2 1{p16e2M4q,b1{p8eM5Cqu, and we select b1 such that b12 ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The second inequality holds because logpnq ď C a n{logppq by Condition (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume that Conditions (C1) and (C7) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If Xi,Ui P Rp, define K ” tv P RMYS : }v}2 ď 1u, K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then Pr ˜ sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK | nÿ i“1 rA2pβTWiqg1pWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wq ´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws ą nt ¯ ď 2exp ˆ ´min „ nt2 324e2M6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' nt 36eM7logpnq ȷ ` 2pm ` kqlogp9q ˙ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Pr ˜ sup vPK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK | nÿ i“1 rA2pβTWiqg1pWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wq ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 59 ´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| ą nt ˙ ď 2exp " ´min „ nt2 36e2M6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' nt 12eM7logpnq ȷ ` pm ` kqlogp9q ` logppq ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' and Pr ˜ sup vPK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK1 | nÿ i“1 rA2pβTWiqg1pWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wq ´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| ą nt ˙ ď 2exp " ´min „ nt2 16e2M6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' nt 8eM7logpnq ȷ ` 2logppq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The lemma holds by using the same arguments as those lead to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume that Conditions (C1), (C4) (C7) hold, m`k “ otn{log2pnqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If Xi,Ui P Rp, define K ” tv P RMYS : }v}2 ď 1u, K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' There are a2 ą 0,a3 ą 0,b3 ą 0 such that Pr ˜ sup v,wPK | nÿ i“1 rA2pβTWiqg1pWi,β,v,wq ´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| ą na2 c pm ` kq n ¸ ď 2expt´pm ` kqu, Pr ˜ sup vPK1,wPK | nÿ i“1 rA2pβTWiqg1pWi,β,v,wq ´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| ą na3 c maxrpm ` kq,logppqs n ¸ ď 2expr´maxtlogppq,pm ` kqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' and Pr ˜ sup vPK1,wPK1 | nÿ i“1 rA2pβTWiqg1pWi,β,v,wq ´vTEtexpp2βTWi ´ βTΩβqpWi ´ Ωβqb2uws| ą nb3 c logppq n ¸ ď 2expt´logppqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 60 Proof: The corollary follows the same arguments as those lead to Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume that Conditions (C1), (C4) (C7) hold, m`k “ otn{log2pnqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If Xi,Ui P Rp, define K ” tv P RMYS : }v}2 ď 1u, K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' There are a21 ą 0,a31 ą 0,b31 ą 0 such that Pr ˜ sup v,wPK | nÿ i“1 rApβTWiqg1pWi,β,v,wq ´vTEtexppβTWi ´ βTΩβ{2qpWi ´ Ωβqb2uws| ą na21 c pm ` kq n ¸ ď 2expt´pm ` kqu, Pr ˜ sup vPK1,wPK | nÿ i“1 rApβTWiqg1pWi,β,v,wq ´vTEtexppβTWi ´ βTΩβ{2qpWi ´ Ωβqb2uws| ą na31 c maxrpm ` kq,logppqs n ¸ ď 2expr´maxtlogppq,pm ` kqus, and Pr ˜ sup vPK1,wPK1 | nÿ i“1 rApβTWiqg1pWi,β,v,wq ´vTEtexppβTWi ´ βTΩβ{2qpWi ´ Ωβqb2uws| ą nb31 c logppq n ¸ ď 2expt´logppqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The corollary follows the same arguments as those that lead to Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume that Conditions (C1) holds, m ` k “ otn{log2pnqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If Xi,Ui P Rp, K ” tv P RMYS : }v}2 “ 1u, K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then there exists constants c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c10 ą 0 Pr ˜ sup v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK | nÿ i“1 tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| ą nt ¸ ď 2exp ` ´c2nminpc3t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c4tq ` 2pm ` kqlogp9q ˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Pr ˜ sup vPK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK | nÿ i“1 tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| ą nt ¸ ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 61 ď 2exp ` ´c7nminpc5t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c6tq ` pm ` kqlogp9q ` logppq ˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' and Pr ˜ sup vPK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='wPK1 | nÿ i“1 tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| ą nt ¸ ď 2exp ` ´c10nminpc8t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='c9tq ` 2logppq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The lemma is a consequence of Lemma 15 in Loh & Wainwright (2012) and using the same arguments as those lead to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume that Conditions (C1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If Xi,Ui P Rp, K ” tv P RMYS : }v}2 ď 1u, K1 ” tv P RpMYSqc : }v}2 “ 1,}v}0 “ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' There are constants a4 ą 0,a5 ą 0,b5 ą 0 such that Pr ˜ sup v,wPK | nÿ i“1 tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| ą na4 c pm ` kq n ¸ ď 2expt´pm ` kqu, Pr ˜ sup vPK1,wPK | nÿ i“1 tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| ą na5 c maxrpm ` kq,logppqs n ¸ ď 2expr´maxtlogppq,pm ` kqus, and Pr ˜ sup vPK1,wPK1 | nÿ i“1 tvTpWi ´ Ωβqb2w ´ EpvTpWi ´ Ωβqb2wqu| ą nb5 c logppq n ¸ ď 2expt´logppqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The corollary follows the same arguments as those lead to Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemmas on Criterion Function and Penalty Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Consider a µ-amenable regularizer ρλ, with µ ă α1 and n ą logppqτ2 1 pm ` kq2{pα1 ´ µq2, where α1,τ1 are defined in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then the func- tion Lpβq´µ}βMc}2 2{2 and Lpβq`ρλpβMcq are strictly convex on β P RMYS, and hence the restricted program (14) is also strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: First define a vector v P Rp with the jth element |vj| ą 0 if j P M Y S, and }vj} “ 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6, we have for β in the feasible set of program 62 (14), vTB2Lpβq BβBβT v ě α1}v}2 2 ´ τ1 c logppq n }v}2 1, and }v}1 ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ` k}v}2, and hence we have vTB2Lpβq BβBβT v ě # α1 ´ τ1pm ` kq c logppq n + }v}2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, vT MYS "B2Lpβq BβBβT pM`SqpM`Sq vMYS ´ µvT SvS ě # α1 ´ µ ´ τ1pm ` kq c logppq n + }v}2 2, where ␣ B2Lpβq{pBβBβTq ( pMYSq,pMYSq is the pm ` kq ˆ pm ` kq block of ␣ B2Lpβq{BβBβT( corresponding to M Y S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence Lpβq ´ µ}βMc}2 2{2 is strictly convex on RMYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Finally, because Lpβq ´ qλpβMcq “ pLpβq ´ µ}βMc}2 2{2q ` tµ}βMc}2 2{2 ´ qλpβMcqu, where the second part is convex over RMYS by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4, hence Lpβq ´ qλpβMcq restricted to RMYS is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Because Lpβq ` ρλpβMcq “ Lpβq ´ qλpβMcq ` λ}βMc}1, the strict convexity of Lpβq ` ρλpβMcq over RMYS follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now as we know C contains r independent columns, without loss of generality, we write C “ pCr,Cm´rq where Cr is a full rank square matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let A,B P Rpˆp be invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' For any matrix norm }}, we have }A´1 ´ B´1} ď }A´1}2}A ´ B} 1 ´ }A´1}}A ´ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In particular, if }A´1}}A ´ B} ď 1{2, then }A´1 ´ B´1} “ Op}A´1}2}A ´ B}q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: This lemma is Lemma 11 in Loh & Wainwright (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose x˚ is feasible for the program min x tfpxq ´ gpxMcq ` λ}xMc}1u, such that }x}1 ď R1,}x}2 ď R2, and CxM “ t, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1) where f P C2, g P C1 and gpxMcq ´ κ{2}xMc}2 2 is concave and C is an r ˆ m ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu and A1 “ tImˆm,0mˆpp´mqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume there are v˚ P B}x˚ Mc}1, w˚ 1 P B}x˚}1, w˚ 2 P B}x˚}2, µ˚ 1 ě 0, µ˚ 2 ě 0, µ˚ 3 such that µ˚ 1pR1 ´ }x˚}1q “ 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2) µ˚ 2pR2 ´ }x˚}2q “ 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3) ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 63 Bfpx˚q Bx˚ ´ ATBgpx˚ Mcq Bx˚ Mc ` λATv˚ ` µ˚ 1w˚ 1 ` µ˚ 2w˚ 2 ` AT 1 CTµ˚ 3 “ 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4) sTB2fpxq BxBxT s ą κ,@s P G˚, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5) where G˚ :“ # s P Rp : }s}2 “ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' sup w1PB}x˚}1 sTw1 ď 0 if }x˚}1 “ R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' sup w2PB}x˚}2 sTw2 ď 0 if }x˚}2 “ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' sup vPB}x˚ Mc}1 sT ˆBfpx˚q Bx˚ ´ ATBgpx˚ Mcq Bx˚ Mc ˙ ` λsTATv “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' µ˚ 1 sup w1PB}x˚}1 sTw1 “ 0,µ˚ 2 sup w2PB}x˚}2 sTw2 “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='CsM “ 0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then x˚ is an isolated local minimum of the program (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The proof of this lemma is similar to the proof of Theorem 3 in Fletcher & Watson (1980) and that of Lemma 10 in Loh & Wainwright (2017), except that we allow additional constraints }x}2 ď R2 and CxM “ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose x˚ is not an isolated local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then there is a sequence pxkq, so that xk Ñ x˚ and φpxkq ď φpx˚q, where φpxq “ fpxq ´ gpxMcq ` λ}xMc}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let sk :“ pxk ´ x˚q{}xk ´ x˚}2, so pskq is a set of feasible directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Since pskq Ă B2p1q, where B2p1q is the ball with radius 1, the set must possess a point of accumulation s P B2p1q, and we can extract a con- vergence subsequence such that pskq Ñ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' With a slight abuse of notation, we still use pskq to denote the subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We will show that s P G˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' First of all by the construction, xk’s are all feasible, and hence Cxk M “ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' There- fore, Csk M “ 0, take the limits on the left and right of the equation implies CsM “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6) Further, because the feasible region is closed, s is also in the feasible direction at x˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' If }x˚}1 “ R1, by the sub-gradient of convexity function }x˚}1 we have 0 ě }xk}1 ´ }x˚}1 “ }x˚ ` }xk ´ x˚}2sk}1 ´ }x˚}1 ě }xk ´ x˚}2skTw1, for any w1 P B}x˚}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' When k Ñ 8, this leads to sup w1PB}x˚}1 sTw1 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7) Further (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2) also implies that if }x˚}1 ‰ R1, then µ˚ 1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Since µ˚ 1 ě 0, hence we have µ˚ 1 sup w1PB}x˚}1 sTw1 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8) 64 Similarly, by the sub-gradient of convexity function }x˚}2 we have 0 ě }xk}2 ´ }x˚}2 “ }x˚ ` }xk ´ x˚}2sk}2 ´ }x˚}2 ě }xk ´ x˚}2skTw2, for any w2 P B}x˚}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' When k Ñ 8, this leads to sup w2PB}x˚}2 sTw2 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='9) Further (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3) also implies that if }x˚}2 ‰ R2, then µ˚ 2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Since µ˚ 2 ě 0, hence we have µ˚ 2 sup w2PB}x˚}2 sTw2 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='10) Further by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4), we have sTBfpx˚q Bx˚ ´ sTATBgpx˚ Mcq Bx˚ Mc ` sTλATv˚ ` µ˚ 1sTw˚ 1 ` µ˚ 2sTw˚ 2 ` sTAT 1 CTµ˚T 3 “ 0 which, together with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6), implies sTBfpx˚q Bx˚ ´ sTATBgpx˚ Mcq Bx˚ Mc ` sTλATv˚ “ ´µ˚ 1sTw˚ 1 ´ µ˚ 2sTw˚ 2 ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='11) By the definition of sub-gradient, we have }x˚ Mc ` }xk ´ x˚}2Ask}1 ´ }x˚ Mc}1 ě }xk ´ x˚}2skTATv for all v P B}x˚ Mc}1 and all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further because }x˚ Mc `}xk ´x˚}2Ask}1 ´}x˚ Mc}1 “ }xk Mc}1 ´ }x˚ Mc}1, we have sTATv “ lim kÑ8skTATv ď lim kÑ8 }xk Mc}1 ´ }x˚ Mc}1 }xk ´ x˚}2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='12) for all v P B}x˚ Mc}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Furthermore, sTBfpx˚q Bx˚ ´ sTATBgpx˚ Mcq Bx˚ Mc “ lim kÑ8skTBfpx˚q Bx˚ ´ skTATBgpx˚ Mcq Bx˚ Mc “ lim kÑ8 xxk ´ x˚, Bfpx˚q Bx˚ ´ AT Bgpx˚ Mcq Bx˚ Mc y }xk ´ x˚}2 “ lim kÑ8 fpxkq ´ gpxk Mcq ´ fpx˚q ` gpx˚ Mcq }xk ´ x˚}2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='13) since xk Ñ x˚ and fpxkq´gpxk Mcq P C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Combining (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='12) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='13), we conclude that sup vPB}x˚ Mc}1 sTBfpx˚q Bx˚ ´ sTATBgpx˚ Mcq Bx˚ Mc ` λsTATv ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 65 ď lim kÑ8 φpxkq ´ φpx˚q }xk ´ x ˚ } ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Combining with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='11), we have 0 ď sTBfpx˚q Bx˚ ´ sTATBgpx˚ Mcq Bx˚ Mc ` λsTATv˚ ď sup vPB}x˚ Mc}1 sTBfpx˚q Bx˚ ´ sTATBgpx˚ Mcq Bx˚ Mc ` λsTATv ď 0, hence sup vPB}x˚ Mc}1 sTBfpx˚q Bx˚ ´ sTATBgpx˚ Mcq Bx˚ Mc ` λsTATv “ 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='14) Now together with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4), we have µ˚ 1sTw˚ 1 ` µ˚ 2sTw˚ 2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='8) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='10), we have µ˚ 1 sup w1PB}x˚}1 sTw1 “ µ˚ 2 sup w2PB}x˚}2 sTw2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='15) Combine (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='9), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='14) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='15), we conclude that s P G˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' By the convexity of the L1 norm, we have }xk}1 ´ }x˚}1 “ }x˚ ` pxk ´ x˚q}1 ´ }x˚}1 ě pxk ´ x˚qTw1 “ xkTw1 ´ }x˚}1 for all w1 P B}x˚}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, xkTw˚ 1 ď }xk}1 ď R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Similarly, by the convexity of the L2 norm, we have }xk}2 ´ }x˚}2 “ }x˚ ` pxk ´ x˚q}2 ´ }x˚}2 ě pxk ´ x˚qTw2 “ xkTw2 ´ }x˚}2 for all w2 P B}x˚}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, xkTw˚ 2 ď }xk}2 ď R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, xkTAT 1 CTµ˚ 3 ´ tTµ˚ 3 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence φpxkq “ fpxkq ´ gpxk Mcq ` λ}xk Mc}1 ě fpxkq ´ gpxk Mcq ` λxkT Mcv˚ ` µ˚ 1pxkTw˚ 1 ´ R1q ` µ˚ 2pxkTw˚ 2 ´ R2q `xkTAT 1 CTµ˚ 3 ´ tTµ˚ 3 for all v˚ P B}x˚ Mc}1, and φpx˚q “ fpx˚q ´ gpx˚ Mcq ` λx˚T Mcv˚ ` µ˚ 1px˚Tw˚ 1 ´ R1q ` µ˚ 2px˚Tw˚ 2 ´ R2q `x˚TAT 1 CTµ˚ 3 ´ tTµ˚ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The equality holds because µ˚ 1px˚Tw˚ 1 ´ R1q “ 0 and µ˚ 2px˚Tw˚ 2 ´ R2q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence lim kÑ8tφpxkq ´ φpx˚qu{}xk ´ x˚}2 2 ě lim kÑ8tfpxkq ´ fpx˚q ´ gpxk Mcq ` gpx˚ Mcq ` xλATv˚ ` µ˚ 1w˚ 1 ` µ˚ 2w˚ 2 66 `AT 1 CTµ˚ 3,xk ´ x˚yu{}xk ´ x˚}2 2 “ lim kÑ8tfpxkq ´ fpx˚q ´ gpxk Mcq ` gpx˚ Mcqu{}xk ´ x˚}2 2 ´ BBfpx˚q Bx˚ ´ ATBgpx˚ Mcq Bx˚ Mc ,xk ´ x˚ F {}xk ´ x˚}2 2 “ lim kÑ8 " fpxkq ´ fpx˚q ´ BBfpx˚q Bx˚ ,xk ´ x˚ F* {}xk ´ x˚}2 2 ´ " gpxk Mcq ´ gpx˚ Mcq ´ B ATBgpx˚ Mcq Bx˚ Mc ,xk ´ x˚ F* {}xk ´ x˚}2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='16) By the concavity of gpxMcq ´ κ{2}xMc}2 2, we have " gpxk Mcq ´ gpx˚ Mcq ´ B ATBgpx˚ Mcq Bx˚ Mc ,xk ´ x˚ F* ď κ{2}xk Mc ´ x˚ Mc}2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further note that φpxkq ´ φpx˚q ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Combine with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='16), we have lim kÑ8 " fpxkq ´ fpx˚q ´ BBfpx˚q Bx˚ ,xk ´ x˚ F* {}xk ´ x˚}2 2 ´κ{2}xk Mc ´ x˚ Mc}2 2{}xk ´ x˚}2 2 ď 0, which by Taylor expansion implies sTB2fpx˚q BxBxT s “ lim kÑ8pxk ´ x˚qTB2fpx˚q BxBxT pxk ´ x˚q{}xk ´ x˚}2 2 ď lim kÑ8tκ}xk Mc ´ x˚ Mc}2 2 ` op}pxk Mc ´ x˚ Mcq}2 2qu{}xk ´ x˚}2 2 ď κ, which contradicts with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence x˚ must be an isolated local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose x˚ is feasible for the program min x tfpxq ´ gpxMcq ` λ}xMc}1u, such that }x}1 ď R1,}x}2 ď R2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='17) where f P C2, g P C1 and gpxMcq ´ κ{2}xMc}2 2 is concave and C is an r ˆ m ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu and A1 “ tImˆm,0mˆpp´mqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume there are v˚ P B}x˚ Mc}1, w˚ 1 P B}x˚}1, w˚ 2 P B}x˚}2, µ˚ 1 ě 0, µ˚ 2 ě 0 such that µ˚ 1pR1 ´ }x˚}1q “ 0 µ˚ 2pR2 ´ }x˚}2q “ 0 Bfpx˚q Bx˚ ´ ATBgpx˚ Mcq Bx˚ Mc ` λATv˚ ` µ˚ 1w˚ 1 ` µ˚ 2w˚ 2 “ 0 sTB2fpxq BxBxT s ą κ,@s P G˚, where G˚ :“ # s P Rp : }s}2 “ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' sup w1PB}x˚}1 sTw1 ď 0 if }x˚}1 “ R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' sup w2PB}x˚}2 sTw2 ď 0 if }x˚}2 “ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 67 sup vPB}x˚ Mc}1 sT ˆBfpx˚q Bx˚ ´ ATBgpx˚ Mcq Bx˚ Mc ˙ ` λsTATv “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' µ˚ 1 sup w1PB}x˚}1 sTw1 “ 0,µ˚ 2 sup w2PB}x˚}2 sTw2 “ 0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then x˚ is an isolated local minimum of the program (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The Corollary holds by using the same argument as those lead to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='12, while ignoring the equality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume Conditions (C1)–(C6) hold, λ ď α1{p8R1q, R1 ď rnα2 1{t64τ2 1 logppqus1{4, δ P r4R1τ1 a logppq{n{λ,1s, n ě 4logppqτ2 1 t?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ´ r ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ku4{pα1 ´µq2, and α1 ą µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Define A “ t0pp´mqˆm,Ipp´mqˆpp´mqu and A1 “ tImˆm,0mˆpp´mqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose rβ is a stationary point of program (7) and pβ is the inte- rior local minimizer of (7) and satisfies suppp pβq Ď M Y S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then suppp rβq Ď M Y S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: Let rv :“ rβ ´ pβ, by the Taylor expansion of the first order derivative tBLp rβq{BβT ´ BLp pβq{BβTurv “ rvTB2Lpβ˚q{BβBβTrv, where β˚ is a point on the line connecting pβ and rβ and hence in the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6 we have tBLp rβq{BβT ´ BLp pβq{BβTurv ě α1}rv}2 2 ´ τ1 a logppq{n}rv}2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We first show that }rv}2 ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose that }rv}2 ą 1, we have tBLp rβq{BβT ´ BLp pβq{BβTurv ě α1}rv}2 ´ 2τ1R1 a logppq{n}rv}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='18) The first order condition implies !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' BLp rβq{Bβ ` ATBρλp rβMcq{BβMc )T p pβ ´ rβq ě 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='19) and hence BLp rβq{BβTrv ď ´ATBρλp rβMcq{BβT Mcrv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Combine with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='18), we have t´ATBρλp rβMcq{BβMc ´ BLp pβq{BβuTrv ě α1}rv}2 ´ 2τ1R1 a logppq{n}rv}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='20) Further because pβ is an interior local minimum, we have BLp pβq{Bβ ` ATBρλp pβMcq{BβMc ` AT 1 CTµ4 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' for some Lagrange multiplier µ4 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Note that pAT 1 CTµ4qTrv “ µT t Cp rβM ´ pβmMq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, plug in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='20), we have α1}rv}2 ´ 2τ1R1 a logppq{n}rv}1 ď tATBρλp pβMcq{BβMc ` AT 1 CTµ4 ´ ATBρλp rβMcq{BβMcuTrv 68 “ tATBρλp pβMcq{BβMc ´ ATBρλp rβMcq{BβMcuTrv ď t}ATBρλp pβMcq{BβMc}8 ` }ATBρλp rβMcq{BβMc}8u}rv}1 ď 2λ}rv}1, where }ATBρλpβMcq{BβMc}8 ď λ holds by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence we have }rv}2 ď t2λ ` 2τ1R1 a logppq{nu}rv}1{α1 ď 2R1t2λ ` 2τ1R1 a logppq{nu{α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The right hand side is at most 1 because λ ď α1{p8R1q, and n ě logppq64τ2 1 R4 1{α2 1, which contradicts with }rv}2 ą 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore }rv}2 ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now note that Lpβq ´ qλpβMcq “ pLpβq ´ µ}βMc}2 2{2q ` tµ}βMc}2 2{2 ´ qλpβMcqu and tµ}βMc}2 2{2 ´ qλpβMcqu is convex by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4, and hence for any β in the feasible set, we have rvTB2tLpβq ´ qλpβMcqu BβBβT rv ě rvTB2Lpβq BβBβT rv ´ µrvTATArv ě pα1 ´ µq}rv}2 2 ´ τ1 c logppq n }rv}2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='21) Further by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='19), we have 0 ď # BLp rβq BβT ´ Bqλp rβMcq BβT Mc A + p pβ ´ rβq ` λrzTAp pβ ´ rβq, where rz P B} rβMc}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, because pβ is an interior local minimum, for pz P B} pβMc}1, we have 0 “ BLp pβq{Bβ ` ATBρλp pβMcq{BβMc ` AT 1 CTµ4 “ # BLp pβq Bβ ´ ATBqλp pβMcq BβMc + ` λATpz ` AT 1 CTµ4, which leads to # BLp pβq BβT ´ Bqλp pβMcq BβT Mc A + rv ` λppzTAqrv “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence 0 ď # BLp pβq BβT ´ Bqλp pβMcq BβT Mc A + rv ´ # BLp rβq BβT ´ Bqλp rβMcq BβT Mc A + rv `λppzTAqp rβ ´ pβq ´ λprzTAqp rβ ´ pβq “ # BLp pβq BβT ´ Bqλp pβMcq BβT Mc A + rv ´ # BLp rβq BβT ´ Bqλp rβMcq BβT Mc A + rv `λppzTAq rβ ´ λ} pβMc}1 ´ λ} rβMc}1 ` λprzTAq pβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 69 which implies λ} rβMc}1 ´ λppzTAq rβ ď # BLp pβq BβT ´ Bqλp pβMcq BβT Mc A + rv ´ # BLp rβq BβT ´ Bqλp rβMcq BβT Mc A + rv ´ λ} pβMc}1 ` λprzTAq pβ ď # BLp pβq BβT ´ Bqλp pβMcq BβT Mc A + rv ´ # BLp rβq BβT ´ Bqλp rβMcq BβT Mc A + rv ď τ1 c logppq n }rv}2 1 ´ pα1 ´ µq}rv}2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='22) The second inequality holds because |przTAq pβ| “ |rzT pβMc| ď }rz}8} pβMc}1 ď } pβMc}1, and last inequality holds by the Taylor expansion and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now we first assume that the following statement holds: If }pATpzqMYS}8 ď 1´δ and λ ě 4R1τ1 a logppq{pnq{δ, then }rv}1 ď t?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ r ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ku}rv}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We then will have λ} rβMc}1 ´ λppzTAq rβ ď « τ1 c logppq n t?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ r ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ku2 ´ pα1 ´ µq ff }rv}2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now because n ě 4logppqτ2 1 t?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ´ r ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ku4{pα1 ´ µq2, we have 0 “ λ} rβMc}1 ´ λ} rβMc}1 ď λ} rβMc}1 ´ λppzTAq rβ ď ´pα1 ´ µq{2}rv}2 2 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The first inequality holds by the fact that ppzTAq rβ “ pzT rβMc ď }pz}8} rβMc}1 ď } rβMc}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence we have λ} rβMc}1 “ λppzTAq rβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now because }ppzTAqpMYSqc}8 ă 1, we conclude that rβpMYSqc “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Hence, suppp rβq Ă M Y S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This would prove the claim in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Thus, we only need to show that if }pATpzqMYS}8 ď 1´δ and λ ě 4R1τ1 a logppq{pnq{δ, then }rv}1 ď tp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ rq ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ku}rv}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' First from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='22), we have pα1 ´ µq}rv}2 2 ´ τ1 c logppq n }rv}2 1 ď # BLp rβq BβT ´ Bqλp rβMcq BβT Mc A + rv ´ # BLp pβq BβT ´ Bqλp pβMcq BβT Mc A + rv ď λppzTAq rβ ´ λ} pβMc}1 ´ λ} rβMc}1 ` λprzTAq pβ “ λprzTAq pβ ´ λ} rβMc}1 ` λppzTAqrv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='23) 70 The second inequality holds by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now since suppp pβq Ď M Y S, we have λprzTAq pβ ´ λ} rβMc}1 ď λ} pβMc}1 ´ λ} rβMc}1 “ λ} pβS}1 ´ λ} rβS}1 ´ λ} rβpMYSqc}1 “ λ} pβS}1 ´ λ} rβS}1 ´ pλ} rβpMYSqc}1 ´ λ} pβpMYSqc}1q “ λ} rβS ` pβS ´ rβS}1 ´ λ} rβS}1 ´ pλ} rβpMYSqc}1 ´ λ} rβpMYSqc ` pβpMYSqc ´ rβpMYSqc}1q ď λ}rvS}1 ´ λ}rvpMYSqc}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='24) In addition, λppzTAqrv “ λtpATpzqSuTrvS ` λtpATpzqpMYSqcuTrvpMYSqc ď λ}pATpzqS}8}rvS}1 ` λ}pATpzqpMYSqc}8}rvpMYSqc}1 ď λ}rvS}1 ` λp1 ´ δq}rvpMYSqc}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='25) Combine (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='23), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='24), and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='25), we have ´τ1 c logppq n }rv}2 1 ď pα1 ´ µq}rv}2 2 ´ τ1 c logppq n }rv}2 1 ď λ}rvS}1 ´ λ}rvpMYSqc}1 ` λ}rvS}1 ` λp1 ´ δq}rvpMYSqc}1 “ λ2}rvS}1 ´ λδ}rvpMYSqc}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now because δλ{2 ě 2τ1R1 a logppq{n ě τ1 a logppq{n}rv}1, the above display im- plies ´2´1δλ}rvS}1 ď λ2}rvS}1 ´ λδ}rvpMYSqc}1 which leads to δ}rvpMYSqc}1 ď p2 ` δ{2q}rvS}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then }rv}1 “ }rvM}1 ` }rvS}1 ` }rvpMYSqc}1 ď }rvM}1 ` }rvS}1 ` p4{δ ` 1q}rvS}1 ď }rvM}1 ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' k}rv}2 ď t?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='r}C´1 r Cm´r}2 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' m ´ r ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ku}rv}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The last equality holds by using the same argument as those lead to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume Conditions (C1)–(C6) hold, λ ď α1{p8R1q, R1 ď rnα2 1{t64τ2 1 logppqus1{4, δ P r4R1τ1 a logppq{n{λ,1s, n ě 4logppqτ2 1 t?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='m ` p2{δ ` 1{2q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ku4{pα1 ´ µq2, and α1 ą µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose rβa is a stationary point of program (8) and pβa is the interior local minimize of (8) and satisfies suppp pβaq Ď M Y S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Then suppp rβaq Ď M Y S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 71 Proof: The Corollary holds by using the same arguments as those lead to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='13, while using the consistency result in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Supporting Results related to Test Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Some Definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We define Σpβq ” ErtYiWi ´ exppβTWi ´ βTΩβ{2qpWi ´ Ωβqub2s By using the relation (2)–(4) and the additional relations Etexpp2βT t Wi ´ βT t Ωβtq | Xiu “ expp2βT t Xiq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Etexpp2βT t Wi ´ βT t ΩβtqpWi ´ Ωβtq | Xiu “ expp2βT t XiqXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Erexpp2βT t Wi ´ βT t ΩβtqtpWi ´ Ωβtqb2 ´ Ω{2u | Xis “ expp2βT t XiqXb2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Σpβtq “ ErtexppβT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t Xiq ´ expp2βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t XiqupXiXT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i ` Ωq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´expp2βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t XiqΩβtXT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i ´ expp2βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t XiqXiβT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t Ω ` expp2βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t Wi ´ βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='t ΩβtqpWi ´ Ωβtqb2s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='and the sample version ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='pΣpβq “ n´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='exppβTWi ´ βTΩβ{2qtpWi ´ Ωβqb2 ´ Ωu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´expp2βTWi ´ βTΩβqtpWi ´ Ωβqb2 ´ Ω{2u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='`exppβTWi ´ βTΩβ{2qΩ ´ expp2βTWi ´ βTΩβqΩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´expp2βTWi ´ βTΩβqΩβpWi ´ ΩβqT ´ expp2βTWi ´ βTΩβqpWi ´ ΩβqβTΩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='`expp2βTWi ´ βTΩβqpWi ´ Ωβqb2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='“ n´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='exppβTWi ´ βTΩβ{2qtpWi ´ Ωβqb2u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´expp2βTWi ´ βTΩβqtpWi ´ Ωβqb2 ` Ω{2u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´expp2βTWi ´ βTΩβqΩβpWi ´ ΩβqT ´ expp2βTWi ´ βTΩβqpWi ´ ΩβqβTΩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='`expp2βTWi ´ βTΩβqpWi ´ Ωβqb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Further, let ΨpΣ,Q,βq “ pCrImˆm,0mˆksQ´1 MYS,MYSpβqΣMYS,MYSpβqQ´1 MYS,MYSpβqrImˆm,0mˆksTCTq, and T0 “ pωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnqTΨ´1pΣ,Q,βtqpωn ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nhnq, where ωn “ ´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nCrImˆm,0mˆksQ´1 MYS,MYSpβtq "BLpβtq Bβt MYS 72 Further define Wald statistics as TW “ npC pβaM ´ tqTΨppΣ, pQ, pβaq´1pC pβaM ´ tq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' the score statistics TS “ n # BLp pβq BβT + MYS pCrImˆm,0mˆksQ´1 MYS,MYSp pβqqT ˆΨ´1ppΣ, pQ, pβqCrImˆm,0mˆksQ´1 MYS,MYSp pβq # BLp pβq Bβ + MYS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Because without knowing the distribution of X, the full likelihood is unknown and hence we do not discuss the likelihood ratio test here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (D1) Assume max 1ďiďnp}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtqWiMYS}3 2 `}Ψ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtqpΩβtqMYS}3 2q´1 ˆEr}YiΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtqWi ´ exppβT t Wi ´ βT t Ωβt{2q ˆΨ´1{2pΣ,Q,βtqCrImˆm,0mˆksQ´1 MYS,MYSpβtqpWi ´ βT t ΩqMYS}3 2|Wis “ Op1q (D2) cΨ ď αmintΨpΣ,Q,βqu ď αmaxtΨpΣ,Q,βqu ď CΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Some Lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Suppose X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=',Xn are independent m dimensional random vec- tor which satisfies, EpXiq “ 0 and řn i“1 covpXiq “ Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Let Z be a m-dimensional standard multivariate normal vector, then sup C |Prp nÿ i“1 Xi P Cq ´ PrpZ P Cq| “ Opm1{2 nÿ i“1 E}Xi}3 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: The lemma follows Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='1 in Bentkus (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Assume Conditions (C1) – (C7) hold, then }Ψ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtq}2 “ "logppq n 1{4 pm ` kq, and }Ψ´1ppΣ, pQ, pβq ´ Ψ´1pΣ,Q,βtq}2 “ "logppq n 1{4 pm ` kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Proof: First }ΨppΣ, pQ, pβaq ´ ΨpΣ,Q,βtq}2 ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 73 “ Optmaxp}pQMYS,MYSp pβaq´1 ´ QMYS,MYSpβtq´1}2, }pΣMYS,MYSp pβaq ´ ΣMYS,MYSpβtq}2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='26) Using the same arguments as those lead to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='47), we have }pQp pβaq´1´Qpβtq´1}2 “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='logppq n )1{4 pm ` kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Now recall that K ” tv P RMYS : }v}2 ď 1u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' for v P K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' by the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='Taylor expansion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='vTtpΣp pβaq ´ pΣpβtquv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='“ n´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='exppβT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a Wi ´ βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a Ωβa{2qvTtpWi ´ Ωβaqb2uvpWi ´ ΩβaqTp pβa ´ βtq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='`n´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2exppβT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a Wi ´ βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a Ωβa{2qvTpWi ´ ΩβaqvTΩp pβa ´ βtq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´n´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2expp2βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a Wi ´ βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a ΩβaqvTtpWi ´ Ωβaqb2 ` Ω{2uvpWi ´ ΩβaqTp pβa ´ βtq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´n´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2expp2βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a Wi ´ βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a ΩβaqvTpWi ´ ΩβaqvTΩp pβa ´ βtq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´n´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2expp2βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a Wi ´ βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a ΩβaqvTΩβapWi ´ ΩβaqTvpWi ´ ΩβaqTp pβa ´ βtq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='´n´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='nÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2expp2βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2expp2βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a Wi ´ βT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='a ΩβaqtvTpWi ´ ΩβaqvTΩ ´ vTΩβavTΩup pβa ´ βtq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' where βa is a point in between pβa and βt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Using the same arguments as those lead to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='39) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='40), we have }pΣp pβaq ´ pΣpβtq}2 “ Op «"logppq n 1{4 pm ` kq ff , and }pΣpβq ´ Σpβtq}2 “ op «"logppq n 1{4 pm ` kq ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Therefore, }pΣp pβaq ´ pΣpβtq}2 “ Op «"logppq n 1{4 pm ` kq ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Combine with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='26) and the fact that }pQp pβaq´1 ´ Qpβtq´1}2 “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='logppq n )1{4 pm ` kq, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='11 we have }Ψ´1ppΣ, pQ, pβaq ´ Ψ´1pΣ,Q,βtq}2 “ Opt}ΨppΣ, pQ, pβaq ´ ΨpΣ,Q,βtq}2u “ "logppq n 1{4 pm ` kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The second relation in the statement holds by using the same arguments as those lead the above results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' The composite ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Based on Braak & Braak (1991), Landau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2016), Schöll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2016), we define the composite regions as follow, where letter L and R represent the left and right hemispheres, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Braak 1 and 2 composite region (Braak12): L_entorhinal, R_entorhinal Braak 3 and 4 composite region (Braak34): L_parahippocampal, L_fusiform, L_lingual, L_amygdala, R_parahippocampal, R_fusiform, R_lingual, R_amygdala, L_middletemporal, L_caudantcing, L_rostantcing, L_postcing, L_isthmuscing, L_insula, L_inferiortemporal, L_temppole, R_middletemporal, R_caudantcing, R_rostantcing, R_postcing, R_isthmuscing, R_insula, R_inferiortemporal, R_temppole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' ON HIGH DIMENSIONAL POISSON MEASUREMENT ERROR MODELS 75 Braak 5 and 6 composite region (Braak56): L_superior_frontal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_lateral_orbitofrontal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_medial_orbitofrontal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_frontal_pole,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_caudal_middle_frontal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_rostral_middle_frontal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_pars_opercularis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_pars_orbitalis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_pars_triangularis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_lateraloccipital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_parietalsupramarginal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_parietalinferior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_superiortemporal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_parietalsuperior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_precuneus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_bankSuperiorTemporalSulcus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_tranvtemp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_superior_frontal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_lateral_orbitofrontal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_medial_orbitofrontal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_frontal_pole,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_parietalsupramarginal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_parietalinferior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_superiortemporal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_parietalsuperior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_precuneus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_bankSuperiorTemporalSulcus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_tranvtemp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_pericalcarine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_postcentral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_cuneus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_precentral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' L_paracentral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_pericalcarine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_postcentral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_cuneus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_precentral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' R_paracentral B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Simulations when measurement error covariance is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' To evaluate the proposed adjustment estimator in Section 6, we added the following three simulation settings where the covariance matrices of the measurement errors contain different numbers of unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ω is a matrix with p{4 unknown parameters, corresponding to p{4 nonzero diago- nal entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ω is a matrix with p{2 unknown parameters, corresponding to p{2 nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ω is a matrix with 6p´15 unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Specifically, Ω “ pσijqpˆp, where σij “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05p1 ´ |i ´ j|{5q for |i ´ j| ď 5 and σij “ 0 for |i ´ j| ą 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In these settings, the number of unknown parameters in Ω increases, while all other settings are the same as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' We evaluate the empirical sizes and powers of the Wald test for n “ 300, p “ 50 in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='9 and p “ 350 in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In the first two settings, where Ω contains relatively few parameters hence their convergence is sufficiently fast, the empirical sizes are well controlled around the nominal level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='05 in all hypothesis tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' In the last setting, the parameter number is too large to achieve sufficiently fast convergence, hence the Wald test cannot control the Type I errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' These results imply that the proposed adjustment can control Type I error rate when the error covariance matrix contains small or moderate number of unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' REFERENCES Baker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Bentkus, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2005), ‘A lyapunov-type bound in rd’, Theory of Probability & Its Applications 49(2), 311–323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Bickel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' & Levina, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' (2008), ‘Covariance regularization by thresholding’, The Annals of Statistics 36(6), 2577–2604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' 76 TABLE B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='9 The empirical size and power of Wald tests for linear hypothesis testing with n “ 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' NUP standards for the number of unknown parameter in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' NUP=p{4 NUP=p{2 NUP “ 6p ´ 15 p “ 50 β2 H0,1 : β2 “ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content=' Ha,1 : β2 ‰ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} +page_content='262 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf'} diff --git a/WtFPT4oBgHgl3EQfrjVl/content/tmp_files/2301.13145v1.pdf.txt b/WtFPT4oBgHgl3EQfrjVl/content/tmp_files/2301.13145v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd75c65d91df91dab72bae324182733b6a2ecfca --- /dev/null +++ b/WtFPT4oBgHgl3EQfrjVl/content/tmp_files/2301.13145v1.pdf.txt @@ -0,0 +1,820 @@ +Accurate and efficient multiscale simulation of a +heterogeneous elastic beam via computation on +small sparse patches +A.J. Roberts∗ +Thien Tran-Duc† +J.E. Bunder‡ +Yannis Kevrekidis§ +January 31, 2023 +Abstract +Modern ‘smart’ materials have complex microscale structure, often with +unknown macroscale closure. The Equation-Free Patch Scheme empowers +us to non-intrusively, efficiently, and accurately simulate over large scales +through computations on only small well-separated patches of the microscale +system. Here the microscale system is a solid beam of random heterogeneous +elasticity. The continuing challenge is to compute the given physics on +just the microscale patches, and couple the patches across un-simulated +macroscale space, in order to establish efficiency, accuracy, consistency, and +stability on the macroscale. Dynamical systems theory supports the scheme. +This research program is to develop a systematic non-intrusive approach, both +computationally and analytically proven, to model and compute accurately +macroscale system levels of general complex physical and engineering systems. +Contents +1 +Introduction +2 +2 +Equation-free patch scheme +4 +2.1 +Scheme is non-intrusive functional ‘wrapper’ . . . . . . . . . . . . . +4 +2.2 +Scheme embeds macroscale dynamics . . . . . . . . . . . . . . . . . +5 +3 +Scheme has proven accuracy +6 +3.1 +Computation verifies exactness . . . . . . . . . . . . . . . . . . . . . +6 +3.2 +Mathematical analysis proves consistency . . . . . . . . . . . . . . . +8 +∗School of Mathematical Sciences, University of Adelaide, South Australia. +mailto: +ProfAJRoberts@protonmail.com https://orcid.org/0000-0001-8930-1552 +†School of Mathematical Sciences, University of Adelaide, South Australia. https://orcid. +org/0000-0002-2004-5156 +‡Mathematical Sciences, University of South Australia, Australia. +https://orcid.org/ +0000-0001-5355-2288 +§Departments of Chemical and Biomolecular Engineering & Applied Mathematics and +Statistics, +Johns Hopkins University, +Baltimore, +Maryland, +USA. https://orcid.org/ +0000-0003-2220-3522 +1 +arXiv:2301.13145v1 [math.NA] 20 Jan 2023 + +1 +Introduction +2 +4 +Conclusion +8 +1 +Introduction +In structural engineering, microscale lattice materials can be light and highly stiff +with customizable macroscale mechanical properties (e.g., Somnic & Jo 2022). +The challenge we address herein is to accurately and efficiently predict macroscale +characteristics emergent from the microscale lattice. Similarly, composite materials +and structures are inherently heterogeneous and anisotropic across multiple scales. +Multiscale modelling is thus critical to the design of composite structures for +lightweight mechanical performance (e.g., Raju et al. 2021, Lucarini et al. 2021). +Such composite materials are used in electronics, space, medical, transportation, +and other industries (e.g. Matouˇs et al. 2017). Herein we establish that the Equation- +Free Patch Scheme can non-intrusively, efficiently, and accurately simulate over +macroscales through computations on only small well-separated patches of the +microscale system. +Consider an example elastic beam with heterogeneous elasticity in 2D as in Figure 1: +say 628 cm long, 20 cm wide. The beam is heterogeneous because it is constructed +from a modern material with micro-structure of size 3 cm—so that the heterogeneity +is ‘visible’ in Figure 1. With a 3 cm micro-grid, the modelling requires circa 5 000 +variables. This specific scenario is easily computable, ode23 took 14 s cpu time +to simulate one period of beam bending oscillation. But if a more realistic 3 mm +micro-structure is simulated, then the computation time increases by a factor +of 1000. If 3D elasticity modelling is required for the beam, then the computation +time increases by even more orders of magnitude. The patch scheme (e.g., Samaey +et al. 2010) we develop herein potentially reduces macroscale computation time by +orders of magnitude—more reduction in higher-D space and/or smaller micro-scale. +The patch scheme achieves efficiency by only computing on small sparse patches +in space. Section 2.1 discusses how the patch scheme is non-intrusive in that it +just ‘wraps around’ a user’s microscale code—a desirable property also identified +by Biezemans et al. (2022). The patch scheme, alternatively called the gap-tooth +method, “has formal similarity with sp [superparametrization]” (Majda & Grooms +2014, p.62) that was developed in meteorology for weather and climate predictions, +and is also akin to the so-called fe-fft and fe2 methods (Lucarini et al. 2021, +e.g.,§4.7). +Figure 1: movie of a full-domain simulation of a heterogeneous beam showing that +beam bending waves and longitudinal compression waves propagate with some +‘average’ properties. +0 +1 +2 +3 +4 +5 +6 +space x +-0.2 +0 +0.2 +y +time = 0.00, E in 0.39 3 + +1 +Introduction +3 +Figure 2: a small part of the +microscale grid used to code 2D +elasticity. The grid is staggered +on the microscale: ▶, horizontal +displacements and velocities; +▲, vertical displacements and +velocities; ⊚, ⊗, components of +strain and stress tensor (1). +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +i − 1 +i +i + 1 +j − 1 +j +j + 1 +Figure 3: example of the 2D mi- +croscale heterogeneous Young’s +modulus Eij used in computing +the elastic Lam´e parameters (3). +In this example, we choose the +heterogeneity to have microscale +period four along the beam. +AkOnic7VrLbhvJFaUnrxGTMaZTY9kgXYgaSp9wMDAgMEAb +IKJsB4ZgBLMJpkS+qYr3S3LGkI/ku2yTfkR7LNLsg2H5BTV1Fi2wa9DKAaEMsdp9bVf +fce09dtThcTMq6IeSfTz760Y9/8tOfXzQ/kvfvnJrz59+utv6/lNSpejuaTefX9MK ++LSTkrXjZlMym+X1RFPh1Oiu+Gb37n7n/3tqjqcj7prlfFBfT/GpWXpajvMGl108Pz +vOvrku6+ynBTZbV5nI5g3xTgb3mfTvJnkQ9aUb346x/3j8/dqC6aWXHXOPwsnxbL6 +6IpqvnvKT394/ztKSOMrfrnw+KqnC0dfFGOmpuqWPXjxfyurFevjvuLeTlrsmnR5Nm0n +A3IGedSWMG4pVxqxU8eIPK7ATtzl7lkRBAqT7p3lD947b5Iypd1/uPnZfTLK6uZ8Ug ++XlfNYMzj1ly9vrsik+p/Lz4SQfvVmtInawrBf5qMie3T3DtXs3/yk5o27ol3CjD5j0v +p10VM3r+nSIuGTP7t3MjoRsCNRVNb+Zjb84i9OVk8nAzxNBd5kL7W8Hw3nTzKftxfv24 +qS4bLz7f5XV1U5rv1G1x8uPjLT7JSgS+OMluy3FzPdBkiqn8rqf5YrCcwmqxWr6i +i+Yiq6Gz8mieTF4DkoFYydwWxKBN6UIf/GlB9AIkIRbh1DCWomBJlpR0aJYQmtqYMZ +qo0baEkpNy2MR5gSmvnZrNZSO5iVLM0mIkwTJQzu4pbSCgPDFde0hckEwxpu24wjqxze +aGKNamEqwQx3ICmN36KxlqS5dAQZwrmHacP8wDIWQSaBKPfrcEKkJ8MKynVc0K5hxO+L +Mxa8tFKbtC+auNcW7jmcwLvbmtVUpekoXftp/W2OrPfkWsMskRGXgqCsDesa64IKP9Nc +6wgQJpx7gjLuQ4GoUqkjLoVACOJpEFyawIc1GERcigFDqDwOEwdfrWZrX2MQqFWBX2G +Epn5dY6lkERfjQA0XxDErieYtJ/iXfI2hQJaFdSUzJqwLatZ+2IQDzIUM0mP9xEg4Qm3 +M3xgLqowO82nClGePu3yJuBgLKh3B0hIPYTRmOItBoI4I58ilgZHiTZpYywGAsWBSDg +c08obGBCX9hXjQJkwPqZKSBtgCIOIcWAxDpRSJpkvYS59JSi+rtIYBeKc94mrtMshB0R +umeRmDANRBs7da0IlW04GEluxDjAVy/FThcM9QO8ZQoLAaCEFhT7HGRJMLI0TcpJ +wSsmwtOaUhQF0KTnDYyiI4ByZ5u4LGrzSHAUZcSw5rbj13GiPoYyshSmFQ5DgpbGu2 +3EOpt4ioamOuzH+hxRUoUC5iYaEJHofq9wPFZNpWkiWqSNi9CYlnWZopqZJQVkamvI +dSKr1eMQkTKj0wZYO+SmpMSiYeY4DSCkVjCHE1IbRZ6aIEeCW+MQgLOglFECISJiI7A +sh27RFGvs8d3apsERkHwdL2JjlXunSRJF5qaj0ZGrXG3gQTFJKisi9oiuDyBy2y/IFD +QszRfZh5LawKu2zDvAIMRFdnXOPh8JG1gn1pKEmEi8q81D+IHBTZePXAMqrS1SL9BRg +X6hQ3VSiXamDRd5N8oVK6fDoHwKeK8VSL/FtUSqgTzoOUI0Mj/kvYwQst0Fxobfae4 +zatimeMkYAKdGetcyGHEcLtk5bGQMB23DqIsF4i0MhRoKlSDjVEqxkF5GrV3PJ9fzM +RICZoM2oe6hdREXQ2HDaYWzVfpiQerKNSs6eWHafOPtANIu197GWFgcoN4LkN32LSkQ0 +iYQTmivpZb4VEK81gevSpFAfMIZRLnwnlK0S2lNlSIBDQ/tANehQaBMpYpXKRDKhn7FK +tK6QBRPhKgUCGSlPyC9AwSJ92LVNnPDvOpfoNPOx+PFZN6gy65vqsuTfn2dj4tqcDnJm +2w0r2ZFdZKNq/x24DvVk+yxE3zsB87wcdO8LETfOwEHzvBx07wsRP8/+0Es2lRX39Rz +W/rgb3oN/lwUrzCp6wuFoNRNULnt36uOjhvrsad5ej1UW2PO4f9+8O7g9G5+d9cuAe +fB4lwjCaAlaIch2vEPUw4etBxQHFXKaCPRaqG6TgPzh68B3fFIyriROKkKsR1L0IugM +kZ84BKmQxSnVB1vPhD20e040Pui1jGvPUBQRuLlLRFJoEVoaVCJtcRTrM6GJ64c4Wlzm +b2yv4ylhaCVwJqPijRQo1J3QrdUZJBInEPpKi4Lgerfl5jV6RiRVTrORL0rsWHMHiRIN +LTJKapyC6Ee6bTunQwpDYCk6Iiy827CDZfw2ILQyp1ru+x8dhmFmqAMvaAbAfo5nwx +E1EXgimDAYxRB3YDt4f/uVgt2UHoy5fXFIpNE6i23IH71ACYiSYBwg/um07F3XtO/o +lFzg0BDsNt/fhUoTjyHK9tTC7DB2ZCqWHikGlGuheIJ6mBfM+awMcT3f+l7HXNptGco +gYEOZeQe7ZgaHSwGNTN8LjkDvKMwOkt3JFP3P8WvKP2BdQZTnMpiWFraMdu0TO6lgx +IJuQa6tMLZKNxQBpCRgP9zYW26mJXci9NLTsMPHbc3cstxfQ7dM9XULsNtlrc1tsN +uh+bS7arcIbkdyP0Ut8tw0/tOwd0/AC93TdV247Dr43lLbNdYsvep65sfzl+s +n+xDBZHsrJNtbEtl7NJBtLVbETuQe+pjl+W2ix1quWn4Adq5abq3knYdnjZoavbhr +tUduMP9+9R2Q7kfirbZdjhf4fKblp+gMpumu6tsh2G25x3qOy2XafKr7snxezcfgqRh +i+1WN49efHrnK+6VbQ9oOzjqta+vXz98sn5eD6mRazZjTJ6/qVe3Z5scyrphxNCq +xg18/8tGb/Kp4haH74kh9sfSPnVfZMa6Ms8t5lblvU4SH0e9aLPNpXd9Phyd4n+bNtX +t3yBqm7nP9cPrm0lwsy9nipilmozD75c0ka+aZ+wJMNi6rYtRM7jHIR1WJDWaj67zKR0 +1RPZhpWdxMiurt9MHul4urS/dovV6dx1FdNPBkusibway4Lepm1e+DQbrJ1/bgWwahP +hN/YkdfPW+5/Lj3m95h73mP9nTvq94fel/3XvZGhz8c/vXwb4d/P/rH0b+O/n30nwD96 +Elr81nvwevov/8Dw+BQBg= +0 +5 · 10-2 0.1 +0.15 +0.2 +0.25 +-0.1 +-5 · 10-2 +0 +5 · 10-2 +0.1 +space x +cross-beam y +0.5 +1 +1.5 +2 +2.5 +A given microscale discretisation of heterogeneous elasticity +We adopt +a simple robust microscale approximation of 2D elasticity within the beam. On +the staggered microscale xy-grid of Figure 2 define the displacements: ▶, hori- +zontal uij(t); ▲, vertical vij(t). Microscale elasticity here first uses centred finite +differences to compute stresses, for heterogeneous Lam´e parameters λ, µ, at the +labelled microscale grid-points (Figure 2): +⊗ +σxy := µij +� +δjuij/δyj + δivij/δxi +� +; +(1a) +⊚ +σxx := (λij + 2µij)δiuij/δxi + λijδjvij/δyj; +(1b) +⊚ +σyy := λijδiuij/δxi + (λij + 2µij)δjvij/δyj. +(1c) +Second, centred finite differences compute the following acceleration odes +▶ +¨uij = δiσxx/δxi + δjσxy/δyj , +(2a) +▲ +¨vij = δiσxy/δxi + δjσyy/δyj , +(2b) +potentially with optional small phenomenological damping supplied by a discretisa- +tion of κ∇2 ˙uij, κ∇2 ˙vij. The patch scheme wraps around whatever microscale code +a user supplies—here it is the microscale system (1) and (2) +We nondimensionalise the system so that the density is one, and the speed of a +macroscale compression wave along the beam is about one, that is, time in these +simulations is roughly in milli-seconds. +Random periodic heterogeneity +The Lam´e parameters which appear in the +stresses (1) are +λ := +νE +(1 + ν)(1 − 2ν), +µ := +E +2(1 + ν), +(3) + +2 +Equation-free patch scheme +4 +in terms of Young’s modulus E and Poisson ratio ν. To have strong microscale +heterogeneity we choose these parameters randomly so that at each microscale grid- +point (iid): Eij is log-normal (here varies by factor of about ten); and νij is uniform +on [0.25, 0.35]. Figure 3 shows an example Eij. Despite such strong heterogeneity, +the movie of Figure 1 shows the macroscale dynamics appears relatively simple. +2 +Equation-free patch scheme +Instead of computing the entire beam as seen in Figure 1, the patch scheme computes +only in small sparse spatial patches such as Figure 4. In this example case, the +patch scheme reduces compute time by a factor ∝ r := (patch size)/(spacing H), +which here is just a modest factor of 1/4. But with greater scale separation and/or +in higher spatial dimensions, the scheme often reduces computational time by many +orders of magnitude. +The movie of Figure 4 shows a slow progressive wave of beam bending, together +with a not-so-slow compression wave along the beam. These macroscale predictions +are accurate (Section 3) due to the correctness of our simple coupling between +patches—even when heterogeneity is strong. +The patch scheme makes these +accurate macroscale predictions even when the macroscale closure is unknown: +the scheme does not code a closure. Further, ‘the closure’ varies depending upon +human assumptions such as choosing averaged models versus cosserat models—the +patch scheme makes no such closure assumptions. The only assumption is that the +macroscale quantities of importance vary smoothly between neighbouring patches. +2.1 +Scheme is non-intrusive functional ‘wrapper’ +Consider one of the patches of the 2D beam shown in Figure 4. With the given +microscale xy-grid (Figure 2), zooming in to the microscale each patch is like that +of Figure 5. Here each patch extends across the cross-section (y-dimension) of the +beam. Open symbols in Figure 5 are ghost nodes outside the patch and implement +given stress-free top/bottom conditions on the beam. The only addition required +by the patch scheme are the edge values (‘squared’ micro-grid nodes in Figure 5) +on the left/right of each patch. +The patch scheme couples patches together by providing the patch-edge values +through interpolation across the macroscale between patches (e.g., Roberts & +Kevrekidis 2007, Roberts et al. 2014, Cao & Roberts 2016). Here we interpolate +from each of the centre patch values across the beam (i = 4 in Figure 5) of ‘nearby’ +Figure 4: movie of a patch scheme simulation of a heterogeneous beam showing the +macroscale propagation across the patches of beam bending waves and longitudinal +compression waves. +0 +1 +2 +3 +4 +5 +6 +space x +-0.2 +0 +0.2 +y +time = 0.00, E in 0.35 3.2 + +2 +Equation-free patch scheme +5 +Figure 5: one example patch +of the 2D elastic beam show- +ing the microscale staggered +grid (Figure 2). This is case +of nsubpatch = 7 micro-grid in- +tervals along the patch, and +ny = 4 intervals across the +beam. +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▶ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊚ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +⊗ +▷ +▷ +▷ +▷ +▷ +▷ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +▷ +▷ +▷ +▷ +▷ +▷ +⃝ +⃝ +⃝ +⃝ +⃝ +⃝ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +i = 1 +2 +3 +4 +5 +6 +7 +j =1 +2 +3 +4 +patches, to determine the corresponding patch-edge value. Here we implement +spectral (fft) interpolation between the patches for high accuracy (Section 3). +The scheme does not presume that any average is appropriate. +This implementation shows that the patch scheme is non-intrusive (e.g., Biezemans +et al. 2022): it just ‘wraps around’ any micro-grid code a user trusts. Consequently, +we provide a toolbox (Maclean et al. 2021) for others to implement the patch +scheme around their micro-code. +2.2 +Scheme embeds macroscale dynamics +Given the patch scheme does not assume anything about what are ‘correct’ +macroscale variables, a crucial question is the following: how can we be assured +that the patch scheme captures the macroscale slow dynamics? +An answer is +provided by the Whitney (1936) embedding theorem. +Roughly, the theorem is that every mD manifold is parametrisable from almost +every subspace of more than 2mD. Let’s see what this means for us. In essence, +the patch scheme provides the higher-D subspace in which the slow manifold of +the macroscale wave dynamics is embedded. +For beams in two spatial dimensions, the basic macroscale beam models have, at +each cross-section, displacement and velocity of both bending and compression. +Thus the elastic beam dynamics has a slow manifold that is m = 4D at every +cross-section.1 Alternatively, 2D cosserat beam models add a shear mode to the +macroscale model—two more variables—leading to a not-quite-so-slow manifold of +m = 6D at every cross-section. These physically based models are slow manifolds +because they focus on the relatively slow waves of solutions varying slowly in space, +and neglect all the faster high-frequency cross-waves. +In the patch scheme, Figures 1 and 4 show simulations with a cross-section of +ny = 7 micro-grid intervals, but let’s discuss the case of just ny = 4 (Figure 5). +For ny = 4, there are seven microscale nodes across each patch edge. Each node +has a displacement and velocity, and so leads to a 14D subspace for macroscale +communication between patches. +1Such statements, invoking a manifold or subspace “at every cross-section”, are in a sense +developed by the theory of Roberts (2015). That is, in systems of large spatial extent there often +are important, spatially global, invariant manifolds of high-D that are effectively decomposable +into a union of spatially local manifolds/subspaces of relatively lower dimension—a dimension +determined by the spatial cross-section—and that are weakly coupled to neighbouring locales. + +3 +Scheme has proven accuracy +6 +Figure 6: multiscale spectrum +of eigenvalues λ separates +macroscale modes on the right +from sub-patch microscale +modes on the left. The axes +are scaled nonlinearly. Here +the small viscosity is 0.001 +so the microscale decays, but +the macroscale waves are long- +lasting. +-3 +-1 +-0.3 +-0.1 +-0.03 +-0.01 +0 +-100 +-30 +-10 +-3 +-1 +-0.3 +-0.10 +0.1 +0.3 +1 +3 +10 +30 +100 +ℜλ +ℑλ +Because 14 > 2 · 6 > 2 · 4 , the Whitney embedding theorem asserts that the +patch scheme exchanges enough information to almost surely parametrise both +such slow manifolds of the macroscale dynamics. The patch scheme does not need +to explicitly compute and exchange specific assumed macroscale average quantities. +3 +Scheme has proven accuracy +Section 3.2 discusses established theory which generally proves that the patch +scheme makes accurate macroscale predictions. Such proofs are in stark contrast +to the vast machine learning/artificial intelligence developments which prove +very few general results: for example, Brenner & Koumoutsakos (2021) comment +“. . . ml studies, as the lack of rigorous theory does not offer (yet!) guarantees +of convergence”. Before discussing theory, we first report some computational +verification of high accuracy. +3.1 +Computation verifies exactness +Here we restricted attention to linear elasticity so we know that the wrapped +patch system is fully characterised by the resultant Jacobian matrix. We numeri- +cally compute the Jacobian matrix of the patch scheme by elementary numerical +differentiation. +Because of the macroscale translational invariance of the patch scheme, the +macroscale eigenvectors are correctly sinusoidal. Hence the only macroscale er- +rors occur in the eigenvalues of the Jacobian. Figure 6 plots the spectrum of all +eigenvalues for one example of random heterogeneity, in the case of five patches for +simplicity. Observe there are: +• (on the right) four λ = 0 of rigid beam motion; +• four −0.001 ± i 1.057 and four −0.003 ± i 2.111 of compressions waves; +• four −0.001 ± i 0.061 and four −0.004 ± i 0.237 of beam bending waves; +• with the above macroscale eigenvalues separated by a spectral gap from the +following sub-patch microscale eigenvalues; + +3 +Scheme has proven accuracy +7 +Table 1: error in patch scheme’s +macroscale eigenvalues λ for +various patch size ratios r: the +macroscale λs are exact to round- +off error—due to patch coupling by +spectral interpolation. +macro-eigenvalue +r = 1 +2 +r = 1 +4 +r = 1 +8 +−0.001 ± i 0.061 +2e-12 +1e-12 +2e-13 +−0.001 ± i 0.061 +2e-12 +4e-12 +2e-12 +−0.004 ± i 0.237 +1e-12 +8e-13 +3e-12 +−0.004 ± i 0.237 +1e-12 +2e-12 +3e-12 +−0.001 ± i 1.057 +7e-13 +4e-13 +6e-13 +−0.001 ± i 1.057 +6e-13 +5e-13 +6e-13 +−0.003 ± i 2.111 +1e-13 +2e-13 +2e-13 +−0.003 ± i 2.111 +4e-13 +5e-13 +2e-13 +Figure 7: multiscale spectrum +of eigenvalues λ for the patch +scheme in the case of zero viscos- +ity. The horizontal axis shows +that all modes have zero real- +part to numerical round-off er- +ror. That is, in the case of zero +viscosity, this patch scheme pre- +serves the wave nature of the +underlying physics. +-5e-13 +-2e-13 +-1e-130 +1e-13 +2e-13 +5e-13 +-100 +-30 +-10 +-3 +-1 +-0.3 +-0.10 +0.1 +0.3 +1 +3 +10 +30 +100 +ℜλ +ℑλ +• (on the left) many ℜλ < −0.1 of uninteresting sub-patch micro-scale fast- +waves (headed by ten eigenvalues around −0.14 ± i 9.29). +To quantify the accuracy, Table 1 compares eigenvalues obtained from full-domain +code, with the above macroscale eigenvalues obtained by the wrapped patch scheme. +For all patch size ratios and heterogeneities tested, the patch scheme’s macroscale +eigenvalues are exact to numerical round-off error. +Such exactness is due to the spectral interpolation used here. If, instead of spectral, +local polynomial interpolation of degree p is used to couple the patches, then +generally the patch scheme has macroscale errors ∝ Hp where H = inter-patch +spacing (e.g., Roberts & Kevrekidis 2007, Roberts et al. 2014). +Undamped waves? +With zero viscosity, there are only oscillations in the under- +lying physics. In such a scenario computational methods are very delicate. Here, +Figure 7 illustrates that all eigenvalues of the patch scheme have |ℜλ| < 10−12.2 +Hence, even with no viscosity, the patch scheme preserves the oscillatory wave +nature of the heterogeneous physics. +There is a perception that the patch scheme “only works well on problems with +an inertial manifold and for systems in which most modes are strongly decaying” +(Majda & Grooms 2014, p.62). This verification of accuracy for purely elastic +2In some realisations of the heterogeneity, the sensitive multiplicity four eigenvalue λ = 0 +numerically splits into four showing |ℜλ| up to 10−6 due to round-off errors. + +4 +Conclusion +8 +beams shows that this perception is false. Applications and theory for other wave +systems also refute this perception (e.g., Cao & Roberts 2016, Bunder et al. 2021, +Divahar et al. 2022). +3.2 +Mathematical analysis proves consistency +Mathematical analysis has proven properties of the patch scheme in general. Mostly, +the published proofs explicitly address dissipative (nonlinear) systems. However, +as discussed by Bunder et al. (2021), the patch scheme in space only recasts spatial +interactions, so whether the time derivative is ∂/∂t of dissipation or ∂2/∂t2 of +waves makes little difference. +Two complementary types of results have been proven. They involve the spacing +between patch centres H. +First, Centre Manifold Theory may be applied at +finite spacing H by introducing a ‘bookkeeping’ parameter γ to label inter-patch +communication (e.g., Roberts et al. 2014, §2) to prove the existence of a slow +manifold in the patch scheme (including when it is applied to nonlinear systems). +Then the parameter γ structures inter-patch interactions, and their algebraic +expression, to empower theory based at γ = 0, via regular perturbation, to address +finite γ such as the case of full coupling γ = 1 (e.g., Roberts et al. 2014, Cor. 2). +Second, the patch scheme is consistent with the underlying micro-code as the +patch spacing H → 0 (e.g., Roberts et al. 2014, Thm. 7). The consistency is +that the macroscale of the patch scheme is the same as the macroscale of the +given micro-coded system, to errors O +� +Hp� +when using polynomial interpolation +of degree p. For example, spectral interpolation corresponds to ‘p = ∞’ so then +errors vanish to all orders as in Table 1. +These results and general proofs were first done for homogeneous systems (e.g., +Roberts & Kevrekidis 2007, Roberts et al. 2014). They were subsequently ex- +tended to heterogeneous microscales (Bunder et al. 2017), and recently extended +to alternative inter-patch coupling that preserves self-adjointness (Bunder et al. +2021). Interestingly, the extension of the theoretical support to heterogeneous +cases invokes the ensemble of all phase-shifts of the heterogeneity. The ensemble is +spatially homogeneous, so the homogeneous proofs and results apply to establish +the heterogeneous results. +4 +Conclusion +As an initial exploration of the patch scheme for homogenisation of heterogeneous +elasticity, we considered the prototypical case of a 2D elastic beam. The scheme +gives a non-intrusive and efficient computational homogenisation of given microscale +system via spatially sparse small patches. The patch coupling has proven accuracy, +controllable error, at finite scale separation. +The patch scheme makes only one assumption: in the scenarios of interest to a +user, there is no significant spatial structures in the mesoscale between the patch +spacing H and the microscale resolved in the patches. In contrast to most other +multiscale methods, there is: no assumed boundary conditions on Representative +Volume Elements (variously periodic, Dirichlet, Neumann); no explicitly assuming + +References +9 +slow variables; and no presumed necessary variational principle. The scheme is +entirely physically interpretable: there is no hidden mystic machinations of neural +networks (e.g., Brenner & Koumoutsakos 2021) +The patch scheme is simple to apply. In contrast to other multiscale methods +there is: no arbitrary averaging; no oversampling regions; no buffer regions; no +action regions; no guessed fast/slow variables; no epsilons; and no limits. As a +non-intrusive ‘wrapper’, anyone can start using the patch scheme via a Matlab/ +Octave Toolbox (Maclean et al. 2021, Roberts et al. 2019–2023) +Acknowledgements +This research was supported by Australian Research Coun- +cil grants DP220103156 and DP200103097. +References +Biezemans, R. A., Le Bris, C., Legoll, F. & Lozinski, A. (2022), Non-intrusive +implementation of a wide variety of Multiscale Finite Element Methods, Technical +report, http://arxiv.org/abs/2211.17024. +Brenner, M. P. & Koumoutsakos, P. (2021), ‘Editorial: Machine Learning and +Physical Review Fluids: An Editorial Perspective’, Physical Review Fluids +6(7), 070001. +Bunder, J. E., Kevrekidis, I. G. & Roberts, A. J. (2021), ‘Equation-free patch +scheme for efficient computational homogenisation via self-adjoint coupling’, +Numerische Mathematik 149(2), 229–272. +Bunder, J. E., Roberts, A. J. & Kevrekidis, I. G. (2017), ‘Good coupling for the mul- +tiscale patch scheme on systems with microscale heterogeneity’, J. Computational +Physics 337, 154–174. +Cao, M. & Roberts, A. J. (2016), ‘Multiscale modelling couples patches of nonlinear +wave-like simulations’, IMA J. Applied Maths. 81(2), 228–254. +Divahar, J., Roberts, A. J., Mattner, T. W., Bunder, J. E. & Kevrekidis, I. G. +(2022), Two novel families of multiscale staggered patch schemes efficiently +simulate large-scale, weakly damped, linear waves, Technical report, https: +//arxiv.org/abs/2210.15823. +Lucarini, S., Upadhyay, M. V. & Segurado, J. (2021), ‘FFT based approaches +in micromechanics: fundamentals, methods and applications’, Modelling and +Simulation in Materials Science and Engineering 30(2), 023002. +Maclean, J., Bunder, J. E. & Roberts, A. J. (2021), ‘A toolbox of equation-free +functions in matlab/octave for efficient system level simulation’, Numerical +Algorithms 87, 1729–1748. +Majda, A. J. & Grooms, I. (2014), ‘New perspectives on superparameterization for +geophysical turbulence’, Journal of Computational Physics 271, 60–77. +Matouˇs, K., Geers, M. G. D., Kouznetsova, V. G. & Gillman, A. (2017), ‘A review of +predictive nonlinear theories for multiscale modeling of heterogeneous materials’, +Journal of Computational Physics 330, 192–220. + +References +10 +Raju, K., Tay, T.-E. & Tan, V. B. C. (2021), ‘A review of the FE2 method for +composites’, Multiscale and Multidisciplinary Modeling, Experiments and Design +4, 1–24. +Roberts, A. J. (2015), ‘Macroscale, slowly varying, models emerge from the mi- +croscale dynamics in long thin domains’, IMA Journal of Applied Mathematics +80(5), 1492–1518. +Roberts, A. J. & Kevrekidis, I. G. (2007), ‘General tooth boundary conditions for +equation free modelling’, SIAM J. Scientific Computing 29(4), 1495–1510. +Roberts, A. J., MacKenzie, T. & Bunder, J. (2014), ‘A dynamical systems approach +to simulating macroscale spatial dynamics in multiple dimensions’, J. Engineering +Mathematics 86(1), 175–207. +Roberts, A. J., Maclean, J. & Bunder, J. E. (2019–2023), Equation-free function tool- +box for matlab/octave, Technical report, [https://github.com/uoa1184615/ +EquationFreeGit]. +Samaey, G., Roberts, A. J. & Kevrekidis, I. G. (2010), Equation-free computation: +an overview of patch dynamics, in J. Fish, ed., ‘Multiscale methods: bridging the +scales in science and engineering’, Oxford University Press, chapter 8, pp. 216– +246. +Somnic, J. & Jo, B. W. (2022), ‘Status and challenges in homogenization methods +for lattice materials’, Materials 15(2), 605. +Whitney, H. (1936), ‘Differentiable manifolds’, Annals of Mathematics 37(3), 645– +680. + diff --git a/WtFPT4oBgHgl3EQfrjVl/content/tmp_files/load_file.txt b/WtFPT4oBgHgl3EQfrjVl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ca274c242f8dfd49d8a45c9d3953a7b2bb13794 --- /dev/null +++ b/WtFPT4oBgHgl3EQfrjVl/content/tmp_files/load_file.txt @@ -0,0 +1,516 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf,len=515 +page_content='Accurate and efficient multiscale simulation of a heterogeneous elastic beam via computation on small sparse patches A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Roberts∗ Thien Tran-Duc† J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Bunder‡ Yannis Kevrekidis§ January 31, 2023 Abstract Modern ‘smart’ materials have complex microscale structure, often with unknown macroscale closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The Equation-Free Patch Scheme empowers us to non-intrusively, efficiently, and accurately simulate over large scales through computations on only small well-separated patches of the microscale system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Here the microscale system is a solid beam of random heterogeneous elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The continuing challenge is to compute the given physics on just the microscale patches, and couple the patches across un-simulated macroscale space, in order to establish efficiency, accuracy, consistency, and stability on the macroscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Dynamical systems theory supports the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' This research program is to develop a systematic non-intrusive approach, both computationally and analytically proven, to model and compute accurately macroscale system levels of general complex physical and engineering systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Equation-free patch scheme 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 Scheme is non-intrusive functional ‘wrapper’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 Mathematical analysis proves consistency .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 8 ∗School of Mathematical Sciences, University of Adelaide, South Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' mailto: ProfAJRoberts@protonmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='com https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='org/0000-0001-8930-1552 †School of Mathematical Sciences, University of Adelaide, South Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' org/0000-0002-2004-5156 ‡Mathematical Sciences, University of South Australia, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='org/ 0000-0001-5355-2288 §Departments of Chemical and Biomolecular Engineering & Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, Maryland, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='org/ 0000-0003-2220-3522 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='13145v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='NA] 20 Jan 2023 1 Introduction 2 4 Conclusion 8 1 Introduction In structural engineering, microscale lattice materials can be light and highly stiff with customizable macroscale mechanical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Somnic & Jo 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The challenge we address herein is to accurately and efficiently predict macroscale characteristics emergent from the microscale lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Similarly, composite materials and structures are inherently heterogeneous and anisotropic across multiple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Multiscale modelling is thus critical to the design of composite structures for lightweight mechanical performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Raju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2021, Lucarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Such composite materials are used in electronics, space, medical, transportation, and other industries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Matouˇs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Herein we establish that the Equation- Free Patch Scheme can non-intrusively, efficiently, and accurately simulate over macroscales through computations on only small well-separated patches of the microscale system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Consider an example elastic beam with heterogeneous elasticity in 2D as in Figure 1: say 628 cm long, 20 cm wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The beam is heterogeneous because it is constructed from a modern material with micro-structure of size 3 cm—so that the heterogeneity is ‘visible’ in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' With a 3 cm micro-grid, the modelling requires circa 5 000 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' This specific scenario is easily computable, ode23 took 14 s cpu time to simulate one period of beam bending oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' But if a more realistic 3 mm micro-structure is simulated, then the computation time increases by a factor of 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' If 3D elasticity modelling is required for the beam, then the computation time increases by even more orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The patch scheme (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Samaey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2010) we develop herein potentially reduces macroscale computation time by orders of magnitude—more reduction in higher-D space and/or smaller micro-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The patch scheme achieves efficiency by only computing on small sparse patches in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 discusses how the patch scheme is non-intrusive in that it just ‘wraps around’ a user’s microscale code—a desirable property also identified by Biezemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The patch scheme, alternatively called the gap-tooth method, “has formal similarity with sp [superparametrization]” (Majda & Grooms 2014, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='62) that was developed in meteorology for weather and climate predictions, and is also akin to the so-called fe-fft and fe2 methods (Lucarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2021, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=',§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Figure 1: movie of a full-domain simulation of a heterogeneous beam showing that beam bending waves and longitudinal compression waves propagate with some ‘average’ properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 0 1 2 3 4 5 6 space x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 y time = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='00, E in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='39 3 1 Introduction 3 Figure 2: a small part of the microscale grid used to code 2D elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The grid is staggered on the microscale: ▶, horizontal displacements and velocities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' ▲, vertical displacements and velocities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' ⊚, ⊗, components of strain and stress tensor (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ i − 1 i i + 1 j − 1 j j + 1 Figure 3: example of the 2D mi- croscale heterogeneous Young’s modulus Eij used in computing the elastic Lam´e parameters (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' In this example, we choose the heterogeneity to have microscale period four along the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='5 · 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 5 · 10-2 0 5 · 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 space x cross-beam y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='5 A given microscale discretisation of heterogeneous elasticity We adopt a simple robust microscale approximation of 2D elasticity within the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' On the staggered microscale xy-grid of Figure 2 define the displacements: ▶, hori- zontal uij(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' ▲, vertical vij(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Microscale elasticity here first uses centred finite differences to compute stresses, for heterogeneous Lam´e parameters λ, µ, at the labelled microscale grid-points (Figure 2): ⊗ σxy := µij � δjuij/δyj + δivij/δxi � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' (1a) ⊚ σxx := (λij + 2µij)δiuij/δxi + λijδjvij/δyj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' (1b) ⊚ σyy := λijδiuij/δxi + (λij + 2µij)δjvij/δyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' (1c) Second, centred finite differences compute the following acceleration odes ▶ ¨uij = δiσxx/δxi + δjσxy/δyj , (2a) ▲ ¨vij = δiσxy/δxi + δjσyy/δyj , (2b) potentially with optional small phenomenological damping supplied by a discretisa- tion of κ∇2 ˙uij, κ∇2 ˙vij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The patch scheme wraps around whatever microscale code a user supplies—here it is the microscale system (1) and (2) We nondimensionalise the system so that the density is one, and the speed of a macroscale compression wave along the beam is about one, that is, time in these simulations is roughly in milli-seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Random periodic heterogeneity The Lam´e parameters which appear in the stresses (1) are λ := νE (1 + ν)(1 − 2ν), µ := E 2(1 + ν), (3) 2 Equation-free patch scheme 4 in terms of Young’s modulus E and Poisson ratio ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' To have strong microscale heterogeneity we choose these parameters randomly so that at each microscale grid- point (iid): Eij is log-normal (here varies by factor of about ten);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' and νij is uniform on [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Figure 3 shows an example Eij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Despite such strong heterogeneity, the movie of Figure 1 shows the macroscale dynamics appears relatively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2 Equation-free patch scheme Instead of computing the entire beam as seen in Figure 1, the patch scheme computes only in small sparse spatial patches such as Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' In this example case, the patch scheme reduces compute time by a factor ∝ r := (patch size)/(spacing H), which here is just a modest factor of 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' But with greater scale separation and/or in higher spatial dimensions, the scheme often reduces computational time by many orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The movie of Figure 4 shows a slow progressive wave of beam bending, together with a not-so-slow compression wave along the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' These macroscale predictions are accurate (Section 3) due to the correctness of our simple coupling between patches—even when heterogeneity is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The patch scheme makes these accurate macroscale predictions even when the macroscale closure is unknown: the scheme does not code a closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Further, ‘the closure’ varies depending upon human assumptions such as choosing averaged models versus cosserat models—the patch scheme makes no such closure assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The only assumption is that the macroscale quantities of importance vary smoothly between neighbouring patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 Scheme is non-intrusive functional ‘wrapper’ Consider one of the patches of the 2D beam shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' With the given microscale xy-grid (Figure 2), zooming in to the microscale each patch is like that of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Here each patch extends across the cross-section (y-dimension) of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Open symbols in Figure 5 are ghost nodes outside the patch and implement given stress-free top/bottom conditions on the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The only addition required by the patch scheme are the edge values (‘squared’ micro-grid nodes in Figure 5) on the left/right of each patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The patch scheme couples patches together by providing the patch-edge values through interpolation across the macroscale between patches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Roberts & Kevrekidis 2007, Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2014, Cao & Roberts 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Here we interpolate from each of the centre patch values across the beam (i = 4 in Figure 5) of ‘nearby’ Figure 4: movie of a patch scheme simulation of a heterogeneous beam showing the macroscale propagation across the patches of beam bending waves and longitudinal compression waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 0 1 2 3 4 5 6 space x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 y time = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='00, E in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 2 Equation-free patch scheme 5 Figure 5: one example patch of the 2D elastic beam show- ing the microscale staggered grid (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' This is case of nsubpatch = 7 micro-grid in- tervals along the patch, and ny = 4 intervals across the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▶ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊚ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ▷ ▷ ▷ ▷ ▷ ▷ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ ▷ ▷ ▷ ▷ ▷ ▷ ⃝ ⃝ ⃝ ⃝ ⃝ ⃝ □ □ □ □ □ □ □ □ □ □ □ □ □ □ i = 1 2 3 4 5 6 7 j =1 2 3 4 patches, to determine the corresponding patch-edge value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Here we implement spectral (fft) interpolation between the patches for high accuracy (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The scheme does not presume that any average is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' This implementation shows that the patch scheme is non-intrusive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Biezemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2022): it just ‘wraps around’ any micro-grid code a user trusts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Consequently, we provide a toolbox (Maclean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2021) for others to implement the patch scheme around their micro-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 Scheme embeds macroscale dynamics Given the patch scheme does not assume anything about what are ‘correct’ macroscale variables, a crucial question is the following: how can we be assured that the patch scheme captures the macroscale slow dynamics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' An answer is provided by the Whitney (1936) embedding theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Roughly, the theorem is that every mD manifold is parametrisable from almost every subspace of more than 2mD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Let’s see what this means for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' In essence, the patch scheme provides the higher-D subspace in which the slow manifold of the macroscale wave dynamics is embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' For beams in two spatial dimensions, the basic macroscale beam models have, at each cross-section, displacement and velocity of both bending and compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Thus the elastic beam dynamics has a slow manifold that is m = 4D at every cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 Alternatively, 2D cosserat beam models add a shear mode to the macroscale model—two more variables—leading to a not-quite-so-slow manifold of m = 6D at every cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' These physically based models are slow manifolds because they focus on the relatively slow waves of solutions varying slowly in space, and neglect all the faster high-frequency cross-waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' In the patch scheme, Figures 1 and 4 show simulations with a cross-section of ny = 7 micro-grid intervals, but let’s discuss the case of just ny = 4 (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' For ny = 4, there are seven microscale nodes across each patch edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Each node has a displacement and velocity, and so leads to a 14D subspace for macroscale communication between patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 1Such statements, invoking a manifold or subspace “at every cross-section”, are in a sense developed by the theory of Roberts (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' That is, in systems of large spatial extent there often are important, spatially global, invariant manifolds of high-D that are effectively decomposable into a union of spatially local manifolds/subspaces of relatively lower dimension—a dimension determined by the spatial cross-section—and that are weakly coupled to neighbouring locales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 3 Scheme has proven accuracy 6 Figure 6: multiscale spectrum of eigenvalues λ separates macroscale modes on the right from sub-patch microscale modes on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The axes are scaled nonlinearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Here the small viscosity is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='001 so the microscale decays, but the macroscale waves are long- lasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='01 0 100 30 10 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='3 1 3 10 30 100 ℜλ ℑλ Because 14 > 2 · 6 > 2 · 4 , the Whitney embedding theorem asserts that the patch scheme exchanges enough information to almost surely parametrise both such slow manifolds of the macroscale dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The patch scheme does not need to explicitly compute and exchange specific assumed macroscale average quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 3 Scheme has proven accuracy Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 discusses established theory which generally proves that the patch scheme makes accurate macroscale predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Such proofs are in stark contrast to the vast machine learning/artificial intelligence developments which prove very few general results: for example, Brenner & Koumoutsakos (2021) comment “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' ml studies, as the lack of rigorous theory does not offer (yet!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=') guarantees of convergence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Before discussing theory, we first report some computational verification of high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 Computation verifies exactness Here we restricted attention to linear elasticity so we know that the wrapped patch system is fully characterised by the resultant Jacobian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' We numeri- cally compute the Jacobian matrix of the patch scheme by elementary numerical differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Because of the macroscale translational invariance of the patch scheme, the macroscale eigenvectors are correctly sinusoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Hence the only macroscale er- rors occur in the eigenvalues of the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Figure 6 plots the spectrum of all eigenvalues for one example of random heterogeneity, in the case of five patches for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Observe there are: (on the right) four λ = 0 of rigid beam motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' four −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='001 ± i 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='057 and four −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='003 ± i 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='111 of compressions waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' four −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='001 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='061 and four −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='004 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='237 of beam bending waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' with the above macroscale eigenvalues separated by a spectral gap from the following sub-patch microscale eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 3 Scheme has proven accuracy 7 Table 1: error in patch scheme’s macroscale eigenvalues λ for various patch size ratios r: the macroscale λs are exact to round- off error—due to patch coupling by spectral interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' macro-eigenvalue r = 1 2 r = 1 4 r = 1 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='001 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='061 2e-12 1e-12 2e-13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='001 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='061 2e-12 4e-12 2e-12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='004 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='237 1e-12 8e-13 3e-12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='004 ± i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='237 1e-12 2e-12 3e-12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='001 ± i 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='057 7e-13 4e-13 6e-13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='001 ± i 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='057 6e-13 5e-13 6e-13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='003 ± i 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='111 1e-13 2e-13 2e-13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='003 ± i 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='111 4e-13 5e-13 2e-13 Figure 7: multiscale spectrum of eigenvalues λ for the patch scheme in the case of zero viscos- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The horizontal axis shows that all modes have zero real- part to numerical round-off er- ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' That is, in the case of zero viscosity, this patch scheme pre- serves the wave nature of the underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 5e-13 2e-13 1e-130 1e-13 2e-13 5e-13 100 30 10 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='3 1 3 10 30 100 ℜλ ℑλ (on the left) many ℜλ < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='1 of uninteresting sub-patch micro-scale fast- waves (headed by ten eigenvalues around −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='14 ± i 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' To quantify the accuracy, Table 1 compares eigenvalues obtained from full-domain code, with the above macroscale eigenvalues obtained by the wrapped patch scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' For all patch size ratios and heterogeneities tested, the patch scheme’s macroscale eigenvalues are exact to numerical round-off error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Such exactness is due to the spectral interpolation used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' If, instead of spectral, local polynomial interpolation of degree p is used to couple the patches, then generally the patch scheme has macroscale errors ∝ Hp where H = inter-patch spacing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Roberts & Kevrekidis 2007, Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Undamped waves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' With zero viscosity, there are only oscillations in the under- lying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' In such a scenario computational methods are very delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Here, Figure 7 illustrates that all eigenvalues of the patch scheme have |ℜλ| < 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 Hence, even with no viscosity, the patch scheme preserves the oscillatory wave nature of the heterogeneous physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' There is a perception that the patch scheme “only works well on problems with an inertial manifold and for systems in which most modes are strongly decaying” (Majda & Grooms 2014, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' This verification of accuracy for purely elastic 2In some realisations of the heterogeneity, the sensitive multiplicity four eigenvalue λ = 0 numerically splits into four showing |ℜλ| up to 10−6 due to round-off errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 4 Conclusion 8 beams shows that this perception is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Applications and theory for other wave systems also refute this perception (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Cao & Roberts 2016, Bunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2021, Divahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='2 Mathematical analysis proves consistency Mathematical analysis has proven properties of the patch scheme in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Mostly, the published proofs explicitly address dissipative (nonlinear) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' However, as discussed by Bunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' (2021), the patch scheme in space only recasts spatial interactions, so whether the time derivative is ∂/∂t of dissipation or ∂2/∂t2 of waves makes little difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Two complementary types of results have been proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' They involve the spacing between patch centres H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' First, Centre Manifold Theory may be applied at finite spacing H by introducing a ‘bookkeeping’ parameter γ to label inter-patch communication (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2014, §2) to prove the existence of a slow manifold in the patch scheme (including when it is applied to nonlinear systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Then the parameter γ structures inter-patch interactions, and their algebraic expression, to empower theory based at γ = 0, via regular perturbation, to address finite γ such as the case of full coupling γ = 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2014, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Second, the patch scheme is consistent with the underlying micro-code as the patch spacing H → 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2014, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The consistency is that the macroscale of the patch scheme is the same as the macroscale of the given micro-coded system, to errors O � Hp� when using polynomial interpolation of degree p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' For example, spectral interpolation corresponds to ‘p = ∞’ so then errors vanish to all orders as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' These results and general proofs were first done for homogeneous systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Roberts & Kevrekidis 2007, Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' They were subsequently ex- tended to heterogeneous microscales (Bunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2017), and recently extended to alternative inter-patch coupling that preserves self-adjointness (Bunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Interestingly, the extension of the theoretical support to heterogeneous cases invokes the ensemble of all phase-shifts of the heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The ensemble is spatially homogeneous, so the homogeneous proofs and results apply to establish the heterogeneous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 4 Conclusion As an initial exploration of the patch scheme for homogenisation of heterogeneous elasticity, we considered the prototypical case of a 2D elastic beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The scheme gives a non-intrusive and efficient computational homogenisation of given microscale system via spatially sparse small patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The patch coupling has proven accuracy, controllable error, at finite scale separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The patch scheme makes only one assumption: in the scenarios of interest to a user, there is no significant spatial structures in the mesoscale between the patch spacing H and the microscale resolved in the patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' In contrast to most other multiscale methods, there is: no assumed boundary conditions on Representative Volume Elements (variously periodic, Dirichlet, Neumann);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' no explicitly assuming References 9 slow variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' and no presumed necessary variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' The scheme is entirely physically interpretable: there is no hidden mystic machinations of neural networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=', Brenner & Koumoutsakos 2021) The patch scheme is simple to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' In contrast to other multiscale methods there is: no arbitrary averaging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' no oversampling regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' no buffer regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' no action regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' no guessed fast/slow variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' no epsilons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' and no limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' As a non-intrusive ‘wrapper’, anyone can start using the patch scheme via a Matlab/ Octave Toolbox (Maclean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2021, Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' 2019–2023) Acknowledgements This research was supported by Australian Research Coun- cil grants DP220103156 and DP200103097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' References Biezemans, R.' metadata={'source': 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materials’, Materials 15(2), 605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' Whitney, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} +page_content=' (1936), ‘Differentiable manifolds’, Annals of Mathematics 37(3), 645– 680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFPT4oBgHgl3EQfrjVl/content/2301.13145v1.pdf'} diff --git a/Y9E3T4oBgHgl3EQfcgoK/content/tmp_files/2301.04525v1.pdf.txt b/Y9E3T4oBgHgl3EQfcgoK/content/tmp_files/2301.04525v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff8da2bca356013f7fad8d83c947a1c64822679f --- /dev/null +++ b/Y9E3T4oBgHgl3EQfcgoK/content/tmp_files/2301.04525v1.pdf.txt @@ -0,0 +1,1157 @@ +Clustering disease trajectories in contrastive +feature space for biomarker discovery in +age-related macular degeneration +Robbie Holland1(�), Oliver Leingang2, Christopher Holmes3, Philipp Anders4, +Johannes C. Paetzold1, Rebecca Kaye7, Sophie Riedl2, Hrvoje Bogunović2, +Ursula Schmidt-Erfurth2, Lars Fritsche6, Hendrik P. N. Scholl4,5, Sobha +Sivaprasad3, Andrew J. Lotery7, Daniel Rueckert1,8, Martin J. Menten1,8 +1 BioMedIA, Imperial College London, London, United Kingdom +robert.holland15@ic.ac.uk +2 Laboratory for Ophthalmic Image Analysis, Medical University of Vienna +3 Moorfields National Institute for Health and Care Biomedical Research Centre, +Moorfields Eye Hospital, London, United Kingdom +4 Institute of Molecular and Clinical Ophthalmology Basel +5 Department of Ophthalmology, Universitat Basel, Basel, Switzerland +6 Department of Biostatistics, University of Michigan, Ann Arbor, United States +7 Clinical and Experimental Sciences, Faculty of Medicine, University of +Southampton, Southampton, United Kingdom +8 Institute for AI and Informatics in Medicine, Technical University of Munich, +Munich, Germany +Abstract. Age-related macular degeneration (AMD) is the leading cause +of blindness in the elderly. Despite this, the exact dynamics of disease +progression are poorly understood. There is a clear need for imaging +biomarkers in retinal optical coherence tomography (OCT) that aid the +diagnosis, prognosis and management of AMD. However, current grad- +ing systems, which coarsely group disease stage into broad categories +describing early and intermediate AMD, have very limited prognostic +value for the conversion to late AMD. In this paper, we are the first to +analyse disease progression as clustered trajectories in a self-supervised +feature space. Our method first pretrains an encoder with contrastive +learning to project images from longitudinal time series to points in +feature space. This enables the creation of disease trajectories, which +are then denoised, partitioned and grouped into clusters. These clusters, +found in two datasets containing time series of 7,912 patients imaged +over eight years, were correlated with known OCT biomarkers. This re- +inforced efforts by four expert ophthalmologists to investigate clusters, +during a clinical comparison and interpretation task, as candidates for +time-dependent biomarkers that describe progression of AMD. +Keywords: Contrastive learning · Trajectory clustering · Disease pro- +gression · Retina · OCT · Biomarker discovery +arXiv:2301.04525v1 [eess.IV] 11 Jan 2023 + +2 +Holland et al. +1 +Introduction +AMD is the leading cause of blindness in the elderly, affecting nearly 200 mil- +lion people worldwide [22]. Patients with early stages of the disease exhibit +few symptoms until suddenly converting to the late stage, at which point their +sharp central vision rapidly deteriorates [14]. AMD patients are commonly im- +aged with optical coherence tomography (OCT), which provides low-cost and +high-resolution retinal images that depict fine physiological details. Several OCT +biomarkers have been tentatively linked to the pathogenesis of AMD, such as reti- +nal layer thickness, photoreceptor atrophy and the presence of drusen, which are +lipidic deposits that build up inside the retina [16]. However, the exact relation +of these biomarkers to the progression from early to late stages of AMD remains +unclear. Current grading systems only coarsely group patients into broad cate- +gories for early, intermediate and late AMD and have limited prognostic value +[8]. There is an unmet need for biomarkers that describe and predict the pro- +gression of AMD. +Large studies that image populations of AMD patients over time are a power- +ful resource to discover novel biomarkers for disease progression. If candidate +biomarkers are already identified a priori, they can be extracted from images +and mapped against disease progression [16,19,3]. However, this approach is not +feasible if the biomarkers are partially or fully unknown. Self-supervised learn- +ing is the most promising method to automatically discover new biomarkers by +clustering or finding anomalous OCT images in lower-dimensional representation +space [18,21,17]. However, by grouping single scans acquired at single points in +time, these biomarkers are by definition static and cannot capture concepts such +as the speed of disease progression or transitions between multiple states of the +disease. Clustering whole time series of images is also problematic, as patients +enter and leave the study at different points in their overall progression. There- +fore, in order to discover biomarkers that describe population-level patterns in +disease progression, it is necessary to analyse and compare portions of patient +time series. +Our contribution In this work, we develop a strategy to automatically discover +biomarkers that capture disease progression in time series of images. This is +illustrated in Figure 1. Our method represents the time series of images as a +trajectory in a self-supervised latent feature space built by contrastive learning. +This representation allows a new partitioning of time series of OCT images +into consecutive subsequences (sub-trajectories) that exhibit distinct units of +disease progression. By clustering these sub-trajectories, we are now able to +detect patterns of disease progression that are common among the population +of patients. +Experimentally, we test our method on a large cohort from two, longitudinal +retinal OCT datasets totalling 160,558 images from 7,912 patients. In doing so we +categorise 3,218 total years of disease progression with time-dependent clusters. +We then correlate our clusters with the known set of biomarkers which reinforces + +Clustering disease trajectories for biomarker discovery in AMD +3 +Fig. 1: We analyse disease progression as clustered trajectories in feature space. +We first train a feature extractor with contrastive learning and then form tra- +jectories by projecting time series of images to feature space. These are then +partitioned into a set of sub-trajectories which are clustered. Finally, clusters are +related to known disease stages before being interpreted as candidate biomarkers +for disease state and progression. PCA space is coloured by visual acuity. +their potential to contain potentially new, time-dependent biomarkers. Finally, +we closed the loop on automated biomarker discovery by working directly with +four ophthalmologists to interpret these clusters in a clinical comparison task. + +1. Contrastive pretraining +Contrastive +PCA feature +Retinal OCT datasets (170k scans) +loss +space +ResNet-50 +CNN +00 +0 +00 +0 +Copy pretrained +feature extractor +0 +0 +0 +PCA reduction +00 +ResNet-50 +00 +CNN +00 +00 +2. Patient trajectories in feature space +Resample +Longitudinal information +with temporal +denoising kernel +Eye +Scan times +04/2017 +08/2017 +11/2017 +2 +07/2012 +12/2012 +04/2013 +3 +06/2019 +09/2020 +10/2020 +01/2015 +03/2016 +04/2016 +5 +08/2017 +09/2017 +02/2018 +6 +09/2012 +03/2013 +07/2013 +3. Clustering and proposing biomarkers +Cluster 1 vs. Cluster 2 +Series A +K-means +Query series +Count biomarkers +clustering +Known biomarkers +Series B +per cluster +Healthy Drusen cRORA +CNV +0 +Clusters +Experts compare clusterst with +similar known biomarkers +2 +but different latent features +tclusters are candidates for +time-dependent biomarkers4 +Holland et al. +2 +Related work +2.1 +Current AMD grading systems +Ophthalmologists’ current understanding of progression from early to late AMD +largely involves drusen. Drusen can grow until suddenly regressing and disap- +pearing, which often precedes the onset of late AMD [16]. While there have +been attempts to group drusen based on their morphology [11], current grading +systems stratify early and intermediate stages solely by drusen size [2,12,7,8]. +Late AMD is classified as either choroidal neovascularisation (CNV), identified +by fluid under the retina, or geographic atrophy by progressive loss of photore- +ceptors and retinal thinning. The degree of atrophy can be staged using cRORA +(complete retinal pigment epithelium and outer retinal atrophy), which measures +the width of focal atrophy in OCT [15]. So far grading systems offer limited di- +agnostic value and little to no prognostic value. +2.2 +Tracking evolution of known biomarkers +Few research efforts have aimed at quantifying and tracking known AMD biomark- +ers over time, such as reticular pseudodrusen [19] and drusen volume [16]. More +work has explored Alzheimer’s disease (AD), which offers a greater array of +quantitative imaging biomarkers, such as levels of tau protein and hippocampal +volume. Young et al. [23] fit an event-based model that rediscovers the order +in which these biomarkers become anomalous as AD progresses. Vogel et al. +[20] find four distinct spatiotemporal trajectories for tau pathology in the brain. +However, mapping biomarkers that evolve during disease progression requires +prior annotation of entire time series. Thus, these biomarkers must be known or +at least suspected a priori. +2.3 +Automated discovery of unknown biomarkers +In order to discover new biomarkers, efforts to find them have turned to au- +tomated biomarker discovery. Imaging biomarkers are proposed by analysing +anomalous scans [17], clusters of scans [25], or a combination of these [18] in +feature space. To build these, neural networks are trained with supervised or +unsupervised proxy tasks. These tasks include image reconstruction [21], seg- +mentation [25] and generative adversarial networks [17]. However, networks are +prone to overfit on their specific task and lose semantic information regarding +the disease. Contrastive learning has recently advanced the state-of-the-art in +training generalisable and unbiased feature extractors. Chen et al. popularised +this paradigm with SimCLR [4], which was later improved on by Grill et al. [9] in +Bootstrap Your Own Latent (BYOL). Contrastive methods encode invariance to +a set of transformations typically uncorrelated with disease features, including +rotation, translation and global shifts in image brightness and contrast. Zhao et +al. leverage contrastive feature spaces to identify high-risk clusters of CT image +patches [24]. + +Clustering disease trajectories for biomarker discovery in AMD +5 +However, all biomarkers discovered by the aforementioned methods work by +grouping single images acquired at single points in time, and in doing so neglect +temporal relationships between images of the same subject. One work that tack- +les this challenge, and the most related to ours, categorises the time-dependent +response of cancer cells to drugs, measured by the changing distance in con- +trastive feature space from healthy controls [5]. +2.4 +Trajectory clustering +The separate field of trajectory clustering is largely focussed on discovering move- +ment patterns taken by cars, animals and hurricanes [6,1,13]. Lee et al., in their +state-of-the-art work TRACLUS [13], assume these trajectories are composed of +consecutive series of common sub-trajectories. For example, different car jour- +neys may at some point travel down the same road. Using this principle, they +develop a partition-and-group framework to cluster segments that are repeated +across multiple trajectories. Similarly, we assume that disease progression can +be divided into multiple, common disease pathways. Firstly, this allows us to +work seamlessly with temporally unaligned scanning series. Secondly, we can +automatically discover novel disease pathways by interpreting sub-trajectories +that are shared by multiple AMD patients. +3 +Materials and Methods +3.1 +Self-supervised feature space using contrastive learning +We adapt BYOL [9] with update coefficient τ = 0.9995 for contrastive pretrain- +ing of a ResNet50 (4x) model. As several of the contrastive transformations +designed for natural images are inapplicable to medical images, we use the set +tailored for retinal OCT images by Holland et al. [10]. Models were trained on +the entire dataset for 120,000 steps with momentum 0.9 and a learning rate of +5 · 10−4 using the Adam optimiser. +After pretraining, we first remove any final linear layers before projecting all +labelled images to the feature space of 2048 dimensions. We then reduce the di- +mension further using principle component analysis (PCA) with D components. +Using PCA allows us to interpolate the feature space and results in fewer di- +mensions, which is advantageous for clustering. To validate that the contrastive +feature space encodes meaningful information for AMD biomarkers, and find +the optimal dimension D ∈ {2, 10, 20, 50}, we perform multi-class classification +of known biomarkers (including healthy controls). Firstly, we split the dataset +into train and test partitions using 85% and 15% of the data, respectively, en- +suring that all scans from each patient belong to the same set. Then to perform +the classification, we fit a class-balanced support vector machine (SVM) on the +training set and report performance on the test set. + +6 +Holland et al. +Fig. 2: Longitudinal scans of a single eye, imaged over four years, projected to +PCA space. PCA space is plotted as a hexmap coloured by the local average +in visual acuity, where higher values indicate poorer quality of vision. Each +row depicts two principle components up to 20. The rightmost columns show +resampled trajectories. Using smaller T captures short-term variation in disease +progression but cannot model long-term changes. MDL aims to optimise this +tradeoff by partitioning trajectories only at points of inflection. +3.2 +Extracting and clustering common sub-trajectories +For each eye, we first form piecewise-linear trajectories by linking points in PCA +space that were derived from consecutively acquired OCT images (see left column +in Figure 2). We then assume, in analogy to TRACLUS [13], that trajectories +encoding disease progression can be partitioned into sub-trajectories that are +common among multiple patients. We compare two methods to achieve this, +shown in the right-most columns in Figure 2. The first method resamples time- +points at regular intervals of T years. For each resampled time t, we find the +corresponding point in feature space by taking a weighted average of all points +in the trajectory. The weights are calculated by convolution of a Gaussian kernel +N(t, σT ) with the acquisition dates of the entire scanning series. We then define + +Resampled trajectories +(T = 0.5 years) +T= +T=1.0 +T= 2.0 +T= 0.5 +Original +Cumulative +MDL Partition +(years) +trajectory +(years) +(years) +variance +Dims 1-2 +PCA #2 +PCA #2 ++18% +CA#2 +18% +PCA #1 +PCA #1 +PCA #1 +PCA #1 +PCA #1 +1 +Dims 3-4 +PCA #4 +PCA #4 +PCA#4 ++13% +7# +CA +31% +PCA #3 +PCA #3 +PCA #3 +PCA #3 +PCA #3 +Dims 5-6 +PCA #6 +PCA #6 ++10% +PCA #6 +PCA #6 +CA #6 +41% +PCA #5 +PCA #5 +PCA #5 +PCA #5 +PCA #5 +Dims 7-20 +PCA #20 +PCA #20 +PCA #20 +PCA #20 +#20 ++31% +PCA +72% +PCA #19 +PCA #19 +PCA #19 +PCA #19 +PCA #19 +Colour map for +visual acuity +0.4 +0.6 +0.8 +1.2 +1.4 +0.2 +1.0 +1.6 +1.8Clustering disease trajectories for biomarker discovery in AMD +7 +sub-trajectories as vectors between consecutive points that are less than or equal +to T years apart. +The second method aims to describe trajectories using the fewest points. It be- +gins by resampling trajectories using the first method (with intervals of T = 0.5 +years). Then, using the minimum description length (MDL) principle, a mini- +mal subset of points are chosen that best preserve changes in disease over time. +To achieve this we find the trajectory H, containing a subset of the points in +the resampled trajectory O, which minimises the following objective (using the +greedy solution from [13]) +L(O|H) = +� +p∈O +d⊥(p, H) − λ|H| +where d⊥(p, H) is the perpendicular distance from a point p to the piece-wise +linear trajectory H, and |H| is the number of points in H. The coefficient λ is +proportional to the total standard deviation in the feature space explained by +D PCA dimensions. +Clustering sub-trajectories To cluster common sub-trajectories we require +a distance function that measures the similarity between vectors. Given two +sub-trajectories U = (ustart, uend) and V = (vstart, vend) their distance is simply +d(U, V ) = ∥ustart − vstart∥2 + ∥uend − vend∥2 +Finally, using d we separate sub-trajectories into K clusters using k-means clus- +tering. +3.3 +Finding optimal hyperparameters using the set of known +biomarkers +We now search for optimal values for the sub-trajectory time interval T ∈ +{0.5, 1.0, 2.0} years, resampling kernel width σT ∈ {0.25, 0.5, 1.0} years and the +number of clusters K ∈ {5, 10, 15, 30} using five random seeds for k-means clus- +tering. To quantitatively compare configurations, we use the conditional entropy +H(B|C) = H(B, C) − H(C) as a scalar measure of how well the clusters C redis- +cover the known biomarkers B detailed in section 3.5. This is calculated directly +from their joint distribution P(B, C), which is found by counting all biomarkers +recorded within sub-trajectories of each cluster. To ensure the equal contribution +of all biomarkers, we reweight their marginal distribution P(B) to be uniformly +distributed. As the number of clusters K increases even randomly permuted as- +signments p(C) will result in reduced values of H(B|p(C)). We address this by +using the adjusted reduction in conditional entropy, H′ (using r = 5 random +trials), where higher values correspond to better rediscovery of B +H′(B|C) = 1 +r +r +� +i +H(B|p(C)) − H(B|C) +As our ultimate goal is to detect biomarkers beyond the known set, we use H′ +only as an indication for the most suitable configuration. + +8 +Holland et al. +3.4 +Proposing clusters as candidate biomarkers +We first relabel the clusters C in order of median visual acuity, so that higher +cluster numbers indicate poorer quality of vision. In order to see which disease +stages each cluster describes, we calculate the conditional probability P(B|C) = +P(B, C)/P(C). Then, to discover new biomarkers beyond the existing set, we +compare clusters that differ maximally in feature space but minimally in the set +of known biomarkers. To find these, we explore eleven pairs of distinct clusters +Ci and Cj with a high degree of cosine similarity P(B|Ci) · P(B|Cj). +Interpreting candidate biomarkers To examine clusters for candidate biomark- +ers, we collaborate with four expert ophthalmologists. For each pair of clusters, +we generate four random ‘A or B’ single-choice questions. Clinicians are shown +one query series in image space from cluster Ci and two further series denoted +A and B, one from Ci and the other from Cj. For each question, clinicians are +tasked with determining which of A or B belongs to the same cluster as the +query. After completing four questions they are prompted to explain on what +basis they matched A or B to the query. This format allows us to both assess +whether the clusters are visually distinguishable by experts and, if so, potentially +extract descriptions of novel biomarkers. +3.5 +OCT datasets +We apply our method to two independent retinal OCT datasets called Dataset +A and Dataset B. We developed our method on Dataset A but run experiments +on both datasets. In both, images were acquired using Topcon 3D OCT devices +(Topcon Corporation, Tokyo, Japan). After strict quality control, Dataset A +consists of 46,496 scans of 6,236 eyes from 3,456 patients. Eyes were scanned 7.7 +times over 1.9 years on average at irregular time intervals. The second dataset, +Dataset B, is larger, containing 114,062 scans of 7,253 eyes from 3,819 patients. +Eyes were scanned 16.6 times over 3.5 years on average. Of each 3D OCT vol- +ume, we extracted the transverse 2D slice centred at the fovea and resampled +to 208×256 pixels with a pixel size of 7.0×23.4 µm2, half the median resolution. +Each scan is labelled with visual acuity, a functional measure assessing the qual- +ity of vision measured in LogMAR. +To record conversions to a comprehensive set of known biomarkers B, we used +established AMD grading protocols described in section 2.1. Early AMD is char- +acterised by small drusen between 63-125µm in diameter. We also recorded CNV, +cRORA (of at least 250µm but smaller than 1000 µm) and cRORA (of at least +1000 µm) [15]. Overall, 861 conversion times t0 were recorded, and any sub- +sequent visits at times t+ before the next conversion were automatically as- +signed with a separate label. Visits prior to any biomarker were labelled as +‘NoBiomarker’. Finally, in each dataset, an additional 150 healthy images that +exhibit no pathology were recorded. Combining these, the known set of biomark- +ers B includes 10 biomarkers and labels. + +Clustering disease trajectories for biomarker discovery in AMD +9 +4 +Results +Fig. 3: Confusion matrices for multi-class classification of known biomarkers and +healthy images using D numbers of PCA dimensions. In general, performance +increases with the number of dimensions D. We find that using D = 20 PCA +dimensions achieves linear separability between known biomarkers. +4.1 +Finding the optimal set of hyperparameters using the known +biomarkers +In both Dataset A and Dataset B 20 principal dimensions achieves linear sepa- +rability between known biomarkers (see Figure 3). Both the healthy stage and +the only extractable early biomarker, drusen, were found to be highly linearly +separable. Thus, we use D = 20 for the remainder of our analysis. +We find that K = 15 clusters of sub-trajectories best explained the set of known +biomarkers B (Figure 4a) as measured by higher values of H′(B|C). Greater +values of T, in addition to MDL partitioning, result in decreased H′(B|C). +We suspect that this is because the known set of biomarkers describe disease +states rather than state transitions, so they are better captured by shorter sub- +trajectories. In order to select a configuration that finds clusters evidencing pro- +gression, we choose T = 1.0, σT = 0.5 and K = 15 for the remainder of our +analysis. +4.2 +Sub-trajectory clusters go beyond the known set of biomarkers +We find that our clusters encode the set of known biomarkers in Dataset A. As +seen in Figure 5, clusters effectively separate healthy, early stage and late-stage + +Dataset A +10D +20D +50D +2D +0.28 +0.05 +0.07 +0.94 +0.04 +0.06 +0.01 +Healthy +0.57 +0.10 +0.92 +0.01 +0.00 +0.01 +0.00 +0.00 +0.92 +0.01 +0.01 +0.00 +0.00 +0.18 +0.01 +0.01 +0.12 +0.72 +0.12 +Drusen +0.42 +0.37 +0.01 +0.01 +0.87 +0.06 +0.01 +0.04 +0.90 +0.04 +0.00 +0.04 +0.00 +0.04 +0.12 +0.36 +0.16 +0.30 +0.01 +0.05 +0.12 +0.23 +0.26 +0.20 +0.29 +0.28 +0.24 +0.33 +0.24 +0.07 +0.19 +0.23 +cRORA (250 μm) +0.26 +0.06 +0.02 +0.62 +0.02 +cRORA (1000 μm) +0.04 +0.16 +0.16 +0.62 +0.02 +0.07 +0.22 +0.01 +0.29 +0.02 +0.63 +0.07 +0.62 +0.06 +0.05 +0.06 +0.21 +0.19 +0.42 +CNV +0.05 +0.09 +0.19 +0.62 +0.05 +0.05 +0.08 +0.25 +0.03 +0.09 +0.20 +0.52 +0.08 +0.14 +0.57 +0.20 +0.16 +0.02Dataset B +0.49 +0.24 +0.00 +0.92 +0.08 +0.00 +0.00 +0.84 +0.03 +0.00 +0.97 +0.00 +0.00 +0.00 +0.03 +Healthy +0.16 +0.11 +0.00 +0.08 +0.05 +0.08 +Drusen +0.36 +0.39 +0.07 +0.30 +0.04 +0.05 +0.14 +0.06 +0.14 +0.62 +0.19 +0.15 +0.16 +0.61 +0.11 +0.02 +0.16 +0.34 +0.00 +0.42 +0.13 +0.00 +0.18 +0.00 +cRORA (250 μm) +0.26 +0.21 +0.11 +0.32 +0.39 +0.16 +0.10 +0.00 +0.45 +0.31 +0.06 +0.39 +0.35 +0.13 +0.03 +0.04 +0.33 +0.08 +cRORA (1000 μm) +0.33 +0.21 +0.08 +0.00 +0.00 +0.04 +0.29 +0.33 +0.33 +0.38 +0.00 +0.00 +0.71 +0.12 +0.17 +0.00 +0.54 +0.61 +0.40 +0.29 +0.23 +0.08 +0.00 +0.02 +0.13 +0.13 +0.19 +0.53 +CNV +0.00 +0.17 +0.14 +0.12 +0.57 +0.00 +0.17 +0.10 +0.11 +ANO +CRORA +CRORA ( +(250 +A (250 μm)10 +Holland et al. +(a) Adjusted conditional entropy H′(B|C) of known biomarkers B given subtrajectory +clusters C against hyperparameters K (left), T (center), σT (right) +Fig. 4: Results of the hyperparameter search, measured by H′(B|C) which is a +scalar measure of how well the clusters C rediscover existing biomarkers B. As +we aim to discover biomarkers beyond the known set, we also consider the level +of disease progression captured by our clusters when choosing our configuration. +Fig. 5: Conditional probabilities P(B|C) of the known set of biomarkers B given +cluster assignments C by our method. Cluster pairs highlighted in blue were, +due to their similarity under the known set of biomarkers B, chosen for further +analysis by clinicians. One cluster pair, highlighted in pink, was used as a trial +task and validation experiment. + +Entropy +0.6 +0.5 +Adjusted reduction in Conditional +0.4 +0.3 +0.2 +0.1 +0.0 +Variants +Dataset A +-0.1 +Dataset B +5 +10 +15 +30 +Number of clusters (K)Entropy +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +Variants +Dataset A +-0.1 +Dataset B +0.25 +0.5 +1.0 +OTEntropy +0.6 +0.5 +Adjusted reduction in Conditional +0.4 +多 +0.3 +0.2 +0.1 +0.0 +Variants +Dataset A +-0.1 +Dataset B +T= 0.5 +T= 1.0 +T= 2.0 +T= MDL +partition +Progression vector sampling strategyDataset A +Cosine similarities over clusters pairs +Conditional probability of known biomarkers +P(BICi) · P(BC)) +given cluster assignments: P(B|C) +0.94 +D.55 0.74 0.58 0.75 C +0.56 0.63 0.43 0.82 +2 0.400.15 0.05 0.07 +0.19 +0.16 +0.24 +0.20 +0.00 +0.00 +0.00 +0.00 +0.09 +0.05 +0.25 +1.00 +0.21 +0.06 +0.04 +0.00 +0.00 +0.09 +0.04 +0.94 1.00 +0.34 +0.84 0.67 0.77 0.67 0.72 0.44 0.71 +0.49 0.17 0.07 0.08 0.18 +2 +0.07 +0.19 +0.31 +2 +0.70 +0.12 +0.07 +0.07 +0.00 +0.00 +0.00 +0.00 +0.02 +0.02 +0.55 0.34 +1.00 0.16 0.14 +0.46 0.16 0.17 0.37 +0.74 +0.08 0.17 0.01 0.06 0.03 +3 +3 +0.00 +0.15 +0.22 +0.04 +0.07 +0.74 0.84 +40.16 +1.00 +0.91 +0.86 0.89 0.96 +4 +0.14 +0.17 +0.11 +0.00 +0.10 +0.60 0.58 0.73 0.33 0.21 0.23 0.27 +4 +5 +0.00 +0.17 +0.07 +0.12 +0.16 +0.21 +0.07 +0.03 +0.11 +0.06 +5 +0.58 0.67 +0.14 0.91 1.00 0.86 0.97 0.94 +0.65 0.47 0.82 0.510.34 0.36 0.32 +9 +0.11 +0.09 +0.10 +0.11 +0.08 +0.04 +0.12 +0.75 0.77 0.46 +6 0.86 0.86 1.00 +0.91 +0.89 0.82 0.82 0.82 0.60 0.44 0.46 0.52 +0.12 +0.11 +0.12 +6 +6 0.89 0.97 0.91 1.00 +0.02 +0.17 +0.19 +0.11 +0.04 +0.08 +0.56 0.67( +0.92 +D.710.51 +0.83 0.61 0.45 0.47 0.36 +7 +0.11 +0.12 +0.09 +0.07 +7 +0.16 +0.01 +0.20 +0.03 +0.63 0.72 0.17 0.96 0.940.89 0.92 1.00 +0.76 0.57 0.87 0.44 0.29 0.31 0.43 +8 +0.13 +0.08 +0.16 +0.13 +0.03 +0.12 +0.11 +8 +0.13 +0.03 +0.07 +0.03 +0.16 +0.14 +0.00 +0.00 +0.16 +0.29 +0.43 0.44 0.37 0.60 0.65 0.82 0.71 0.76 1.00 0.66 0.90 0.42 0.26 0.27 0.66 +9 +9 +0.82 0.71 0.74 0.58 0.47 0.82 0.51 0.57 0.66 +0.25 +0.07 +0.14 +0.15 +0.00 +0.05 +0.13 +1.00 0.52 0.41 0.29 0.31 0.51 +0 +0.00 +0.06 +0.15 +0 +0.08 0.73 0.82 0.82 0.83 0.87 0.90 0.52 +0.06 +0.15 +0.20 +0.21 +0.40 0.49 +1.00 +0.53 0.36 0.360.70 +0.00 +0.06 +0.06 +0.00 +0.06 +0.20 +0.96 +0.07 +0.01 +0.02 +0.01 +0.07 +0.14 +0.25 +0.29 +0.07 +0.08 +0.15 0.17 0.17 0.33 0.51 0.60 0.61 0.44 0.42 0.41 0.53 +D.61 +2 +2 +1.00 0.95 +0.00 +0.00 +0.00 +0.00 +0.05 +0.05 +0.35 +0.37 +0.06 +0.12 +0.05 0.07 0.01 0.21 0.34 0.44 0.45 0.29 0.26 0.29 0.36 +50.951.00 +1.00 +0.62 +3 +3 +0.03 +0.00 +0.00 +0.00 +0.08 +0.04 +0.34 +0.36 +0.05 +0.10 +0.07 0.08 0.06 0.23 0.36 0.46 0.47 0.31 0.27 0.31 0.36 +0.96 1.00 1.00 0.60 +4 +4 +0.70 0.61 0.62 0.60 +0.00 +0.00 +0.00 +0.00 +0.00 +0.06 +0.28 +0.30 +0.36 +0.19 0.18 0.03( +0.27 0.32 0.52 0.36 0.43 0.66 +0.51 +1.00 +5 +0.00 +5 +13 14 15 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +(to) +(to) +CNV +(to) +(to) +Healthy + μm) +Drusen +NoBiomarker +CRORA +(250 +(1000 | +(1000μm) +RAClustering disease trajectories for biomarker discovery in AMD +11 +biomarkers. We find two clusters (1-2) describing drusen, five (4-8) describing the +transition from drusen to cRORA (250 µm) and three (12-14) describing cRORA +(1000 µm). Most healthy samples were in cluster 3, and CNV was spread amongst +most late-stage clusters. In Dataset B we find three clusters of healthy looking +scans, three of early stages and six in the late stage that capture atrophy. +Interpreting candidate biomarkers We now report the results of the single- +choice clinical clustering task (described in section 3.4), comparing cluster pairs +which are highlighted in blue and pink boxes in Figure 5. In Dataset A, clinicians +were easily able to separate clusters 1 and 15 by using known biomarkers, such as +hypertransmission and photoreceptor degeneration. The result of this validation +experiment was expected, as these clusters were already highly separable under +the known set of biomarkers. More interestingly, clinicians were able to exactly +differentiate between early AMD clusters 1 and 2 in Dataset A despite their high +similarity in known biomarkers. When prompted, all clinicians cite differences +in drusen, with two finding differences in the number of small drusen. Their +ability to distinguish some of the pairs was mixed, as they sometimes found no +consistent or visible differences or had a low inter-rater agreement. +5 +Discussion and Conclusion +In this paper, we proposed a method to automatically discover time-dependent +biomarkers that detect periods of disease progression common among groups of +patients. By partitioning entire time series into representative sub-trajectories, +and then clustering them, we categorised 3,218 total years of disease progression +across two datasets. We showed that these clusters rediscovered the established +set of OCT biomarkers for AMD, which reinforced the use of our clusters as can- +didate biomarkers. Then, by working directly with ophthalmologists, we closed +the loop in our automated biomarker discovery. To this end ophthalmologists +compared clusters that were indistinguishable using current grading systems, +yet were separable in contrastive feature space. We envision that further inves- +tigation into sub-trajectory clusters could advance understanding of how AMD +progresses, and potentially lead to grading systems with greater prognostic value. +Our method is applicable to any dataset studying any disease with time series +of images. While our method identified two clusters that described drusen in +Dataset A and three that described healthy-looking scans in Dataset B, most +clusters were associated with intermediate and late-stage AMD. This is due +to the overrepresentation of patients with late disease in our datasets. In or- +der to find more clusters categorising progression in early AMD, we aim to +repeat this analysis in datasets that begin imaging patients earlier in their over- +all progression. Moreover, due to the slow progression of AMD, a large number +of sub-trajectories captured unchanging disease states. In order to isolate sub- +trajectories capturing the periods of greatest disease progression, we intend to +increase the number of clusters K and interpret those that convert to late disease +the fastest. + +12 +Holland et al. +Conclusion Inspired by inadequate grading systems for disease progression +in early AMD, we proposed the first method to analyse disease progression as +clustered trajectories in self-supervised feature space. By correlating our clusters +with known OCT biomarkers for AMD, we reinforced their potential as time- +dependent biomarkers for disease progression. After this, we closed the loop on +automated biomarker discovery by working directly with ophthalmologists to +investigate our clusters. We envision that self-supervised learning can enable +detection of patterns of disease progression in time series of patient populations, +which can lead to grading systems with greater prognostic value. +References +1. Bian, J., et al.: A survey on trajectory clustering analysis. CoRR abs/1802.06971 +(2018), http://arxiv.org/abs/1802.06971 +2. Bird, A.C., et al.: An international classification and grading system for age- +related maculopathy and age-related macular degeneration. Survey of ophthal- +mology 39(5), 367–374 (1995) +3. Chen, K.G., et al.: Longitudinal study of dark adaptation as a functional outcome +measure for age-related macular degeneration. Ophthalmology 126(6), 856–865 +(2019) +4. Chen, T., et al.: A simple framework for contrastive learning of visual represen- +tations. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='uk 2 Laboratory for Ophthalmic Image Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Medical University of Vienna 3 Moorfields National Institute for Health and Care Biomedical Research Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Moorfields Eye Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' United Kingdom 4 Institute of Molecular and Clinical Ophthalmology Basel 5 Department of Ophthalmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Universitat Basel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Basel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Switzerland 6 Department of Biostatistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' United States 7 Clinical and Experimental Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Faculty of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' University of Southampton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Southampton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' United Kingdom 8 Institute for AI and Informatics in Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Technical University of Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Germany Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Age-related macular degeneration (AMD) is the leading cause of blindness in the elderly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Despite this, the exact dynamics of disease progression are poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' There is a clear need for imaging biomarkers in retinal optical coherence tomography (OCT) that aid the diagnosis, prognosis and management of AMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' However, current grad- ing systems, which coarsely group disease stage into broad categories describing early and intermediate AMD, have very limited prognostic value for the conversion to late AMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In this paper, we are the first to analyse disease progression as clustered trajectories in a self-supervised feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Our method first pretrains an encoder with contrastive learning to project images from longitudinal time series to points in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' This enables the creation of disease trajectories, which are then denoised, partitioned and grouped into clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' These clusters, found in two datasets containing time series of 7,912 patients imaged over eight years, were correlated with known OCT biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' This re- inforced efforts by four expert ophthalmologists to investigate clusters, during a clinical comparison and interpretation task, as candidates for time-dependent biomarkers that describe progression of AMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Keywords: Contrastive learning · Trajectory clustering · Disease pro- gression · Retina · OCT · Biomarker discovery arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='04525v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='IV] 11 Jan 2023 2 Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 1 Introduction AMD is the leading cause of blindness in the elderly, affecting nearly 200 mil- lion people worldwide [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Patients with early stages of the disease exhibit few symptoms until suddenly converting to the late stage, at which point their sharp central vision rapidly deteriorates [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' AMD patients are commonly im- aged with optical coherence tomography (OCT), which provides low-cost and high-resolution retinal images that depict fine physiological details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Several OCT biomarkers have been tentatively linked to the pathogenesis of AMD, such as reti- nal layer thickness, photoreceptor atrophy and the presence of drusen, which are lipidic deposits that build up inside the retina [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' However, the exact relation of these biomarkers to the progression from early to late stages of AMD remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Current grading systems only coarsely group patients into broad cate- gories for early, intermediate and late AMD and have limited prognostic value [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' There is an unmet need for biomarkers that describe and predict the pro- gression of AMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Large studies that image populations of AMD patients over time are a power- ful resource to discover novel biomarkers for disease progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' If candidate biomarkers are already identified a priori, they can be extracted from images and mapped against disease progression [16,19,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' However, this approach is not feasible if the biomarkers are partially or fully unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Self-supervised learn- ing is the most promising method to automatically discover new biomarkers by clustering or finding anomalous OCT images in lower-dimensional representation space [18,21,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' However, by grouping single scans acquired at single points in time, these biomarkers are by definition static and cannot capture concepts such as the speed of disease progression or transitions between multiple states of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Clustering whole time series of images is also problematic, as patients enter and leave the study at different points in their overall progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' There- fore, in order to discover biomarkers that describe population-level patterns in disease progression, it is necessary to analyse and compare portions of patient time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Our contribution In this work, we develop a strategy to automatically discover biomarkers that capture disease progression in time series of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' This is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Our method represents the time series of images as a trajectory in a self-supervised latent feature space built by contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' This representation allows a new partitioning of time series of OCT images into consecutive subsequences (sub-trajectories) that exhibit distinct units of disease progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' By clustering these sub-trajectories, we are now able to detect patterns of disease progression that are common among the population of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Experimentally, we test our method on a large cohort from two, longitudinal retinal OCT datasets totalling 160,558 images from 7,912 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In doing so we categorise 3,218 total years of disease progression with time-dependent clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We then correlate our clusters with the known set of biomarkers which reinforces Clustering disease trajectories for biomarker discovery in AMD 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 1: We analyse disease progression as clustered trajectories in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We first train a feature extractor with contrastive learning and then form tra- jectories by projecting time series of images to feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' These are then partitioned into a set of sub-trajectories which are clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Finally, clusters are related to known disease stages before being interpreted as candidate biomarkers for disease state and progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' PCA space is coloured by visual acuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' their potential to contain potentially new, time-dependent biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Finally, we closed the loop on automated biomarker discovery by working directly with four ophthalmologists to interpret these clusters in a clinical comparison task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Contrastive pretraining Contrastive PCA feature Retinal OCT datasets (170k scans) loss space ResNet-50 CNN 00 0 00 0 Copy pretrained feature extractor 0 0 0 PCA reduction 00 ResNet-50 00 CNN 00 00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Patient trajectories in feature space Resample Longitudinal information with temporal denoising kernel Eye Scan times 04/2017 08/2017 11/2017 2 07/2012 12/2012 04/2013 3 06/2019 09/2020 10/2020 01/2015 03/2016 04/2016 5 08/2017 09/2017 02/2018 6 09/2012 03/2013 07/2013 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Clustering and proposing biomarkers Cluster 1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Cluster 2 Series A K-means Query series Count biomarkers clustering Known biomarkers Series B per cluster Healthy Drusen cRORA CNV 0 Clusters Experts compare clusterst with similar known biomarkers 2 but different latent features tclusters are candidates for time-dependent biomarkers4 Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 2 Related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='1 Current AMD grading systems Ophthalmologists’ current understanding of progression from early to late AMD largely involves drusen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Drusen can grow until suddenly regressing and disap- pearing, which often precedes the onset of late AMD [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' While there have been attempts to group drusen based on their morphology [11], current grading systems stratify early and intermediate stages solely by drusen size [2,12,7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Late AMD is classified as either choroidal neovascularisation (CNV), identified by fluid under the retina, or geographic atrophy by progressive loss of photore- ceptors and retinal thinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' The degree of atrophy can be staged using cRORA (complete retinal pigment epithelium and outer retinal atrophy), which measures the width of focal atrophy in OCT [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' So far grading systems offer limited di- agnostic value and little to no prognostic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='2 Tracking evolution of known biomarkers Few research efforts have aimed at quantifying and tracking known AMD biomark- ers over time, such as reticular pseudodrusen [19] and drusen volume [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' More work has explored Alzheimer’s disease (AD), which offers a greater array of quantitative imaging biomarkers, such as levels of tau protein and hippocampal volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' [23] fit an event-based model that rediscovers the order in which these biomarkers become anomalous as AD progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Vogel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' [20] find four distinct spatiotemporal trajectories for tau pathology in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' However, mapping biomarkers that evolve during disease progression requires prior annotation of entire time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Thus, these biomarkers must be known or at least suspected a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='3 Automated discovery of unknown biomarkers In order to discover new biomarkers, efforts to find them have turned to au- tomated biomarker discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Imaging biomarkers are proposed by analysing anomalous scans [17], clusters of scans [25], or a combination of these [18] in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' To build these, neural networks are trained with supervised or unsupervised proxy tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' These tasks include image reconstruction [21], seg- mentation [25] and generative adversarial networks [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' However, networks are prone to overfit on their specific task and lose semantic information regarding the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Contrastive learning has recently advanced the state-of-the-art in training generalisable and unbiased feature extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' popularised this paradigm with SimCLR [4], which was later improved on by Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' [9] in Bootstrap Your Own Latent (BYOL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Contrastive methods encode invariance to a set of transformations typically uncorrelated with disease features, including rotation, translation and global shifts in image brightness and contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' leverage contrastive feature spaces to identify high-risk clusters of CT image patches [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Clustering disease trajectories for biomarker discovery in AMD 5 However, all biomarkers discovered by the aforementioned methods work by grouping single images acquired at single points in time, and in doing so neglect temporal relationships between images of the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' One work that tack- les this challenge, and the most related to ours, categorises the time-dependent response of cancer cells to drugs, measured by the changing distance in con- trastive feature space from healthy controls [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='4 Trajectory clustering The separate field of trajectory clustering is largely focussed on discovering move- ment patterns taken by cars, animals and hurricanes [6,1,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=', in their state-of-the-art work TRACLUS [13], assume these trajectories are composed of consecutive series of common sub-trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' For example, different car jour- neys may at some point travel down the same road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Using this principle, they develop a partition-and-group framework to cluster segments that are repeated across multiple trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Similarly, we assume that disease progression can be divided into multiple, common disease pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Firstly, this allows us to work seamlessly with temporally unaligned scanning series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Secondly, we can automatically discover novel disease pathways by interpreting sub-trajectories that are shared by multiple AMD patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 3 Materials and Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='1 Self-supervised feature space using contrastive learning We adapt BYOL [9] with update coefficient τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='9995 for contrastive pretrain- ing of a ResNet50 (4x) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' As several of the contrastive transformations designed for natural images are inapplicable to medical images, we use the set tailored for retinal OCT images by Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Models were trained on the entire dataset for 120,000 steps with momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='9 and a learning rate of 5 · 10−4 using the Adam optimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' After pretraining, we first remove any final linear layers before projecting all labelled images to the feature space of 2048 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We then reduce the di- mension further using principle component analysis (PCA) with D components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Using PCA allows us to interpolate the feature space and results in fewer di- mensions, which is advantageous for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' To validate that the contrastive feature space encodes meaningful information for AMD biomarkers, and find the optimal dimension D ∈ {2, 10, 20, 50}, we perform multi-class classification of known biomarkers (including healthy controls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Firstly, we split the dataset into train and test partitions using 85% and 15% of the data, respectively, en- suring that all scans from each patient belong to the same set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Then to perform the classification, we fit a class-balanced support vector machine (SVM) on the training set and report performance on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 6 Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 2: Longitudinal scans of a single eye, imaged over four years, projected to PCA space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' PCA space is plotted as a hexmap coloured by the local average in visual acuity, where higher values indicate poorer quality of vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Each row depicts two principle components up to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' The rightmost columns show resampled trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Using smaller T captures short-term variation in disease progression but cannot model long-term changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' MDL aims to optimise this tradeoff by partitioning trajectories only at points of inflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='2 Extracting and clustering common sub-trajectories For each eye, we first form piecewise-linear trajectories by linking points in PCA space that were derived from consecutively acquired OCT images (see left column in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We then assume, in analogy to TRACLUS [13], that trajectories encoding disease progression can be partitioned into sub-trajectories that are common among multiple patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We compare two methods to achieve this, shown in the right-most columns in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' The first method resamples time- points at regular intervals of T years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' For each resampled time t, we find the corresponding point in feature space by taking a weighted average of all points in the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' The weights are calculated by convolution of a Gaussian kernel N(t, σT ) with the acquisition dates of the entire scanning series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We then define Resampled trajectories (T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='5 years) T= T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='0 T= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='0 T= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='Original ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='Cumulative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='MDL Partition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='(years) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='trajectory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='(years) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='(years) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='PCA #19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='PCA #19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='PCA #19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='Colour map for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='visual acuity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='8Clustering disease trajectories for biomarker discovery in AMD 7 sub-trajectories as vectors between consecutive points that are less than or equal to T years apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' The second method aims to describe trajectories using the fewest points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' It be- gins by resampling trajectories using the first method (with intervals of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='5 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Then, using the minimum description length (MDL) principle, a mini- mal subset of points are chosen that best preserve changes in disease over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' To achieve this we find the trajectory H, containing a subset of the points in the resampled trajectory O, which minimises the following objective (using the greedy solution from [13]) L(O|H) = � p∈O d⊥(p, H) − λ|H| where d⊥(p, H) is the perpendicular distance from a point p to the piece-wise linear trajectory H, and |H| is the number of points in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' The coefficient λ is proportional to the total standard deviation in the feature space explained by D PCA dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Clustering sub-trajectories To cluster common sub-trajectories we require a distance function that measures the similarity between vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Given two sub-trajectories U = (ustart, uend) and V = (vstart, vend) their distance is simply d(U, V ) = ∥ustart − vstart∥2 + ∥uend − vend∥2 Finally, using d we separate sub-trajectories into K clusters using k-means clus- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='3 Finding optimal hyperparameters using the set of known biomarkers We now search for optimal values for the sub-trajectory time interval T ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='0} years, resampling kernel width σT ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='0} years and the number of clusters K ∈ {5, 10, 15, 30} using five random seeds for k-means clus- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' To quantitatively compare configurations, we use the conditional entropy H(B|C) = H(B, C) − H(C) as a scalar measure of how well the clusters C redis- cover the known biomarkers B detailed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' This is calculated directly from their joint distribution P(B, C), which is found by counting all biomarkers recorded within sub-trajectories of each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' To ensure the equal contribution of all biomarkers, we reweight their marginal distribution P(B) to be uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' As the number of clusters K increases even randomly permuted as- signments p(C) will result in reduced values of H(B|p(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We address this by using the adjusted reduction in conditional entropy, H′ (using r = 5 random trials), where higher values correspond to better rediscovery of B H′(B|C) = 1 r r � i H(B|p(C)) − H(B|C) As our ultimate goal is to detect biomarkers beyond the known set, we use H′ only as an indication for the most suitable configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 8 Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='4 Proposing clusters as candidate biomarkers We first relabel the clusters C in order of median visual acuity, so that higher cluster numbers indicate poorer quality of vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In order to see which disease stages each cluster describes, we calculate the conditional probability P(B|C) = P(B, C)/P(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Then, to discover new biomarkers beyond the existing set, we compare clusters that differ maximally in feature space but minimally in the set of known biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' To find these, we explore eleven pairs of distinct clusters Ci and Cj with a high degree of cosine similarity P(B|Ci) · P(B|Cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Interpreting candidate biomarkers To examine clusters for candidate biomark- ers, we collaborate with four expert ophthalmologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' For each pair of clusters, we generate four random ‘A or B’ single-choice questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Clinicians are shown one query series in image space from cluster Ci and two further series denoted A and B, one from Ci and the other from Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' For each question, clinicians are tasked with determining which of A or B belongs to the same cluster as the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' After completing four questions they are prompted to explain on what basis they matched A or B to the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' This format allows us to both assess whether the clusters are visually distinguishable by experts and, if so, potentially extract descriptions of novel biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='5 OCT datasets We apply our method to two independent retinal OCT datasets called Dataset A and Dataset B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We developed our method on Dataset A but run experiments on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In both, images were acquired using Topcon 3D OCT devices (Topcon Corporation, Tokyo, Japan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' After strict quality control, Dataset A consists of 46,496 scans of 6,236 eyes from 3,456 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Eyes were scanned 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='7 times over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='9 years on average at irregular time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' The second dataset, Dataset B, is larger, containing 114,062 scans of 7,253 eyes from 3,819 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Eyes were scanned 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='6 times over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='5 years on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Of each 3D OCT vol- ume, we extracted the transverse 2D slice centred at the fovea and resampled to 208×256 pixels with a pixel size of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='0×23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='4 µm2, half the median resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Each scan is labelled with visual acuity, a functional measure assessing the qual- ity of vision measured in LogMAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' To record conversions to a comprehensive set of known biomarkers B, we used established AMD grading protocols described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Early AMD is char- acterised by small drusen between 63-125µm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We also recorded CNV, cRORA (of at least 250µm but smaller than 1000 µm) and cRORA (of at least 1000 µm) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Overall, 861 conversion times t0 were recorded, and any sub- sequent visits at times t+ before the next conversion were automatically as- signed with a separate label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Visits prior to any biomarker were labelled as ‘NoBiomarker’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Finally, in each dataset, an additional 150 healthy images that exhibit no pathology were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Combining these, the known set of biomark- ers B includes 10 biomarkers and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Clustering disease trajectories for biomarker discovery in AMD 9 4 Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 3: Confusion matrices for multi-class classification of known biomarkers and healthy images using D numbers of PCA dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In general, performance increases with the number of dimensions D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We find that using D = 20 PCA dimensions achieves linear separability between known biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='1 Finding the optimal set of hyperparameters using the known biomarkers In both Dataset A and Dataset B 20 principal dimensions achieves linear sepa- rability between known biomarkers (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Both the healthy stage and the only extractable early biomarker, drusen, were found to be highly linearly separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Thus, we use D = 20 for the remainder of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We find that K = 15 clusters of sub-trajectories best explained the set of known biomarkers B (Figure 4a) as measured by higher values of H′(B|C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Greater values of T, in addition to MDL partitioning, result in decreased H′(B|C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We suspect that this is because the known set of biomarkers describe disease states rather than state transitions, so they are better captured by shorter sub- trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In order to select a configuration that finds clusters evidencing pro- gression, we choose T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='0, σT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='5 and K = 15 for the remainder of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='2 Sub-trajectory clusters go beyond the known set of biomarkers We find that our clusters encode the set of known biomarkers in Dataset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' As seen in Figure 5, clusters effectively separate healthy, early stage and late-stage Dataset A 10D 20D 50D 2D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='05 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='12 Drusen 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='01 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='00 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='00 5 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 (to) (to) CNV (to) (to) Healthy μm) Drusen NoBiomarker CRORA (250 (1000 | (1000μm) RAClustering disease trajectories for biomarker discovery in AMD 11 biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We find two clusters (1-2) describing drusen, five (4-8) describing the transition from drusen to cRORA (250 µm) and three (12-14) describing cRORA (1000 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Most healthy samples were in cluster 3, and CNV was spread amongst most late-stage clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In Dataset B we find three clusters of healthy looking scans, three of early stages and six in the late stage that capture atrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Interpreting candidate biomarkers We now report the results of the single- choice clinical clustering task (described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content='4), comparing cluster pairs which are highlighted in blue and pink boxes in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In Dataset A, clinicians were easily able to separate clusters 1 and 15 by using known biomarkers, such as hypertransmission and photoreceptor degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' The result of this validation experiment was expected, as these clusters were already highly separable under the known set of biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' More interestingly, clinicians were able to exactly differentiate between early AMD clusters 1 and 2 in Dataset A despite their high similarity in known biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' When prompted, all clinicians cite differences in drusen, with two finding differences in the number of small drusen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Their ability to distinguish some of the pairs was mixed, as they sometimes found no consistent or visible differences or had a low inter-rater agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 5 Discussion and Conclusion In this paper, we proposed a method to automatically discover time-dependent biomarkers that detect periods of disease progression common among groups of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' By partitioning entire time series into representative sub-trajectories, and then clustering them, we categorised 3,218 total years of disease progression across two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We showed that these clusters rediscovered the established set of OCT biomarkers for AMD, which reinforced the use of our clusters as can- didate biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Then, by working directly with ophthalmologists, we closed the loop in our automated biomarker discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' To this end ophthalmologists compared clusters that were indistinguishable using current grading systems, yet were separable in contrastive feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' We envision that further inves- tigation into sub-trajectory clusters could advance understanding of how AMD progresses, and potentially lead to grading systems with greater prognostic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Our method is applicable to any dataset studying any disease with time series of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' While our method identified two clusters that described drusen in Dataset A and three that described healthy-looking scans in Dataset B, most clusters were associated with intermediate and late-stage AMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' This is due to the overrepresentation of patients with late disease in our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In or- der to find more clusters categorising progression in early AMD, we aim to repeat this analysis in datasets that begin imaging patients earlier in their over- all progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Moreover, due to the slow progression of AMD, a large number of sub-trajectories captured unchanging disease states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' In order to isolate sub- trajectories capturing the periods of greatest disease progression, we intend to increase the number of clusters K and interpret those that convert to late disease the fastest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' 12 Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Conclusion Inspired by inadequate grading systems for disease progression in early AMD, we proposed the first method to analyse disease progression as clustered trajectories in self-supervised feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' By correlating our clusters with known OCT biomarkers for AMD, we reinforced their potential as time- dependent biomarkers for disease progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' After this, we closed the loop on automated biomarker discovery by working directly with ophthalmologists to 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Pathological cluster identification by unsupervised analysis in 3,822 uk biobank cardiac mris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' cardiovasc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} +page_content=' med 7, 539788 (2020)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E3T4oBgHgl3EQfcgoK/content/2301.04525v1.pdf'} diff --git a/Z9E5T4oBgHgl3EQfDg58/content/tmp_files/2301.05406v1.pdf.txt b/Z9E5T4oBgHgl3EQfDg58/content/tmp_files/2301.05406v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9646c9eca41132261a9893ee36dd96044db4012f --- /dev/null +++ b/Z9E5T4oBgHgl3EQfDg58/content/tmp_files/2301.05406v1.pdf.txt @@ -0,0 +1,1689 @@ +Non-Gaussianity in the cosmic microwave background from loop quantum +cosmology +Roshna K∗ and V. Sreenath† +Department of Physics, National Institute of Technology Karnataka, Surathkal, Mangaluru 575025, India. +Primordial non-Gaussianity has set strong constraints on models of the early universe. +Studies have shown that Loop Quantum Cosmology (LQC), which is an attempt to extend +inflationary scenario to planck scales, leads to a strongly scale dependent and oscillatory non- +Gaussianity. In particular, the non-Gaussianity function fNL(k1, k2, k3) generated in LQC, +though similar to that generated during slow roll inflation at small scales, is highly scale +dependent and oscillatory at large wavelengths. In this work, we investigate the imprints of +such a primordial bispectrum in the bispectrum of Cosmic Microwave Background (CMB). +Inspired by earlier works, we propose an analytical template for the primordial bispectrum +in LQC and compute the corresponding reduced bispectra of temperature and electric po- +larisation and their three-point cross-correlations. We show that CMB bispectra generated +in LQC is consistent with the observations from Planck. We conclude with a discussion of +our results and its implications to LQC. +I. +INTRODUCTION +Numerous theoretical insights together with several observational efforts, spanned over a cen- +tury, have enabled us to arrive at a compelling model of our Universe referred to as the standard +model or the Lambda Cold Dark Matter (ΛCDM) model [1]. According to this model, the seeds +of the current distribution of galaxies spread over the fabric of spacetime known as the large scale +structure were sown during the earliest phase of the universe. Tiny perturbations generated in the +early universe lead to tiny anisotropies in the Cosmic Microwave Background (CMB) which in turn +lead to the inhomegeneous large scale distribution of galaxies that we see today. Though we have +a good level of understanding of this evolution, several details are yet to be worked out. One such +detail concerns the origin of these perturbations in our Universe. +Inflation, see, for instance, [2–5], due to its simplicity, provides the most popular explanation +for the origin of these perturbations [6, 7] (For a discussion on alternate views, see [8, 9].). In +inflationary scenario, quantum fluctuations in the inflaton leads to the primordial perturbations. +Appealing to the nearly de Sitter symmetry of the spacetime during inflation, we assume that at +a time when the perturbations are sufficiently sub-horizon, quantum perturbations are generated +in the Bunch-Davies vacuum. Such a prescription has been highly successful, in that, it leads to +primordial perturbations that are nearly Gaussian and scale invariant as demanded by observations +[1, 6, 10]. Even though inflation is successful, it is still an incomplete theory. We do not take in to +account the evolution of perturbations before the time at which the initial conditions are imposed. +In fact, inflation does not account for the physics in the planck regime close to the big bang +singularity. There have been several attempts to address these issues. In this work, we will concern +ourselves with loop quantum cosmology (LQC) [11–15]. +Loop quantum cosmology is an attempt to extend inflationary scenario to the planck regime +using principles of loop quantum gravity [13–19]. In LQC, quantum gravitational effects in the +planck regime leads to a quantum bounce [11, 12]. Thus in LQC, a quantum bounce precedes +the inflationary phase. Generation and evolution of perturbations in LQC have been extensively +∗ roshnak.217ph005@nitk.edu.in +† sreenath@nitk.edu.in +arXiv:2301.05406v1 [astro-ph.CO] 13 Jan 2023 + +2 +studied at the level of primordial power spectra [20–46] and primordial non-Gaussianity [47–50]. In +general, studies show that the effect of the bounce is to introduce an additional scale corresponding +to the curvature at the bounce. Modes of perturbations which have comparable length to this new +scale gets modified leading to a highly scale dependent power spectrum. At smaller wavelengths, the +perturbations are not affected by the bounce and the power spectrum is nearly scale invariant as in +slow roll inflation [29]. Perturbations show a similar behaviour at second order in perturbations [47, +49]. Studies show that primordial non-Gaussianity quantified using the function fNL(k1, k2, k3), +at scales comparable to the curvature at the bounce, is strongly scale dependent and oscillatory +with a very large amplitude. At smaller scales, the fNL(k1, k2, k3) is similar to that in slow roll. +Studies also show that the bispectrum is more sensitive to the bounce than the power spectrum. +Assuming sixty or so e-folds of inflation, the scale at which the imprints of the bounce, on +primordial perturbations, occur depends on the amount of expansion between the bounce and the +onset of inflation. Observational constraints from the CMB temperature power spectrum demand +that any departure from scale invariance should happen only at multipoles of ℓ ≲ 30 [1, 51]. If we +assume that, the effects of primordial power spectrum on the CMB is observable at ℓ ≲ 30, then, +since the bispectrum is more sensitive to the effects of the bounce than the power spectrum [47, 49], +there is a possibility that the imprints of large, scale dependent and oscillatory primordial non- +Gaussianity is observable at larger multipoles. Hence it is important to investigate the consistency +of LQC with observations by Planck. With this motivation, in this work, we compute the imprints +of such a non-Gaussianity in the temperature (T) and electric polarisation (E) of the CMB. We +assume an analytical template for primordial non-Gaussianity generated in LQC, compute the +⟨TTT⟩, ⟨TTE⟩, ⟨TEE⟩ and ⟨EEE⟩ correlations and show that they are similar to those generated +in slow roll inflation and hence is consistent with observations by Planck. +The rest of the paper is organised as follows. In the next section, we briefly introduce the essen- +tials of LQC and present analytical templates for the primordial power spectrum and bispectrum. +In section III, we discuss the essential formulae to compute the three-point correlation functions +of anisotropies in temperature and electric polarisation. In section IV, we apply these formulae to +LQC. We present the numerical techniques and our calculation of reduced bispectra of tempera- +ture fluctuations and electric polarisation and their three-point cross-correlations in section V. We +conclude the paper with a summary and discussion of our results and their consequences to LQC +in section VI. +II. +LOOP QUANTUM COSMOLOGY +In this section, we will discuss the essentials of LQC that is relevant to this paper (for reviews, +see, for instance, [13–15]). In particular, we will discuss LQC as applied to FLRW geometries +sourced by a scalar field φ and scalar perturbations δφ(⃗x) living on this background. +A. +Background +In LQC, FLRW background geometry is described by a wavefunction ΨFLRW(v, φ), which satis- +fies the equation ˆHFLRWΨFLRW(v, φ) = 0, where ˆHFLRW is the Hamiltonian operator corresponding +to the classical background Hamiltonian and v is the volume factor which is proportional to the +cube of scale factor a. Numerical investigations of such a system has shown that the scale factor +undergoes a bounce [11, 12, 52, 53]. It turns out, if the wave function is sharply peaked over the +values of scale factor, the behaviour of scale factor can be described by certain effective equations + +3 +−104 +−102 +0 +102 +104 +106 +t (TPl) +103 +108 +1013 +1018 +1023 +1028 +1033 +a(t) +Inflation +−10 +−5 +0 +5 +10 +0 +2 +4 +6 +8 +100 +101 +102 +103 +104 +105 +106 +107 +t (TPl) +−5 +0 +5 +10 +15 +φ(t) +FIG. 1. +Figure illustrates the behaviour of scale factor (left) and scalar field(right) in LQC. As mentioned +in the text, scale factor undergoes a bounce preceding inflation. Scalar field starts rolling up the potential +until its kinetic energy becomes zero and then starts slowly rolling down the potential leading to inflation. +In making this plot, we have worked with the mass of scalar field to be consistent with the constraints on +the amplitude of the primordial power spectrum and with ρsup = 0.41m4 +Pl. +[11, 12, 30, 52, 53], namely +� ˙a +a +�2 += κ +3ρ +� +1 − +ρ +ρsup +� +, +¨a +a = −κ +6 ρ +� +1 − 4 +ρ +ρsup +� +− κ +2 P +� +1 − 2 +ρ +ρsup +� +, +(2.1) +where ρ, P are the energy density and pressure of the scalar field and κ = 8 π G. From the above +expression, it is clear that at ρ = ρsup, Hubble parameter H = ˙a/a = 0 and ¨a/a > 0 i.e. +scale +factor is at minimum. In other words, the universe undergoes a bounce at ρ = ρsup. Further, if +we assume that the scalar field is governed by a potential V (φ), then the evolution of scalar field +is given by +¨φ + 3 H ˙φ + Vφ = 0, +(2.2) +where Vφ = dV/dφ. For a suitable potential, inflationary phase will set in after the bounce [34, 54– +58]. The background dynamics in LQC with a scalar field governed by a quadratic potential is +illustrated in Figure 1. +B. +Perturbations +We will follow dressed metric approach to describe primordial perturbations in LQC [22–24, 29, +47, 49]. In this approach, we assume that the wavefunction takes the form Ψ = ΨFLRW(v, φ) ⊗ +δΨ(v, φ, δφ), which satisfies the equation ˆHΨ = 0, where ˆH = ˆHFLRW + ˆHpert. As mentioned +earlier, ΨFLRW(v, φ) satisfies the equation ˆHFLRWΨFLRW(v, φ) = 0. Perturbations are treated as +test fields living on the background FLRW geometries described by ΨFLRW(v, φ). In practice, this +implies that perturbations can be evolved using the classical Hamiltonian but with the background +functions in them described by the effective equations. This is similar to perturbations living as test +fields on a curved space time described by a ‘dressed’ metric which satisfies the effective equations. + +4 +In order to compute primordial bispectrum, we need to consider Hamiltonian up to third order +in perturbations, i.e. +we need Hpert = H(2) + H(3). There are two approaches to arrive at the +Hamiltonian describing perturbations, one can either use gauge invariant variables or rather work +with a fixed gauge. We follow the latter approach. In particular, we will work with spatially flat +gauge [47, 59]. +The second order Hamiltonian describing perturbations δφ in the spatially flat gauge is +H(2) = +� +d3x N S(2)(⃗x) = N 1 +2 +� +d3x +� 1 +a3 δpφ2 + a3 (∂δφ)2 + a3 U δφ2 +� +, +(2.3) +with the potential U given by +U = −9 +p4 +φ +a8π2a ++ 3 +2κ +p2 +φ +a6 − 6 pφ +a πa +Vφ + Vφφ + 6 pφ ˙pφ +a4 πa +− 3 +p2 +φ ˙πa +a4 π2a +− 3 +˙a p2 +φ +a5 πa +. +(2.4) +In the above expressions, πa, pφ and δpφ are momenta conjugate to a, φ and δφ respectively. Setting +lapse N = 1 will imply cosmic time and N = a corresponds to conformal time. Hamiltonian at +third order in perturbations is +H(3) = N +� +d3x +�� +9 κ p3 +φ +4 a4 πa +− +27 p5 +φ +2 a6π3a +− 3 a2 pφ Vφφ +2 πa ++ a3 Vφφφ +6 +� +δφ3 +− +3 pφ +2 a4 πa +δp2 +φ δφ − +9 p3 +φ +a5π2a +δpφδφ2 − 3 a2 pφ +2 πa +δφ (⃗∂δφ)2 + +3 p2 +φ +N a πa +δφ2∂2χ + 3 +2 +a2 pφ +N2 κ πa +δφ ∂2χ ∂2χ ++ 3 +p2 +φ +N a πa +δφ ∂iχ∂iδφ + 1 +N δpφ ∂iδφ ∂iχ − 3 +2 +a2 pφ +N2 κ πa +δφ ∂i∂jχ ∂i∂jχ +� +, +(2.5) +where ∂2χ = (−3 N κ/a) +�� +pφ +2 − a5 Vφ +κ πa +� +δφ − +pφ +κ a πa δpφ +� +. +From the second order Hamiltonian H(2), one can derive the free evolution of the scalar pertur- +bation, given by, +(□ − U(t)) δφ(⃗x, t) = 0, +(2.6) +where □ is the d’Alembertian of the FLRW background metric. The third order Hamiltonian H(3) +provides the self-interaction of the scalar perturbations. +The perturbations, since they evolve through the bounce and then through the inflationary +phase, carry signatures of the early universe which they imprint on the CMB. Perturbations are +quantified using correlation functions. +In order to compute correlation functions, one need to +promote δφ to an operator ˆδφ. The field operator ˆδφ is then expanded in terms of annihilation +and creation operators as +ˆδφ(⃗x, η) = +� +d3k +(2π)3 ˆδφ⃗k(η) ei⃗k·⃗x = +� +d3k +(2π)3 +� +ˆA⃗k ϕk(η) + ˆA† +−⃗k ϕ∗ +k(η) +� +ei⃗k·⃗x +(2.7) +where [ ˆA⃗k, ˆA† +⃗k′] = ℏ (2π)3 δ(3)(⃗k + ⃗k′), [ ˆA⃗k, ˆA⃗k′] = 0 = [ ˆA† +⃗k, ˆA† +⃗k′]. The mode functions ϕk(η) satisfy +the equation +ϕ′′ +k + 2a′ +a ϕ′ +k + (k2 + a2 U) ϕk = 0 , +(2.8) + +5 +where k2 ≡ kikj δij is the comoving wavenumber, and prime indicates derivative with respect to +conformal time. The scalar power spectrum of ˆδφ is a dimensionless function that quantifies the +two-point correlation in momentum space via +⟨0| ˆδφ⃗k(η) ˆδφ⃗k′(η)|0⟩ ≡ (2π)3δ(3)(⃗k + ⃗k′)2π2 +k3 Pδφ(k, η) , +(2.9) +where |0⟩ is the vacuum annihilated by the operators ˆA⃗k for all ⃗k. Power spectrum, in terms of +mode functions, is Pδφ(k, η) = (ℏ k3/2π2) |ϕk(η)|2. +The three-point function of ˆδφ at tree level is given by [47, 59] +⟨ ˆδφ⃗k1(η) ˆδφ⃗k2(η) ˆδφ⃗k3(η) ⟩ = − i/ℏ +� +dη′⟨0| +� +ˆδφ +I +⃗k1(η) ˆδφ +I +⃗k2(η) ˆδφ +I +⃗k3(η), ˆHI +int(η′) +� +|0⟩ + O(H2 +int), +(2.10) +where ˆHI +int(η) is the operator corresponding to H(3) in the interaction picture. +Even though we worked in spatially flat gauge, it is convenient to compute correlation functions +in terms of curvature perturbations R. This is because, curvature perturbations have a unique +property that they stop evolving after they cross the horizon and remain constant till they re-enter +horizon towards late radiation domination or during early matter domination epoch, saving us a +lot of computational time. Curvature perturbations are related to perturbations in scalar field +through the relation [47, 59] +R(⃗x, η) = −a +z δφ(⃗x, η) + +� +−3 +2 + 3 Vφ a5 +κ Pφ πa ++ κ +4 +z2 +a2 +� �a +z δφ(⃗x, η) +�2 ++ · · · , +(2.11) +where trailing dots indicates terms that leads to subdominant terms in the three-point functions +when evaluated towards the end of inflation. +The power spectrum of curvature perturbation is related to that of scalar modes ˆδφ⃗k(η) through +the relation +PR(k) = +�a(ηend) +z(ηend) +�2 +Pδφ(k, η), +(2.12) +where z = −6 pφ/(κ πa). +The three-point function of curvature perturbation can be obtained in terms of ˆδφ⃗k(η) by using +Eq. (2.11) as +⟨0| ˆR⃗k1 ˆR⃗k2 ˆR⃗k3|0⟩ = +� +−a +z +�3 +⟨0| ˆδφ⃗k1 ˆδφ⃗k2 ˆδφ⃗k3|0⟩ ++ +� +−3 +2 + 3 Vφ a5 +κ pφ πa ++ κ +4 +z2 +a2 +� � +−a +z +�4 � � +d3p +(2π)3 ⟨0| ˆδφ⃗k1 ˆδφ⃗k2 ˆδφ⃗p ˆδφ⃗k3−⃗p|0⟩ + (⃗k1 ↔ ⃗k3) + (⃗k2 ↔ ⃗k3) ++ · · · +� +. +(2.13) +The wave numbers of three modes in the three-point function are constrained by a Dirac delta +function. We define the scalar bispectrum as the three-point function sans Dirac delta function as +⟨0| ˆR⃗k1 ˆR⃗k2 ˆR⃗k3|0⟩ ≡ (2π)3δ(3)(⃗k1 + ⃗k2 + ⃗k3) BR(k1, k2, k3) . +(2.14) +The amplitude of bispectrum can be quantified using a dimensionless function fNL(k1, k2 , k3), akin +to the dimensionless power spectrum PR(k) that quantifies two-point correlations, as +fNL(k1, k2, k3) ≡ −5 +6BR(k1, k2, k3) × (∆k1∆k2 + ∆k1∆k3 + ∆k2∆k3)−1 +(2.15) +where ∆k ≡ 2 π2 +k3 PR(k). + +6 +−104 −103 −102 −101 +−1000 100 +101 +102 +103 +104 +105 +106 +t (TPl) +10−5 +10−3 +10−1 +101 +103 +105 +� +|Ω(η)| (MPl) +k⋆ +kLQC +kI +FIG. 2. +The figure represents the relevant scales in LQC. kLQC is the scale corresponding to the value of +curvature at the bounce. kI corresponds to the smallest scale that is sub-horizon during inflation. As can +be seen, only modes kLQC ≳ k > kI are excited during the bounce and hence are scale dependent. Modes +with larger wavenumbers are excited only during horizon crossing towards the end of inflation and hence +will be scale invariant. +C. +Templates of scalar power spectrum and bispectrum +In order to understand the evolution of perturbations in LQC, let us rewrite Eqn. (2.8), +v′′ +k + +� +k2 + Ω(η) +� +vk = 0, +(2.16) +where vk = a ϕk is the Mukhanov-Sasaki variable and Ω(η) = a2 U − +a′′ +a . +We compare the +behaviour of +� +|Ω(η)| as a function of time with relevant wavenumbers in figure 2. As shown in the +figure, all observationally relevant wavenumbers are adiabatic much before the bounce and hence +we can impose adiabatic initial conditions. From the figure, it is also clear that there are two +relevant scales in the problem. The value of curvature at the bounce defines a scale kLQC and the +value of curvature at the onset of inflation defines a scale kI. Wavenumbers which are much larger +than kLQC, are not effected by the bounce and they will be in Bunch-Davies vacuum at the onset +of inflation. This implies that power spectrum of modes k >> kLQC will be nearly scale invariant +as in slow roll inflation. Modes which are comparable to kLQC and larger than kI will be excited +both during the bounce as well as during the horizon exit during inflation. These modes are in +excited non-Gaussian states during the onset of inflation and hence they will be further amplified +as they exit the horizon during inflation. Hence, the modes kI < k < kLQC will be strongly scale +dependent. Modes whose wavenumbers are smaller than kI are always superhorizon and hence +they are never excited. The primordial power spectrum and bispectrum are evaluated towards the +end of inflation when all the relevant modes are well outside the horizon. +The primordial power spectrum and bispectrum can be calculated numerically. +Given the +background dynamics described in Eqns. (2.1, 2.2), the evolution of perturbations are found by + +7 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +k +� +Mpc−1� +10−10 +10−9 +10−8 +10−7 +10−6 +PR(k) +kLQC +k⋆ +analytical result +numerical result +10−5 +10−4 +10−3 +10−2 +k +� +Mpc−1� +−106 +−105 +−104 +−103 +−102 +−101 +−1000 +100 +101 +102 +103 +104 +105 +106 +fNL(k, k, k) +kLQC +k⋆ +analytical result +numerical result +FIG. 3. +The primordial power spectrum and the non-Gaussianity function generated in LQC obtained +numerically (in black). Analytical templates for power spectrum and non-Gaussianity given in Eqns. (2.17) +and (2.19) (in grey). +solving Eqn. (2.8). The power spectrum of curvature perturbation can then be calculated using +Eqn. (2.12). Calculation of ⟨ ˆδφ⃗k1(η) ˆδφ⃗k2(η) ˆδφ⃗k3(η) ⟩ requires one to perform integrals in Eqn. +(2.10). The ⟨0| ˆR⃗k1 ˆR⃗k2 ˆR⃗k3|0⟩ three-point function of curvature perturbation can then be calculated +using Eqn. (2.13). The dimensionless non-Gaussianity function of curvature perturbation is then +calculated by using Eqn. (2.15). This numerical calculation of primordial power spectrum and +non-Gaussianity has been implemented in class_lqc [47, 49]. We present the results obtained +using that code in figure 3. +For calculating the three-point functions involving temperature and electric polarisation, one +needs to convolve the primordial bispectrum with the CMB transfer functions. For performing +this calculation, it is convenient to have analytical templates of primordial power spectrum and +bispectrum. Following [46, 60, 61], we will use the following template for describing the power +spectrum. It is given by +PR(k) = As +� +� +� +� +� +� +� +� +� +( k +kI )2( +kI +kLQC )q +if k ≤ kI, +( +k +kLQC )q +if kI < k ≤ kLQC, +( +k +kLQC )(ns−1) if k > kLQC, +(2.17) +where we work with q = −0.7, kI = 5 × 10−5 k⋆, kLQC = 0.1 k⋆ and k⋆ = 0.002Mpc−1 represents +the pivot scale. The amplitude of power spectrum As and the spectral index ns have been set to +their values obtained by Planck. The analytical template for power spectra is drawn along with +the exact numerical calculation in figure 3. +As is evident from figure 3, the primordial non-Gaussianity fNL(k1, k2, k3) is scale dependent +and oscillatory. The exponential decay in the value of fNL(k1, k2, k3) as k ≈ kLQC was explained +in [47, 49] by analysing the poles of the integrand in Eqn. (2.10). In particular, by analysing the +pole of scale factor around the bounce, the analytical behaviour of fNL sans oscillations was found +to be +fNL(k1, k2, k3) ∝ e +−α k1 + k2 + k3 +kb +, +(2.18) +where α = 0.647. To the above scale dependent form, we incorporate the oscillations and also add +the fact that for k > kb the shape of bispectrum approaches that of slow roll. Thus, we obtain the + +8 +analytical template for LQC to be +fNL(k1, k2, k3) = f +bounce +NL +e +−α k1+k2+k3 +kb +sin +�k1 + k2 + k3 +kI +� ++ f +loc +NL. +(2.19) +The above analytical template is plotted along with the exact numerical of fNL(k, k, k) result in +figure 3, where we have worked with kb = 1.5 kLQC, f +loc +NL = 10−2 and f +bounce +NL += 80000. The value +of f +loc +NL that we work with is similar to that produced in slow roll inflation. As is evident, from the +figure, the template qualitatively captures the essential features of the primordial non-Gaussianity +in LQC. +III. +CMB BISPECTRA +Primordial perturbations leave their imprints in the CMB radiation as temperature fluctuations +and as electric and magnetic polarisations, often referred to as E and B modes respectively (see, +for instance, [62–64]). The temperature fluctuations and E modes are produced from primordial +scalar perturbations, whereas B modes are not. +Since we are interested in understanding the +imprints of scalar bispectrum, we will focus on the bispectra of temperature fluctuations and +electric polarisation and their three-point cross-correlations. In this section, we will discuss the +essential aspects of computing these bispectra. +Since CMB is observed on a sphere, namely the surface of last scattering, it is convenient to +decompose it in terms of spherical harmonics, +X(ˆn) = +� +ℓ,m +aX +ℓm Yℓm(ˆn) +(3.1) +where X could be either fluctuation in temperature defined as (T(ˆn) − ¯T)/ ¯T, where ¯T is the +mean temperature of the CMB, or electric polarisation E(ˆn). The multipole aX +ℓm corresponding to +anisotropies in the temperature and electric polarisation is related to the curvature perturbation +through the relation +aX +ℓm = 4π (−i)ℓ +� +d3k +(2π)3 Rk ∆X +ℓ (k) Yℓm(k). +(3.2) +In the above, ∆X +ℓ is the transfer function which captures the physics post horizon exit of perturba- +tions towards the end of inflation. We are interested in calculating the three-point function of these +multipoles of the form ⟨aX +ℓ1m1 aY +ℓ2m2 aZ +ℓ3m3⟩, where X, Y and Z can be either temperature fluctuations +or E mode polarisation and where the average is over different realisations of the Universe. +The three-point function of multipole coefficients can be expressed in terms of three-point +functions of primordial perturbations as [62, 65–68] +⟨aX +ℓ1m1 aY +ℓ2m2 aZ +ℓ3m3 ⟩ = (4π)3 (−i)ℓ1+ℓ2+ℓ3 +� +d3k1 +(2π)3 +� +d3k2 +(2π)3 +� +d3k3 +(2π)3 ∆X +ℓ1∆Y +ℓ2∆Z +ℓ3 +× ⟨Rk1Rk2Rk3⟩ Yℓ1m1(ˆk1) Yℓ2m2(ˆk2) Yℓ3m3(ˆk3). +(3.3) +Using Eqn. (2.14) and expressing the Dirac-Delta function in its exponential form, we obtain +⟨aX +ℓ1m1 aY +ℓ2m2 aZ +ℓ3m3 ⟩ = (4π)3 (−i)ℓ1+ℓ2+ℓ3 +� +d3k1 +(2π)3 +� +d3k2 +(2π)3 +� +d3k3 +(2π)3 ∆X +ℓ1∆Y +ℓ2∆Z +ℓ3 +× +� +d3x ei(⃗k1 +⃗k2 +⃗k3).⃗x BR(k1, k2, k3) Yℓ1m1(ˆk1) Yℓ2m2(ˆk2) Yℓ3m3(ˆk3). (3.4) + +9 +Up on using plane wave expansion, +ei⃗k·⃗x = +∞ +� +ℓ=0 +ℓ +� +m=−ℓ +iℓ jℓ(k x) Yℓm(ˆx) Y ∗ +ℓm(ˆk), +(3.5) +and the orthonormal property of spherical harmonics, we obtain +⟨aX +ℓ1m1 aY +ℓ2m2 aZ +ℓ3m3 ⟩ = bXYZ +ℓ1 ℓ2 ℓ3 Gm1 m2 m3 +ℓ1 ℓ2 ℓ3 +, +(3.6) +where all the dependence on m indices are captured in the Gaunt integral +Gm1 m2 m3 +ℓ1 ℓ2 ℓ3 += +� +dˆx Yℓ1m1(ˆx) Yℓ2m2(ˆx) Yℓ3m3(ˆx). +(3.7) +The quantity bXYZ +ℓ1 ℓ2 ℓ3 is called the reduced bispectrum and is given by +bXYZ +ℓ1 ℓ2 ℓ3 = +� 2 +π +�3 � +x2dx +� +dk1 +� +dk2 +� +dk3 (k1 k2 k3)2 BR(k1, k2, k3) +× ∆X +ℓ1∆Y +ℓ2∆Z +ℓ3 jℓ1(k1 x) jℓ2(k2 x) jℓ3(k3 x). +(3.8) +The presence of Gaunt integral implies that the reduced bispectra is non-zero only when the +multipoles satisfies the triangle inequality |ℓ1 − ℓ2| ≤ ℓ3 ≤ |ℓ1 + ℓ2| and when ℓ1 + ℓ2 + ℓ3 is even. +For isotropic theories, it suffices to work with the reduced bispectrum. +IV. +REDUCED BISPECTRA FROM LOOP QUANTUM COSMOLOGY +We will now compute the reduced bispectrum generated in LQC. The reduced bispectrum +corresponding to a primordial bispectrum can be computed using Eqn. (3.8). The primordial +bispectrum corresponding to the non-Gaussianity function Eqn. (2.19) is +BR(k1, k2, k3) = −6 +5 (2π2)2 +� +f +bounce +NL +e +−α k1+k2+k3 +kb +sin +�k1 + k2 + k3 +kI +� +× +�PR(k1) +k3 +1 +PR(k2) +k3 +2 ++ PR(k2) +k3 +2 +PR(k3) +k3 +3 ++ PR(k3) +k3 +3 +PR(k1) +k3 +1 +� ++ f +loc +NL +� ¯PR(k1) +k3 +1 +¯PR(k2) +k3 +2 ++ +¯PR(k2) +k3 +2 +¯PR(k3) +k3 +3 ++ +¯PR(k3) +k3 +3 +¯PR(k1) +k3 +1 +�� +. +(4.1) +In the above, the bispectrum contains a scale dependent part arising from the bounce and +a nearly scale invariant part which we have taken to be in the local form. In the latter part, +we take ¯PR(k) = As (k/k⋆)ns−1. We can substitute Eqn. (4.1) in Eqn. (3.8) to compute the +reduced bispectrum generated in LQC. However, this calculation involves four integrals, three over +wavenumbers and one over x variable, which is computationally expensive. +This calculation can however be simplified, essentially just to two integrals, if we use the separa- +ble property of the above primordial bispectrum. The total primordial bispectrum is not separable, +however, the contribution due to the bounce and that due to the scale invariant local template are +separable. Hence, the reduced bispectrum in LQC can be expressed as +bXYZ +ℓ1ℓ2ℓ3 = bbounce +ℓ1ℓ2ℓ3 + bloc +ℓ1ℓ2ℓ3, +(4.2) + +10 +where +bbounce +ℓ1ℓ2ℓ3 += − +� 2 +π +�3 6 +5 (2π2)2 f +bounce +NL +� ∞ +0 +dx x2 +� +A +X +ℓ1(x) B +Y +ℓ2(x) D +Z +ℓ3(x) + C +X +ℓ1(x) B +Y +ℓ2(x)B +Z +ℓ3(x) ++A +X +ℓ1(x) D +Y +ℓ2(x) B +Z +ℓ3(x) + B +X +ℓ1(x) A +Y +ℓ2(x) D +Z +ℓ3(x) + D +X +ℓ1(x) A +Y +ℓ2(x) B +Z +ℓ3(x) ++B +X +ℓ1(x) C +Y +ℓ2(x) B +Z +ℓ3(x) + B +X +ℓ1(x) B +Y +ℓ2(x) C +Z +ℓ3(x) + D +X +ℓ1(x) B +Y +ℓ2(x) A +Z +ℓ3(x) ++B +X +ℓ1(x) D +Y +ℓ2(x) A +Z +ℓ3(x) − A +X +ℓ1(x) A +Y +ℓ2(x) C +Z +ℓ3(x) − C +X +ℓ1(x) A +Y +ℓ2(x) A +Z +ℓ3(x) +−A +X +ℓ1(x) C +Y +ℓ2(x) A +Z +ℓ3(x) +� +(4.3) +and +bloc +ℓ1ℓ2ℓ3 = − +� 2 +π +�3 6 +5 (2π2)2 f +loc +NL +� ∞ +0 +dx x2 +� +E +X +ℓ1(x) E +Y +ℓ2(x) G +Z +ℓ3(x) + G +X +ℓ1(x) E +Y +ℓ2(x) E +Z +ℓ3(x) ++ E +X +ℓ1(x) G +Y +ℓ2(x) E +Z +ℓ3(x) +� +. +(4.4) +In the above expressions, the functions A +X +ℓ (x), B +X +ℓ (x), C +X +ℓ (x), D +X +ℓ (x), E +X +ℓ (x) and G +X +ℓ (x) are +A +X +ℓ (x) = +� ∞ +0 +dk ∆X +ℓ (k) jℓ(kx) PR(k) +k +e +− αk +kb sin( k +kI +), +(4.5a) +B +X +ℓ (x) = +� ∞ +0 +dk ∆X +ℓ (k) jℓ(kx) PR(k) +k +e +− αk +kb cos( k +kI +), +(4.5b) +C +X +ℓ (x) = +� ∞ +0 +dk ∆X +ℓ (k) jℓ(kx) k2 e +− αk +kb sin( k +kI +), +(4.5c) +D +X +ℓ (x) = +� ∞ +0 +dk ∆X +ℓ (k) jl(kx) k2e +− αk +kb cos( k +kI +), +(4.5d) +E +X +ℓ (x) = +� ∞ +0 +dk ∆X +ℓ (k) jℓ(kx) k−1As(k/k∗)ns−1, +(4.5e) +G +X +ℓ (x) = +� ∞ +0 +dk ∆X +ℓ (k) jℓ(kx) k2. +(4.5f) +Note that, each of the functions A +X +ℓ (x), B +X +ℓ (x), C +X +ℓ (x), D +X +ℓ (x), E +X +ℓ (x) and G +X +ℓ (x) involve an +integral over the wavenumber. The reduced bispectrum can now be calculated by evaluating these +functions for all the required values of multipoles and then finally performing the integrals Eqns. +(4.3, 4.4). +V. +NUMERICAL PROCEDURE AND RESULTS +We now discuss the numerical procedure we have followed for computing the reduced bispectrum +generated in LQC. The first step in calculating reduced bispectrum is the evaluation of functions +Eqns. (4.5). In order to compute these functions, we require the transfer functions ∆X +ℓ , where X +can be either temperature fluctuations or electric polarisation. We use publicly available Boltzmann +code class [69] to generate both the transfer functions. We perform the integral using Simpson’s +rule. We choose this method since the integrand is highly oscillatory and this gives better accuracy +when we work with sufficiently small step size. +Since the scale of oscillations occur at kI = +10−7 Mpc−1, we have worked with a step size of ∆k = 10−8 Mpc−1. This leads to an accuracy +of O(10−32). The behaviour of functions A +X +ℓ (x), B +X +ℓ (x), C +X +ℓ (x), D +X +ℓ (x), E +X +ℓ (x) and G +X +ℓ (x) for + +11 +102 +103 +104 +x (MPc) +10−29 +10−26 +10−23 +10−20 +10−17 +10−14 +10−11 +10−8 +���A +T +4(x) +��� +���B +T +4(x) +��� +���C +T +4 (x) +��� +���D +T +4(x) +��� +���E +T +4(x) +��� +���G +T +4(x) +��� +102 +103 +104 +x (MPc) +10−29 +10−26 +10−23 +10−20 +10−17 +10−14 +���A +E +4(x) +��� +���B +E +4 (x) +��� +���C +E +4 (x) +��� +���D +E +4(x) +��� +���E +E +4 (x) +��� +���G +E +4(x) +��� +102 +103 +104 +x (MPc) +10−97 +10−85 +10−73 +10−61 +10−49 +10−37 +10−25 +10−13 +���A +T +40(x) +��� +���B +T +40(x) +��� +���C +T +40(x) +��� +���D +T +40(x) +��� +���E +T +40(x) +��� +���G +T +40(x) +��� +102 +103 +104 +x (MPc) +10−97 +10−85 +10−73 +10−61 +10−49 +10−37 +10−25 +10−13 +���A +E +40(x) +��� +���B +E +40(x) +��� +���C +E +40(x) +��� +���D +E +40(x) +��� +���E +E +40(x) +��� +���G +E +40(x) +��� +FIG. 4. The behaviour of functions in Eqn. (4.5) with x for multipoles ℓ = 4 and 40. The contribution from +the local part of the bispectrum viz. +E +X +ℓ (x) and G +X +ℓ (x) are clearly dominant compared to those arising +from the bounce part viz. +A +X +ℓ (x), B +X +ℓ (x), C +X +ℓ (x) and D +X +ℓ (x). +multipoles ℓ = 4 and 40 are shown in figure 4. From the figure, it is clear that the functions +E +X +ℓ (x) and G +X +ℓ (x) are dominant compared to A +X +ℓ (x), B +X +ℓ (x), C +X +ℓ (x) and D +X +ℓ (x). +This is an +indication of the fact that local part of the bispectrum is dominant compared to the oscillatory +part. The next step in computing reduced bispectrum is the evaluation of Eqns. (4.3, 4.4). We +perform these integrals over x with a step size of 50 in the range x ∈ [0, 40000]. We have made the +calculations faster by using vectorization available in NumPy and by parallelizing the computation +wherever possible. +Reduced bispectra bTTT +ℓ1, ℓ2, ℓ3, bTTE +ℓ1, ℓ2, ℓ3, bTEE +ℓ1, ℓ2, ℓ3 and bEEE +ℓ1, ℓ2, ℓ3 generated in LQC are shown in figures +5, 6, 7 and 8 respectively. We have illustrated two different configurations of the bispectra. In +these figures, we have separately plotted the contribution from the local and bounce parts of the +bispectrum. The figures show that the contribution to the bispectrum from the oscillatory part +of the template is negligible compared to that from the local part. This shows that the reduced +bispectra generated in LQC will be similar to that produced in slow roll inflation and hence will +be consistent with observations by Planck [10]. + +12 +0 +10 +20 +30 +40 +50 +ℓ +10−60 +10−55 +10−50 +10−45 +10−40 +10−35 +10−30 +10−25 +10−20 +���bTTT +ℓ,ℓ,ℓ +��� +local +bounce +0 +10 +20 +30 +40 +50 +ℓ +10−50 +10−46 +10−42 +10−38 +10−34 +10−30 +10−26 +10−22 +���bTTT +2,ℓ,ℓ +��� +local +bounce +FIG. 5. +The reduced bispectra bTTT +ℓ1, ℓ2, ℓ3 in two different configurations. The plots illustrate that bloc +ℓ1ℓ2ℓ3 >> +bbounce +ℓ1ℓ2ℓ3 . +0 +10 +20 +30 +40 +50 +ℓ +10−54 +10−46 +10−38 +10−30 +10−23 +���bTTE +ℓ,ℓ,ℓ +��� +local +bounce +0 +10 +20 +30 +40 +50 +ℓ +10−51 +10−44 +10−37 +10−30 +10−23 +���bTTE +2,ℓ,ℓ +��� +local +bounce +FIG. 6. The plots of reduced bispectra bTTE +ℓ1, ℓ2, ℓ3 in two different configurations. Note that that bloc +ℓ1ℓ2ℓ3 >> +bbounce +ℓ1ℓ2ℓ3 . +0 +10 +20 +30 +40 +50 +ℓ +10−57 +10−49 +10−41 +10−33 +10−25 +���bTEE +ℓ,ℓ,ℓ +��� +local +bounce +0 +10 +20 +30 +40 +50 +ℓ +10−50 +10−43 +10−37 +10−31 +10−25 +���bTEE +2,ℓ,ℓ +��� +local +bounce +FIG. 7. The plots of reduced bispectra bTEE +ℓ1, ℓ2, ℓ3 in two different configurations. Clearly, the bloc +ℓ1ℓ2ℓ3 is much +larger than bbounce +ℓ1ℓ2ℓ3 . + +13 +0 +10 +20 +30 +40 +50 +ℓ +10−55 +10−48 +10−41 +10−34 +10−27 +���bEEE +ℓ,ℓ,ℓ +��� +local +bounce +0 +10 +20 +30 +40 +50 +ℓ +10−55 +10−48 +10−41 +10−34 +10−27 +���bEEE +2,ℓ,ℓ +��� +local +bounce +FIG. 8. +The plots of reduced bispectra bEEE +ℓ1, ℓ2, ℓ3 in two different configurations. Note that, the reduced +bispectrum is dominated by contribution from the local part of the template. +VI. +SUMMARY AND DISCUSSION +State of the art measurements of CMB by Planck has put strong constraints on primordial +non-Gaussianity [10]. Observations by Planck point towards a small primordial non-Gaussianity +which is consistent with the one generated in slow roll models of inflation. In LQC, primordial +perturbations originate in an adiabatic vacuum before the bounce. These then evolve through +the bounce, then through the inflationary epoch before their amplitude freezes upon horizon exit +during inflation. +The quantum bounce sets a scale kLQC in the problem. +Modes which have +wavenumbers comparable to or smaller than kLQC are excited during the bounce and modes with +larger wavenumbers are not. +This implies that modes with k ≲ kLQC are in an excited and +non-Gaussian state at the onset of inflation. This non-Gaussianity is then further enhanced as +the modes exit the horizon during inflation. +However, modes with longer wavenumbers, since +they are not excited during the bounce, behave very similarly to modes in slow roll inflation. +Hence, we have a situation where longer wavelength modes are non-Gaussian where as the shorter +ones remain Gaussian. In order to establish the viability of LQC as a model for pre-inflationary +universe, it is important to answer whether LQC is compatible with the constraints on primordial +non-Gaussianity set by Planck. +With this goal, we investigated the imprints of primordial non-Gaussianity in the bispectrum of +temperature and electric polarisation and their cross-correlations generated in LQC. In particular, +motivated by previous efforts, we proposed a template which captures the essential features of +the primordial bispectrum generated in LQC. We then used the template to compute the bTTT +ℓ1 ℓ2 ℓ3, +bTTE +ℓ1 ℓ2 ℓ3, bTEE +ℓ1 ℓ2 ℓ3 and bEEE +ℓ1 ℓ2 ℓ3 bispectra. To simplify the calculation, we used the separable property +of the proposed template. We considered the bispectra in LQC as consisting of two terms, one +scale dependent and oscillatory part arising from the bounce and the other nearly scale invariant +part arising from slow roll inflation. We find that, the contribution from the bounce to the reduced +bispectra is negligible compared to that arising from the part corresponding to slow roll inflation. +This implies that the reduced bispectrum generated in LQC is similar to that generated in slow roll +inflation. Hence, we conclude that the primordial non-Gaussianity generated in LQC is compatible +with the constraints from Planck. This is the central result of this paper. +The primordial perturbations generated in LQC is non-Gaussian in nature, yet our computation +illustrates that the reduced bispectra of temperature and electric polarisation are similar to that of +slow roll. This seemingly contradicting finding is because of the oscillatory nature of the primordial + +14 +bispectrum. The reduced bispectra involves integrals over wavenumbers which average over these +oscillations. Our result could be compared with those of [70, 71] where they had worked with a +non-oscillatory template. While they found that, in the absence of oscillations, the contribution +from the bounce is significant enough to be observed by Planck, we find that presence of oscillations +in the primordial bispectra dilutes any imprints of non-Gaussianity on the reduced bispectra. Thus, +it should be highlighted that a small reduced bispectra need not necessarily imply the absence of +primordial non-Gaussianity. Hence, it would also be interesting to look for any other measurable +imprints of such oscillatory and scale dependent primordial non-Gaussianity. +Finally, our findings are relevant for constraints on the amount of pre-inflationary expansion in +LQC. In LQC, the amount of expansion before inflationary epoch is set by the value of scalar field +at the bounce. The scalar field rolls up the potential after the bounce, comes to rest momentarily +before it rolls down and settles in to the inflationary attractor. Hence, the value of the scalar field +at the bounce determines the amount of expansion between the bounce and the onset of inflation. +This epoch of expansion is relevant as it determines whether the scales that are sensitive to the +effects of the bounce are visible today. If this epoch of pre-inflationary expansion is very large, then +the imprints of the bounce will not be visible in the Universe today. However, if the pre-inflationary +expansion is small, then the primordial power spectrum and bispectrum will be scale dependent at +observable scales. The power spectrum of temperature fluctuations fits extremely well to those due +to a nearly scale invariant primordial power spectrum at multipoles ℓ > 30 [1, 51]. This imposes +a lower limit to the amount of pre-inflationary expansion and hence a lower limit to the value of +scalar field at the bounce [29]. +Compared to the primordial power spectrum, the primordial non-Gaussianity is more sensitive +to the bounce, i.e. +fNL(k1, k2, k3) is scale dependent at larger wavenumbers than the primordial +power spectrum. This leads to a question whether imprints of primordial non-Gaussianity leads +to a stronger lower limit to the pre-inflationary expansion. Our calculations, carried out in this +work, answer this question in the negative. 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Durrer, Constraining the bispectrum from bouncing cosmologies +with Planck, (2022), arXiv:2212.05977 [astro-ph.CO]. + diff --git a/Z9E5T4oBgHgl3EQfDg58/content/tmp_files/load_file.txt b/Z9E5T4oBgHgl3EQfDg58/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1caa7001bc82794ea760cb495763ce1f8e956c19 --- /dev/null +++ b/Z9E5T4oBgHgl3EQfDg58/content/tmp_files/load_file.txt @@ -0,0 +1,1100 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf,len=1099 +page_content='Non-Gaussianity in the cosmic microwave background from loop quantum cosmology Roshna K∗ and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Sreenath† Department of Physics, National Institute of Technology Karnataka, Surathkal, Mangaluru 575025, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Primordial non-Gaussianity has set strong constraints on models of the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Studies have shown that Loop Quantum Cosmology (LQC), which is an attempt to extend inflationary scenario to planck scales, leads to a strongly scale dependent and oscillatory non- Gaussianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In particular, the non-Gaussianity function fNL(k1, k2, k3) generated in LQC, though similar to that generated during slow roll inflation at small scales, is highly scale dependent and oscillatory at large wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In this work, we investigate the imprints of such a primordial bispectrum in the bispectrum of Cosmic Microwave Background (CMB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Inspired by earlier works, we propose an analytical template for the primordial bispectrum in LQC and compute the corresponding reduced bispectra of temperature and electric po- larisation and their three-point cross-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We show that CMB bispectra generated in LQC is consistent with the observations from Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We conclude with a discussion of our results and its implications to LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' INTRODUCTION Numerous theoretical insights together with several observational efforts, spanned over a cen- tury, have enabled us to arrive at a compelling model of our Universe referred to as the standard model or the Lambda Cold Dark Matter (ΛCDM) model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' According to this model, the seeds of the current distribution of galaxies spread over the fabric of spacetime known as the large scale structure were sown during the earliest phase of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Tiny perturbations generated in the early universe lead to tiny anisotropies in the Cosmic Microwave Background (CMB) which in turn lead to the inhomegeneous large scale distribution of galaxies that we see today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Though we have a good level of understanding of this evolution, several details are yet to be worked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' One such detail concerns the origin of these perturbations in our Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Inflation, see, for instance, [2–5], due to its simplicity, provides the most popular explanation for the origin of these perturbations [6, 7] (For a discussion on alternate views, see [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In inflationary scenario, quantum fluctuations in the inflaton leads to the primordial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Appealing to the nearly de Sitter symmetry of the spacetime during inflation, we assume that at a time when the perturbations are sufficiently sub-horizon, quantum perturbations are generated in the Bunch-Davies vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Such a prescription has been highly successful, in that, it leads to primordial perturbations that are nearly Gaussian and scale invariant as demanded by observations [1, 6, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Even though inflation is successful, it is still an incomplete theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We do not take in to account the evolution of perturbations before the time at which the initial conditions are imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In fact, inflation does not account for the physics in the planck regime close to the big bang singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' There have been several attempts to address these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In this work, we will concern ourselves with loop quantum cosmology (LQC) [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Loop quantum cosmology is an attempt to extend inflationary scenario to the planck regime using principles of loop quantum gravity [13–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In LQC, quantum gravitational effects in the planck regime leads to a quantum bounce [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Thus in LQC, a quantum bounce precedes the inflationary phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Generation and evolution of perturbations in LQC have been extensively ∗ roshnak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='217ph005@nitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='in † sreenath@nitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='05406v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='CO] 13 Jan 2023 2 studied at the level of primordial power spectra [20–46] and primordial non-Gaussianity [47–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In general, studies show that the effect of the bounce is to introduce an additional scale corresponding to the curvature at the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Modes of perturbations which have comparable length to this new scale gets modified leading to a highly scale dependent power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' At smaller wavelengths, the perturbations are not affected by the bounce and the power spectrum is nearly scale invariant as in slow roll inflation [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Perturbations show a similar behaviour at second order in perturbations [47, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Studies show that primordial non-Gaussianity quantified using the function fNL(k1, k2, k3), at scales comparable to the curvature at the bounce, is strongly scale dependent and oscillatory with a very large amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' At smaller scales, the fNL(k1, k2, k3) is similar to that in slow roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Studies also show that the bispectrum is more sensitive to the bounce than the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Assuming sixty or so e-folds of inflation, the scale at which the imprints of the bounce, on primordial perturbations, occur depends on the amount of expansion between the bounce and the onset of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Observational constraints from the CMB temperature power spectrum demand that any departure from scale invariance should happen only at multipoles of ℓ ≲ 30 [1, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' If we assume that, the effects of primordial power spectrum on the CMB is observable at ℓ ≲ 30, then, since the bispectrum is more sensitive to the effects of the bounce than the power spectrum [47, 49], there is a possibility that the imprints of large, scale dependent and oscillatory primordial non- Gaussianity is observable at larger multipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Hence it is important to investigate the consistency of LQC with observations by Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' With this motivation, in this work, we compute the imprints of such a non-Gaussianity in the temperature (T) and electric polarisation (E) of the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We assume an analytical template for primordial non-Gaussianity generated in LQC, compute the ⟨TTT⟩, ⟨TTE⟩, ⟨TEE⟩ and ⟨EEE⟩ correlations and show that they are similar to those generated in slow roll inflation and hence is consistent with observations by Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In the next section, we briefly introduce the essen- tials of LQC and present analytical templates for the primordial power spectrum and bispectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In section III, we discuss the essential formulae to compute the three-point correlation functions of anisotropies in temperature and electric polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In section IV, we apply these formulae to LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We present the numerical techniques and our calculation of reduced bispectra of tempera- ture fluctuations and electric polarisation and their three-point cross-correlations in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We conclude the paper with a summary and discussion of our results and their consequences to LQC in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' LOOP QUANTUM COSMOLOGY In this section, we will discuss the essentials of LQC that is relevant to this paper (for reviews, see, for instance, [13–15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In particular, we will discuss LQC as applied to FLRW geometries sourced by a scalar field φ and scalar perturbations δφ(⃗x) living on this background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Background In LQC, FLRW background geometry is described by a wavefunction ΨFLRW(v, φ), which satis- fies the equation ˆHFLRWΨFLRW(v, φ) = 0, where ˆHFLRW is the Hamiltonian operator corresponding to the classical background Hamiltonian and v is the volume factor which is proportional to the cube of scale factor a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Numerical investigations of such a system has shown that the scale factor undergoes a bounce [11, 12, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' It turns out, if the wave function is sharply peaked over the values of scale factor, the behaviour of scale factor can be described by certain effective equations 3 −104 −102 0 102 104 106 t (TPl) 103 108 1013 1018 1023 1028 1033 a(t) Inflation −10 −5 0 5 10 0 2 4 6 8 100 101 102 103 104 105 106 107 t (TPl) −5 0 5 10 15 φ(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Figure illustrates the behaviour of scale factor (left) and scalar field(right) in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' As mentioned in the text, scale factor undergoes a bounce preceding inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Scalar field starts rolling up the potential until its kinetic energy becomes zero and then starts slowly rolling down the potential leading to inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In making this plot, we have worked with the mass of scalar field to be consistent with the constraints on the amplitude of the primordial power spectrum and with ρsup = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='41m4 Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' [11, 12, 30, 52, 53], namely � ˙a a �2 = κ 3ρ � 1 − ρ ρsup � , ¨a a = −κ 6 ρ � 1 − 4 ρ ρsup � − κ 2 P � 1 − 2 ρ ρsup � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='1) where ρ, P are the energy density and pressure of the scalar field and κ = 8 π G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' From the above expression, it is clear that at ρ = ρsup, Hubble parameter H = ˙a/a = 0 and ¨a/a > 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' scale factor is at minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In other words, the universe undergoes a bounce at ρ = ρsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Further, if we assume that the scalar field is governed by a potential V (φ), then the evolution of scalar field is given by ¨φ + 3 H ˙φ + Vφ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='2) where Vφ = dV/dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' For a suitable potential, inflationary phase will set in after the bounce [34, 54– 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The background dynamics in LQC with a scalar field governed by a quadratic potential is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Perturbations We will follow dressed metric approach to describe primordial perturbations in LQC [22–24, 29, 47, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In this approach, we assume that the wavefunction takes the form Ψ = ΨFLRW(v, φ) ⊗ δΨ(v, φ, δφ), which satisfies the equation ˆHΨ = 0, where ˆH = ˆHFLRW + ˆHpert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' As mentioned earlier, ΨFLRW(v, φ) satisfies the equation ˆHFLRWΨFLRW(v, φ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Perturbations are treated as test fields living on the background FLRW geometries described by ΨFLRW(v, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In practice, this implies that perturbations can be evolved using the classical Hamiltonian but with the background functions in them described by the effective equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This is similar to perturbations living as test fields on a curved space time described by a ‘dressed’ metric which satisfies the effective equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 4 In order to compute primordial bispectrum, we need to consider Hamiltonian up to third order in perturbations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' we need Hpert = H(2) + H(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' There are two approaches to arrive at the Hamiltonian describing perturbations, one can either use gauge invariant variables or rather work with a fixed gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We follow the latter approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In particular, we will work with spatially flat gauge [47, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The second order Hamiltonian describing perturbations δφ in the spatially flat gauge is H(2) = � d3x N S(2)(⃗x) = N 1 2 � d3x � 1 a3 δpφ2 + a3 (∂δφ)2 + a3 U δφ2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='3) with the potential U given by U = −9 p4 φ a8π2a + 3 2κ p2 φ a6 − 6 pφ a πa Vφ + Vφφ + 6 pφ ˙pφ a4 πa − 3 p2 φ ˙πa a4 π2a − 3 ˙a p2 φ a5 πa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='4) In the above expressions, πa, pφ and δpφ are momenta conjugate to a, φ and δφ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Setting lapse N = 1 will imply cosmic time and N = a corresponds to conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Hamiltonian at third order in perturbations is H(3) = N � d3x �� 9 κ p3 φ 4 a4 πa − 27 p5 φ 2 a6π3a − 3 a2 pφ Vφφ 2 πa + a3 Vφφφ 6 � δφ3 − 3 pφ 2 a4 πa δp2 φ δφ − 9 p3 φ a5π2a δpφδφ2 − 3 a2 pφ 2 πa δφ (⃗∂δφ)2 + 3 p2 φ N a πa δφ2∂2χ + 3 2 a2 pφ N2 κ πa δφ ∂2χ ∂2χ + 3 p2 φ N a πa δφ ∂iχ∂iδφ + 1 N δpφ ∂iδφ ∂iχ − 3 2 a2 pφ N2 κ πa δφ ∂i∂jχ ∂i∂jχ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5) where ∂2χ = (−3 N κ/a) �� pφ 2 − a5 Vφ κ πa � δφ − pφ κ a πa δpφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' From the second order Hamiltonian H(2), one can derive the free evolution of the scalar pertur- bation, given by, (□ − U(t)) δφ(⃗x, t) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='6) where □ is the d’Alembertian of the FLRW background metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The third order Hamiltonian H(3) provides the self-interaction of the scalar perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The perturbations, since they evolve through the bounce and then through the inflationary phase, carry signatures of the early universe which they imprint on the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Perturbations are quantified using correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In order to compute correlation functions, one need to promote δφ to an operator ˆδφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The field operator ˆδφ is then expanded in terms of annihilation and creation operators as ˆδφ(⃗x, η) = � d3k (2π)3 ˆδφ⃗k(η) ei⃗k·⃗x = � d3k (2π)3 � ˆA⃗k ϕk(η) + ˆA† −⃗k ϕ∗ k(η) � ei⃗k·⃗x (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='7) where [ ˆA⃗k, ˆA† ⃗k′] = ℏ (2π)3 δ(3)(⃗k + ⃗k′), [ ˆA⃗k, ˆA⃗k′] = 0 = [ ˆA† ⃗k, ˆA† ⃗k′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The mode functions ϕk(η) satisfy the equation ϕ′′ k + 2a′ a ϕ′ k + (k2 + a2 U) ϕk = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='8) 5 where k2 ≡ kikj δij is the comoving wavenumber, and prime indicates derivative with respect to conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The scalar power spectrum of ˆδφ is a dimensionless function that quantifies the two-point correlation in momentum space via ⟨0| ˆδφ⃗k(η) ˆδφ⃗k′(η)|0⟩ ≡ (2π)3δ(3)(⃗k + ⃗k′)2π2 k3 Pδφ(k, η) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='9) where |0⟩ is the vacuum annihilated by the operators ˆA⃗k for all ⃗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Power spectrum, in terms of mode functions, is Pδφ(k, η) = (ℏ k3/2π2) |ϕk(η)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The three-point function of ˆδφ at tree level is given by [47, 59] ⟨ ˆδφ⃗k1(η) ˆδφ⃗k2(η) ˆδφ⃗k3(η) ⟩ = − i/ℏ � dη′⟨0| � ˆδφ I ⃗k1(η) ˆδφ I ⃗k2(η) ˆδφ I ⃗k3(η), ˆHI int(η′) � |0⟩ + O(H2 int), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='10) where ˆHI int(η) is the operator corresponding to H(3) in the interaction picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Even though we worked in spatially flat gauge, it is convenient to compute correlation functions in terms of curvature perturbations R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This is because, curvature perturbations have a unique property that they stop evolving after they cross the horizon and remain constant till they re-enter horizon towards late radiation domination or during early matter domination epoch, saving us a lot of computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Curvature perturbations are related to perturbations in scalar field through the relation [47, 59] R(⃗x, η) = −a z δφ(⃗x, η) + � −3 2 + 3 Vφ a5 κ Pφ πa + κ 4 z2 a2 � �a z δφ(⃗x, η) �2 + · · · , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='11) where trailing dots indicates terms that leads to subdominant terms in the three-point functions when evaluated towards the end of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The power spectrum of curvature perturbation is related to that of scalar modes ˆδφ⃗k(η) through the relation PR(k) = �a(ηend) z(ηend) �2 Pδφ(k, η), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='12) where z = −6 pφ/(κ πa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The three-point function of curvature perturbation can be obtained in terms of ˆδφ⃗k(η) by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='11) as ⟨0| ˆR⃗k1 ˆR⃗k2 ˆR⃗k3|0⟩ = � −a z �3 ⟨0| ˆδφ⃗k1 ˆδφ⃗k2 ˆδφ⃗k3|0⟩ + � −3 2 + 3 Vφ a5 κ pφ πa + κ 4 z2 a2 � � −a z �4 � � d3p (2π)3 ⟨0| ˆδφ⃗k1 ˆδφ⃗k2 ˆδφ⃗p ˆδφ⃗k3−⃗p|0⟩ + (⃗k1 ↔ ⃗k3) + (⃗k2 ↔ ⃗k3) + · · · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='13) The wave numbers of three modes in the three-point function are constrained by a Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We define the scalar bispectrum as the three-point function sans Dirac delta function as ⟨0| ˆR⃗k1 ˆR⃗k2 ˆR⃗k3|0⟩ ≡ (2π)3δ(3)(⃗k1 + ⃗k2 + ⃗k3) BR(k1, k2, k3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='14) The amplitude of bispectrum can be quantified using a dimensionless function fNL(k1, k2 , k3), akin to the dimensionless power spectrum PR(k) that quantifies two-point correlations, as fNL(k1, k2, k3) ≡ −5 6BR(k1, k2, k3) × (∆k1∆k2 + ∆k1∆k3 + ∆k2∆k3)−1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='15) where ∆k ≡ 2 π2 k3 PR(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 6 −104 −103 −102 −101 −1000 100 101 102 103 104 105 106 t (TPl) 10−5 10−3 10−1 101 103 105 � |Ω(η)| (MPl) k⋆ kLQC kI FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The figure represents the relevant scales in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' kLQC is the scale corresponding to the value of curvature at the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' kI corresponds to the smallest scale that is sub-horizon during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' As can be seen, only modes kLQC ≳ k > kI are excited during the bounce and hence are scale dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Modes with larger wavenumbers are excited only during horizon crossing towards the end of inflation and hence will be scale invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Templates of scalar power spectrum and bispectrum In order to understand the evolution of perturbations in LQC, let us rewrite Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='8), v′′ k + � k2 + Ω(η) � vk = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='16) where vk = a ϕk is the Mukhanov-Sasaki variable and Ω(η) = a2 U − a′′ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We compare the behaviour of � |Ω(η)| as a function of time with relevant wavenumbers in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' As shown in the figure, all observationally relevant wavenumbers are adiabatic much before the bounce and hence we can impose adiabatic initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' From the figure, it is also clear that there are two relevant scales in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The value of curvature at the bounce defines a scale kLQC and the value of curvature at the onset of inflation defines a scale kI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Wavenumbers which are much larger than kLQC, are not effected by the bounce and they will be in Bunch-Davies vacuum at the onset of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This implies that power spectrum of modes k >> kLQC will be nearly scale invariant as in slow roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Modes which are comparable to kLQC and larger than kI will be excited both during the bounce as well as during the horizon exit during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' These modes are in excited non-Gaussian states during the onset of inflation and hence they will be further amplified as they exit the horizon during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Hence, the modes kI < k < kLQC will be strongly scale dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Modes whose wavenumbers are smaller than kI are always superhorizon and hence they are never excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The primordial power spectrum and bispectrum are evaluated towards the end of inflation when all the relevant modes are well outside the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The primordial power spectrum and bispectrum can be calculated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Given the background dynamics described in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='2), the evolution of perturbations are found by 7 10−7 10−6 10−5 10−4 10−3 10−2 k � Mpc−1� 10−10 10−9 10−8 10−7 10−6 PR(k) kLQC k⋆ analytical result numerical result 10−5 10−4 10−3 10−2 k � Mpc−1� −106 −105 −104 −103 −102 −101 −1000 100 101 102 103 104 105 106 fNL(k, k, k) kLQC k⋆ analytical result numerical result FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The primordial power spectrum and the non-Gaussianity function generated in LQC obtained numerically (in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Analytical templates for power spectrum and non-Gaussianity given in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='19) (in grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' solving Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The power spectrum of curvature perturbation can then be calculated using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Calculation of ⟨ ˆδφ⃗k1(η) ˆδφ⃗k2(η) ˆδφ⃗k3(η) ⟩ requires one to perform integrals in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The ⟨0| ˆR⃗k1 ˆR⃗k2 ˆR⃗k3|0⟩ three-point function of curvature perturbation can then be calculated using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The dimensionless non-Gaussianity function of curvature perturbation is then calculated by using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This numerical calculation of primordial power spectrum and non-Gaussianity has been implemented in class_lqc [47, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We present the results obtained using that code in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' For calculating the three-point functions involving temperature and electric polarisation, one needs to convolve the primordial bispectrum with the CMB transfer functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' For performing this calculation, it is convenient to have analytical templates of primordial power spectrum and bispectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Following [46, 60, 61], we will use the following template for describing the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' It is given by PR(k) = As � � � � � � � � � ( k kI )2( kI kLQC )q if k ≤ kI, ( k kLQC )q if kI < k ≤ kLQC, ( k kLQC )(ns−1) if k > kLQC, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='17) where we work with q = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='7, kI = 5 × 10−5 k⋆, kLQC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='1 k⋆ and k⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='002Mpc−1 represents the pivot scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The amplitude of power spectrum As and the spectral index ns have been set to their values obtained by Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The analytical template for power spectra is drawn along with the exact numerical calculation in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' As is evident from figure 3, the primordial non-Gaussianity fNL(k1, k2, k3) is scale dependent and oscillatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The exponential decay in the value of fNL(k1, k2, k3) as k ≈ kLQC was explained in [47, 49] by analysing the poles of the integrand in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In particular, by analysing the pole of scale factor around the bounce, the analytical behaviour of fNL sans oscillations was found to be fNL(k1, k2, k3) ∝ e −α k1 + k2 + k3 kb , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='18) where α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' To the above scale dependent form, we incorporate the oscillations and also add the fact that for k > kb the shape of bispectrum approaches that of slow roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Thus, we obtain the 8 analytical template for LQC to be fNL(k1, k2, k3) = f bounce NL e −α k1+k2+k3 kb sin �k1 + k2 + k3 kI � + f loc NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='19) The above analytical template is plotted along with the exact numerical of fNL(k, k, k) result in figure 3, where we have worked with kb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5 kLQC, f loc NL = 10−2 and f bounce NL = 80000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The value of f loc NL that we work with is similar to that produced in slow roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' As is evident, from the figure, the template qualitatively captures the essential features of the primordial non-Gaussianity in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' CMB BISPECTRA Primordial perturbations leave their imprints in the CMB radiation as temperature fluctuations and as electric and magnetic polarisations, often referred to as E and B modes respectively (see, for instance, [62–64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The temperature fluctuations and E modes are produced from primordial scalar perturbations, whereas B modes are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Since we are interested in understanding the imprints of scalar bispectrum, we will focus on the bispectra of temperature fluctuations and electric polarisation and their three-point cross-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In this section, we will discuss the essential aspects of computing these bispectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Since CMB is observed on a sphere, namely the surface of last scattering, it is convenient to decompose it in terms of spherical harmonics, X(ˆn) = � ℓ,m aX ℓm Yℓm(ˆn) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='1) where X could be either fluctuation in temperature defined as (T(ˆn) − ¯T)/ ¯T, where ¯T is the mean temperature of the CMB, or electric polarisation E(ˆn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The multipole aX ℓm corresponding to anisotropies in the temperature and electric polarisation is related to the curvature perturbation through the relation aX ℓm = 4π (−i)ℓ � d3k (2π)3 Rk ∆X ℓ (k) Yℓm(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='2) In the above, ∆X ℓ is the transfer function which captures the physics post horizon exit of perturba- tions towards the end of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We are interested in calculating the three-point function of these multipoles of the form ⟨aX ℓ1m1 aY ℓ2m2 aZ ℓ3m3⟩, where X, Y and Z can be either temperature fluctuations or E mode polarisation and where the average is over different realisations of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The three-point function of multipole coefficients can be expressed in terms of three-point functions of primordial perturbations as [62, 65–68] ⟨aX ℓ1m1 aY ℓ2m2 aZ ℓ3m3 ⟩ = (4π)3 (−i)ℓ1+ℓ2+ℓ3 � d3k1 (2π)3 � d3k2 (2π)3 � d3k3 (2π)3 ∆X ℓ1∆Y ℓ2∆Z ℓ3 × ⟨Rk1Rk2Rk3⟩ Yℓ1m1(ˆk1) Yℓ2m2(ˆk2) Yℓ3m3(ˆk3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='3) Using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='14) and expressing the Dirac-Delta function in its exponential form, we obtain ⟨aX ℓ1m1 aY ℓ2m2 aZ ℓ3m3 ⟩ = (4π)3 (−i)ℓ1+ℓ2+ℓ3 � d3k1 (2π)3 � d3k2 (2π)3 � d3k3 (2π)3 ∆X ℓ1∆Y ℓ2∆Z ℓ3 × � d3x ei(⃗k1 +⃗k2 +⃗k3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='⃗x BR(k1, k2, k3) Yℓ1m1(ˆk1) Yℓ2m2(ˆk2) Yℓ3m3(ˆk3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='4) 9 Up on using plane wave expansion, ei⃗k·⃗x = ∞ � ℓ=0 ℓ � m=−ℓ iℓ jℓ(k x) Yℓm(ˆx) Y ∗ ℓm(ˆk), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5) and the orthonormal property of spherical harmonics, we obtain ⟨aX ℓ1m1 aY ℓ2m2 aZ ℓ3m3 ⟩ = bXYZ ℓ1 ℓ2 ℓ3 Gm1 m2 m3 ℓ1 ℓ2 ℓ3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='6) where all the dependence on m indices are captured in the Gaunt integral Gm1 m2 m3 ℓ1 ℓ2 ℓ3 = � dˆx Yℓ1m1(ˆx) Yℓ2m2(ˆx) Yℓ3m3(ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='7) The quantity bXYZ ℓ1 ℓ2 ℓ3 is called the reduced bispectrum and is given by bXYZ ℓ1 ℓ2 ℓ3 = � 2 π �3 � x2dx � dk1 � dk2 � dk3 (k1 k2 k3)2 BR(k1, k2, k3) × ∆X ℓ1∆Y ℓ2∆Z ℓ3 jℓ1(k1 x) jℓ2(k2 x) jℓ3(k3 x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='8) The presence of Gaunt integral implies that the reduced bispectra is non-zero only when the multipoles satisfies the triangle inequality |ℓ1 − ℓ2| ≤ ℓ3 ≤ |ℓ1 + ℓ2| and when ℓ1 + ℓ2 + ℓ3 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' For isotropic theories, it suffices to work with the reduced bispectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' REDUCED BISPECTRA FROM LOOP QUANTUM COSMOLOGY We will now compute the reduced bispectrum generated in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The reduced bispectrum corresponding to a primordial bispectrum can be computed using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The primordial bispectrum corresponding to the non-Gaussianity function Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='19) is BR(k1, k2, k3) = −6 5 (2π2)2 � f bounce NL e −α k1+k2+k3 kb sin �k1 + k2 + k3 kI � × �PR(k1) k3 1 PR(k2) k3 2 + PR(k2) k3 2 PR(k3) k3 3 + PR(k3) k3 3 PR(k1) k3 1 � + f loc NL � ¯PR(k1) k3 1 ¯PR(k2) k3 2 + ¯PR(k2) k3 2 ¯PR(k3) k3 3 + ¯PR(k3) k3 3 ¯PR(k1) k3 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='1) In the above, the bispectrum contains a scale dependent part arising from the bounce and a nearly scale invariant part which we have taken to be in the local form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In the latter part, we take ¯PR(k) = As (k/k⋆)ns−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We can substitute Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='1) in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='8) to compute the reduced bispectrum generated in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' However, this calculation involves four integrals, three over wavenumbers and one over x variable, which is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This calculation can however be simplified, essentially just to two integrals, if we use the separa- ble property of the above primordial bispectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The total primordial bispectrum is not separable, however, the contribution due to the bounce and that due to the scale invariant local template are separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Hence, the reduced bispectrum in LQC can be expressed as bXYZ ℓ1ℓ2ℓ3 = bbounce ℓ1ℓ2ℓ3 + bloc ℓ1ℓ2ℓ3, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='bbounce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1ℓ2ℓ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='= − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='�3 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5 (2π2)2 f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='bounce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='NL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='dx x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) + C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x)B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='+A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) + B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) + D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='+B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) + B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) + D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='+B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) − A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) − C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='−A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ1(x) C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ2(x) A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ3(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='3) and bloc ℓ1ℓ2ℓ3 = − � 2 π �3 6 5 (2π2)2 f loc NL � ∞ 0 dx x2 � E X ℓ1(x) E Y ℓ2(x) G Z ℓ3(x) + G X ℓ1(x) E Y ℓ2(x) E Z ℓ3(x) + E X ℓ1(x) G Y ℓ2(x) E Z ℓ3(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='4) In the above expressions, the functions A X ℓ (x), B X ℓ (x), C X ℓ (x), D X ℓ (x), E X ℓ (x) and G X ℓ (x) are A X ℓ (x) = � ∞ 0 dk ∆X ℓ (k) jℓ(kx) PR(k) k e − αk kb sin( k kI ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5a) B X ℓ (x) = � ∞ 0 dk ∆X ℓ (k) jℓ(kx) PR(k) k e − αk kb cos( k kI ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5b) C X ℓ (x) = � ∞ 0 dk ∆X ℓ (k) jℓ(kx) k2 e − αk kb sin( k kI ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5c) D X ℓ (x) = � ∞ 0 dk ∆X ℓ (k) jl(kx) k2e − αk kb cos( k kI ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5d) E X ℓ (x) = � ∞ 0 dk ∆X ℓ (k) jℓ(kx) k−1As(k/k∗)ns−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5e) G X ℓ (x) = � ∞ 0 dk ∆X ℓ (k) jℓ(kx) k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5f) Note that, each of the functions A X ℓ (x), B X ℓ (x), C X ℓ (x), D X ℓ (x), E X ℓ (x) and G X ℓ (x) involve an integral over the wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The reduced bispectrum can now be calculated by evaluating these functions for all the required values of multipoles and then finally performing the integrals Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' NUMERICAL PROCEDURE AND RESULTS We now discuss the numerical procedure we have followed for computing the reduced bispectrum generated in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The first step in calculating reduced bispectrum is the evaluation of functions Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In order to compute these functions, we require the transfer functions ∆X ℓ , where X can be either temperature fluctuations or electric polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We use publicly available Boltzmann code class [69] to generate both the transfer functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We perform the integral using Simpson’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We choose this method since the integrand is highly oscillatory and this gives better accuracy when we work with sufficiently small step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Since the scale of oscillations occur at kI = 10−7 Mpc−1, we have worked with a step size of ∆k = 10−8 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This leads to an accuracy of O(10−32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The behaviour of functions A X ℓ (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' B X ℓ (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' C X ℓ (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' D X ℓ (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ (x) and G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='ℓ (x) for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='102 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The behaviour of functions in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5) with x for multipoles ℓ = 4 and 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The contribution from the local part of the bispectrum viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' E X ℓ (x) and G X ℓ (x) are clearly dominant compared to those arising from the bounce part viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' A X ℓ (x), B X ℓ (x), C X ℓ (x) and D X ℓ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' multipoles ℓ = 4 and 40 are shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' From the figure, it is clear that the functions E X ℓ (x) and G X ℓ (x) are dominant compared to A X ℓ (x), B X ℓ (x), C X ℓ (x) and D X ℓ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This is an indication of the fact that local part of the bispectrum is dominant compared to the oscillatory part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The next step in computing reduced bispectrum is the evaluation of Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We perform these integrals over x with a step size of 50 in the range x ∈ [0, 40000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We have made the calculations faster by using vectorization available in NumPy and by parallelizing the computation wherever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Reduced bispectra bTTT ℓ1, ℓ2, ℓ3, bTTE ℓ1, ℓ2, ℓ3, bTEE ℓ1, ℓ2, ℓ3 and bEEE ℓ1, ℓ2, ℓ3 generated in LQC are shown in figures 5, 6, 7 and 8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We have illustrated two different configurations of the bispectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In these figures, we have separately plotted the contribution from the local and bounce parts of the bispectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The figures show that the contribution to the bispectrum from the oscillatory part of the template is negligible compared to that from the local part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This shows that the reduced bispectra generated in LQC will be similar to that produced in slow roll inflation and hence will be consistent with observations by Planck [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 12 0 10 20 30 40 50 ℓ 10−60 10−55 10−50 10−45 10−40 10−35 10−30 10−25 10−20 ���bTTT ℓ,ℓ,ℓ ��� local bounce 0 10 20 30 40 50 ℓ 10−50 10−46 10−42 10−38 10−34 10−30 10−26 10−22 ���bTTT 2,ℓ,ℓ ��� local bounce FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The reduced bispectra bTTT ℓ1, ℓ2, ℓ3 in two different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The plots illustrate that bloc ℓ1ℓ2ℓ3 >> bbounce ℓ1ℓ2ℓ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 0 10 20 30 40 50 ℓ 10−54 10−46 10−38 10−30 10−23 ���bTTE ℓ,ℓ,ℓ ��� local bounce 0 10 20 30 40 50 ℓ 10−51 10−44 10−37 10−30 10−23 ���bTTE 2,ℓ,ℓ ��� local bounce FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The plots of reduced bispectra bTTE ℓ1, ℓ2, ℓ3 in two different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Note that that bloc ℓ1ℓ2ℓ3 >> bbounce ℓ1ℓ2ℓ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 0 10 20 30 40 50 ℓ 10−57 10−49 10−41 10−33 10−25 ���bTEE ℓ,ℓ,ℓ ��� local bounce 0 10 20 30 40 50 ℓ 10−50 10−43 10−37 10−31 10−25 ���bTEE 2,ℓ,ℓ ��� local bounce FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The plots of reduced bispectra bTEE ℓ1, ℓ2, ℓ3 in two different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Clearly, the bloc ℓ1ℓ2ℓ3 is much larger than bbounce ℓ1ℓ2ℓ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 13 0 10 20 30 40 50 ℓ 10−55 10−48 10−41 10−34 10−27 ���bEEE ℓ,ℓ,ℓ ��� local bounce 0 10 20 30 40 50 ℓ 10−55 10−48 10−41 10−34 10−27 ���bEEE 2,ℓ,ℓ ��� local bounce FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The plots of reduced bispectra bEEE ℓ1, ℓ2, ℓ3 in two different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Note that, the reduced bispectrum is dominated by contribution from the local part of the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' SUMMARY AND DISCUSSION State of the art measurements of CMB by Planck has put strong constraints on primordial non-Gaussianity [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Observations by Planck point towards a small primordial non-Gaussianity which is consistent with the one generated in slow roll models of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In LQC, primordial perturbations originate in an adiabatic vacuum before the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' These then evolve through the bounce, then through the inflationary epoch before their amplitude freezes upon horizon exit during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The quantum bounce sets a scale kLQC in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Modes which have wavenumbers comparable to or smaller than kLQC are excited during the bounce and modes with larger wavenumbers are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This implies that modes with k ≲ kLQC are in an excited and non-Gaussian state at the onset of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This non-Gaussianity is then further enhanced as the modes exit the horizon during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' However, modes with longer wavenumbers, since they are not excited during the bounce, behave very similarly to modes in slow roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Hence, we have a situation where longer wavelength modes are non-Gaussian where as the shorter ones remain Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In order to establish the viability of LQC as a model for pre-inflationary universe, it is important to answer whether LQC is compatible with the constraints on primordial non-Gaussianity set by Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' With this goal, we investigated the imprints of primordial non-Gaussianity in the bispectrum of temperature and electric polarisation and their cross-correlations generated in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In particular, motivated by previous efforts, we proposed a template which captures the essential features of the primordial bispectrum generated in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We then used the template to compute the bTTT ℓ1 ℓ2 ℓ3, bTTE ℓ1 ℓ2 ℓ3, bTEE ℓ1 ℓ2 ℓ3 and bEEE ℓ1 ℓ2 ℓ3 bispectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' To simplify the calculation, we used the separable property of the proposed template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We considered the bispectra in LQC as consisting of two terms, one scale dependent and oscillatory part arising from the bounce and the other nearly scale invariant part arising from slow roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We find that, the contribution from the bounce to the reduced bispectra is negligible compared to that arising from the part corresponding to slow roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This implies that the reduced bispectrum generated in LQC is similar to that generated in slow roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Hence, we conclude that the primordial non-Gaussianity generated in LQC is compatible with the constraints from Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This is the central result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The primordial perturbations generated in LQC is non-Gaussian in nature, yet our computation illustrates that the reduced bispectra of temperature and electric polarisation are similar to that of slow roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This seemingly contradicting finding is because of the oscillatory nature of the primordial 14 bispectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The reduced bispectra involves integrals over wavenumbers which average over these oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Our result could be compared with those of [70, 71] where they had worked with a non-oscillatory template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' While they found that, in the absence of oscillations, the contribution from the bounce is significant enough to be observed by Planck, we find that presence of oscillations in the primordial bispectra dilutes any imprints of non-Gaussianity on the reduced bispectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Thus, it should be highlighted that a small reduced bispectra need not necessarily imply the absence of primordial non-Gaussianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Hence, it would also be interesting to look for any other measurable imprints of such oscillatory and scale dependent primordial non-Gaussianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Finally, our findings are relevant for constraints on the amount of pre-inflationary expansion in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' In LQC, the amount of expansion before inflationary epoch is set by the value of scalar field at the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The scalar field rolls up the potential after the bounce, comes to rest momentarily before it rolls down and settles in to the inflationary attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Hence, the value of the scalar field at the bounce determines the amount of expansion between the bounce and the onset of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This epoch of expansion is relevant as it determines whether the scales that are sensitive to the effects of the bounce are visible today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' If this epoch of pre-inflationary expansion is very large, then the imprints of the bounce will not be visible in the Universe today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' However, if the pre-inflationary expansion is small, then the primordial power spectrum and bispectrum will be scale dependent at observable scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' The power spectrum of temperature fluctuations fits extremely well to those due to a nearly scale invariant primordial power spectrum at multipoles ℓ > 30 [1, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This imposes a lower limit to the amount of pre-inflationary expansion and hence a lower limit to the value of scalar field at the bounce [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Compared to the primordial power spectrum, the primordial non-Gaussianity is more sensitive to the bounce, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' fNL(k1, k2, k3) is scale dependent at larger wavenumbers than the primordial power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This leads to a question whether imprints of primordial non-Gaussianity leads to a stronger lower limit to the pre-inflationary expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Our calculations, carried out in this work, answer this question in the negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' More specifically, since the reduced bispectra do not carry any imprints of the bounce, we find that it do not provide any constraints on the epoch of pre-inflationary expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' ACKNOWLEDGEMENT We thank Ivan Agullo for his comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' This work was supported by Science and Engineering Research Board (SERB) through Start-up Research Grant SRG/2021/001769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' We acknowledge the use of PU HPC facility of the National Supercomputing Mission project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (Planck), Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Cosmological parameters, (2018), arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='06209 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Riotto, Inflation and the theory of cosmological perturbations, ICTP Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Notes Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 14, 317 (2003), arXiv:hep-ph/0210162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Bassett, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Tsujikawa, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Wands, Inflation dynamics and reheating, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 78, 537 (2006), arXiv:astro-ph/0507632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Sriramkumar, An introduction to inflation and cosmological perturbation theory, (2009), arXiv:0904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='4584 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Baumann, Inflation, in Theoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small (2011) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' 523–686, arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='5424 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Akrami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' (Planck), Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content=' Constraints on inflation, (2018), arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='06211 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} +page_content='CO].' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E5T4oBgHgl3EQfDg58/content/2301.05406v1.pdf'} diff --git a/ZNAyT4oBgHgl3EQfifg0/content/tmp_files/2301.00395v1.pdf.txt b/ZNAyT4oBgHgl3EQfifg0/content/tmp_files/2301.00395v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb4220ab97ed63275cd95daa92afcf6a2033ede2 --- /dev/null +++ b/ZNAyT4oBgHgl3EQfifg0/content/tmp_files/2301.00395v1.pdf.txt @@ -0,0 +1,2656 @@ +CORGI-PM +: A Chinese Corpus For Gender Bias Probing and +Mitigation +Ge Zhang1 3 4 ∗, Yizhi Li2 ∗, Yaoyao Wu5, Linyuan Zhang 6, Chenghua Lin 2 †, Jiayi Geng7, Shi Wang 3 †, Jie Fu 1 +1 Beijing Academy of Artificial Intelligence, China +2 Department of Computer Science, The University of Sheffield, UK +3 Institute of Computing Technology, Chinese Academy of Sciences, China +4 University of Michigan Ann Arbor, USA +5 University of Colorado Boulder, USA +6 Sichuan University, China +7 McGill University, Canada +{yizhi.li, c.lin}@sheffield.ac.uk2, gezhang@umich.edu1, wangshi@ict.ac.cn3 +Abstract +As natural language processing (NLP) for gen- +der bias becomes a significant interdisciplinary +topic, the prevalent data-driven techniques, +such as large-scale language models, suffer +from data inadequacy and biased corpus, espe- +cially for languages with insufficient resources, +such as Chinese. +To this end, we propose +a Chinese cOrpus foR Gender bIas Probing +and Mitigation (CORGI-PM1), which con- +tains 32.9k sentences with high-quality labels +derived by following an annotation scheme +specifically developed for gender bias in the +Chinese context. Moreover, we address three +challenges for automatic textual gender bias +mitigation, which requires the models to de- +tect, classify, and mitigate textual gender bias. +We also conduct experiments with state-of-the- +art language models to provide baselines. To +our best knowledge, CORGI-PM is the first +sentence-level Chinese corpus for gender bias +probing and mitigation. +1 +Introduction +Increasing recognition in consensus is that iden- +tifying and preventing toxic gender attitudes and +stereotypes is essential for society (Blodgett et al., +2020). Since gender-biased information could be +presented and widely propagated in textual format, +it is essential to develop automatic methods for +detecting and mitigating textual gender bias. +Natural language processing (NLP) has been +widely used in text-related applications, which have +a significant influence on gender bias topics (Costa- +jussà, 2019). On the one hand, large-scale language +models (LMs), as a key technique of modern NLP, +are proven to learn the subjective gender bias in +the training corpus or even amplify it (Zhao et al., +2017) On the other hand, it becomes increasingly +∗ The two authors contributed equally to this work. +† Corresponding authors. +1Our code is available at GitHub +promising to apply cutting-edge NLP techniques +for probing and mitigating gender bias. +Language +Models +Gender-related +Vocabulary +Polarity +Calculation +Word +Matching +Sentence-level +Reranking +Potentially +Biased Corpus +Corpus with +Biased Vocabulary +Raw Corpus +Figure 1: Pipeline of Retrieving and Filtering Potentially Bi- +ased Sentences from Raw Corpus for Human Annotation. +Building a high-quality text corpus has been one +of the key tangents in improving NLP applications +for debiasing gender stereotypes in texts (Sun et al., +2019). Some researchers introduce automatic an- +notation techniques, such as gender-swapped based +methods, to create corpora for gender bias mitiga- +tion (Lu et al., 2020; Zhao et al., 2018; Rudinger +et al., 2018). While it is attractive to build a large +corpus without heavy labors, automatic gender- +swapped based methods highly depend on the qual- +ity of base language models and are prone to cre- +ating nonsensical sentences (Sun et al., 2019). To +address this issue, some works devote effort to de- +veloping human-annotated corpora for gender bias +mitigation. However, these corpora either mainly +focus on word- or grammar-level bias (Webster +et al., 2018; Zhu and Liu, 2020; Sahai and Sharma, +2021; Zhou et al., 2019), or only concern about +sexism-related topics (Jiang et al., 2022; Chiril +et al., 2021, 2020; Parikh et al., 2019). +Moreover, existing works on gender bias exclu- +arXiv:2301.00395v1 [cs.CL] 1 Jan 2023 + +sively focus on English (Costa-jussà, 2019), where +few datasets exist for other influential languages +such as Chinese. (N.B. details of generated gen- +der bias corpus with nonsensical Chinese sentences +can be found in Appendix D). We aim to tackle the +aforementioned issues by providing a high-quality +Chinese human-annotated corpus for contextual- +level gender bias probing and mitigation. +To this end, we propose the Chinese cOrpus foR +Gender bIas Probing and Mitigation (CORGI-PM) +dataset, which consists of 32.9k human-annotated +sentences, including both gender-biased and non- +biased samples. +For the initial data collection, +we propose an automatic method that builds a po- +tentially gender-biased sentence set from existing +large-scale Chinese corpora. Inspired by the metric +leveraging language models for gender bias score +calculation proposed in Bolukbasi et al. (2016); +Jiao and Luo (2021), the samples containing words +of high gender bias scores are recalled, and then +reranked and filtered according to their sentence- +level gender-biased probability, as illustrated in +Fig. 1. To ensure the quality of our corpus, the +annotation scheme is carefully designed, and an- +notators with qualified educational backgrounds +are selected to further label and paraphrase the re- +trieved sentences. +Additionally, we address three challenges based +on CORGI-PM, i.e., gender bias detection, classifi- +cation, and mitigation, which provide clear defini- +tions and evaluation protocols for NLP tasks in gen- +der bias probing and mitigation. In order to provide +referential baselines and benchmarks for our pro- +posed challenges, we conduct random data splitting +with balanced labels and implement experiments +on cutting-edge language models in zero-shot, in- +context learning, and fine-tuning paradigms. We +discuss the experimental settings and provide result +analysis in §3. The implementation details can be +referred to in Appendix C. +In summary, we provide a well-annotated Chi- +nese corpus for gender bias probing and mitiga- +tion, along with clearly defined corresponding +challenges. With a properly designed annotation +scheme, CORGI-PM provides a corpus of high +quality that assists models in detecting gender bias +in texts. More importantly, other than the 22.5k +human-annotated non-biased samples, all the 5.2k +biased sentences in our corpus are further labeled +with gender bias subclasses and companies with +parallel bias-free versions provided by the annota- +Sample +Quantity +Category +Train +Valid +Test +Biased +AC +1.90k +235 +237 +DI +2.70k +334 +337 +ANB +2.47k +306 +309 +Non-biased +21.4k +516 +526 +Overall +30.1k +1391 +1409 +Table 1: Overall Statistics of the CORGI-PM Dataset. The +notations, AC, DI, and ANB represent specific bias labels +described in § 2.2. +tors. Our codes and dataset will be released for the +benefit of the community. +2 +Data Collection +2.1 +Sample Filtering +We propose an automatic processing method to +recall, rerank, and filter annotation candidates from +raw corpora using a two-stage filtering from word- +level to sentence-level, as illustrated in Fig. 1. The +Chinese sentence samples are mainly screened out +from the SlguSet (Zhao et al., 2021) and the CCL +corpus (Weidong et al., 2019). +To recall gender-biased words or retrieve candi- +date sentences with gender bias scores, we com- +pare the target word/sentence representations with +the seed direction, which can be calculated by the +subtraction between the word embeddings of she +and he +(Bolukbasi et al., 2016; Jiao and Luo, +2021). We leverage different Chinese LMs includ- +ing ERNIE (Zhang et al., 2019), CBert (Cui et al., +2020), and Chinese word vectors (Qiu et al., 2018) +to acquire the word-level and sentence-level rep- +resentations. For word-level filtering, we use the +mentioned metric to build a vocabulary of high +bias scores and recall sentences containing such +words from the raw corpora with exact matches. +We compute gender bias scores of the crawled sen- +tences and group them by the gender bias keywords +acquired in the previous stage for sentence-level fil- +tering. The final sentences for annotation are then +selected according to a specific global threshold +gender bias score and an in-group threshold rank. +The word-level filtering process presented as word +clouds can be found in Appendix B.1. +2.2 +Annotation Scheme +The annotation scheme is designed for gender bias +probing and mitigation. For gender bias probing, +the annotators are required to provide the follow- +ing information given a sentence: whether gender +bias exists; if so, how the bias is established. For +gender bias mitigation, the corrected non-biased +version of the biased sentences is also required. We + +Linguistic +Non-biased +Biased +Corrected Biased +Info. +Train +Valid +Test +Train +Valid +Test +Train +Valid +Test +Word +724k +18.9k +17.7k +228k +24.8k +28.3k +265k +27.1k +30.0k +Dictionary +574k +14.4k +14.1k +167k +18.4k +20.4k +191k +19.9k +21.5k +Character +1,156k +30.1k +28.1k +358k +39.2k +44.4k +417k +42.8k +46.9k +Sent. Length +53.952 +58.397 +53.473 +85.837 +76.087 +85.214 +99.839 +82.853 +89.939 +Table 2: Linguistic Characteristics of the Corpus. Word, Dictionary, and Character separately denote the total Chinese word +number, total unique Chinese word number, and total character number of the specific categories. The sentence lengths are +defined as the number of containing characters. +further describe the annotation scheme details in +the following paragraphs. +Existence and Categorization. +The annotators are required to annotate whether +the sentence is gender-biased (B) or non-biased (N) +in contextual-level or word-level, and further clar- +ify how the bias is established. Given that our raw +data is collected using gender-related keywords or +from gender-related corpus, the samples annotated +without gender bias are useful human-annotated +negative samples for detecting gender bias. To addi- +tionally provide information about gender bias cate- +gorization, we classify gender bias types into three +subtypes : (1) Gender Stereotyped activity and +career choices (AC); (2) Gender Stereotyped de- +scriptions and inductions (DI); and (3) Expressed +gender-stereotyped attitudes, norms and beliefs +(ANB). The classification standard is inspired by +(King et al., 2021) and further summed up into the +mentioned subtypes. +Bias Mitigation. Annotators are also required to +mitigate the gender bias of selected sentences while +keeping the original semantic information. We +also ask our annotators to diversify the expres- +sions if applicable. The major revision patterns +can be summarized as follows: (1). Replace the +gender-specific pronouns with neutral pronouns. +(2). Replace the gender-specific adjectives with +neutral descriptions with similar semantics defini- +tions. (3). Add additional comments to neutralize +the sentences which cannot be directly mitigated. +2.3 +Corpus Analysis +In this section, we report the linguistic statistics +of CORGI-PM as Tab. 1. We design a balanced +split to create the valid and test set considering the +negative-positive ratio and bias subclass proportion +in the global distribution. As revealed in Tab. 22, +we observe two major differences compared the de- +biased samples with the original ones: longer and +more diverse expressions (N.B. sentence length and +vocabulary size of Tab. 2). We hypothesize that it +2We use the Jieba to parse. +is due to human annotators’ intention to keep the +semantic information unchanged and the sentence +coherent while mitigating gender bias. They may +use more conjunctions and longer descriptions com- +pared to some gender-biased inherent expressions. +More details for quality managing and control can +be referred to Appendix B.1 and B.2. +3 +Gender Bias Mitigation Challenges +To provide a clear definition for automatic textual +gender bias probing and mitigation tasks, we pro- +pose corresponding challenges and standardize the +evaluation protocols. We address two tasks, detec- +tion, and classification, for gender bias probing and +formalize the gender mitigation challenge as a text +mitigation task. +3.1 +Challenges of Detection and +Classification +We regard both the gender bias detection and clas- +sification challenges as supervised classification +tasks and evaluate them with metrics of consensus. +Definition. The gender bias detection challenge +can be regarded as a binary classification task, +where the model is required to predict the prob- +ability that a given sentence contains gender bias. +As described in § 2.2, biased samples are further +categorized into one or more kinds. Therefore, we +can address the gender classification challenge as +a multi-label classification task. The precision, re- +call, and F1-score are selected as the main metrics +in these two challenges. Class-wise metrics and +macro average summarized evaluation are required +through both valid and test sets to show the perfor- +mance of language models. +Baselines. We finetune Chinese language mod- +els from three representative different pretrained +paradigms, i.e., Chinese BERT, Electra, and XL- +Net Cui et al. (2020), for both the detection and +classification tasks by adding an additional dense +prediction layer. 3 We also provide GPT-3 (Brown +et al., 2020) curie’s few-shot performance for both +3Pretrained models can be found at theHFL Anthology. + +Model +Metrics +Classification (Val.) +Classification (Test) +Detection (Val.) +Detection (Test.) +AC +DI +ANB +Avg. +AC +DI +ANB +Avg. +N +B +Avg. +N +B +Avg. +BERT +Precision +.609 +.729 +.533 +.624 +.556 +.615 +.521 +.564 +.699 +.950 +.824 +.742 +.980 +.861 +Recall +.594 +.665 +.543 +.601 +.493 +.652 +.585 +.577 +.971 +.591 +.781 +.985 +.662 +.823 +F1-Score +.602 +.695 +.538 +.612 +.522 +.633 +.551 +.567 +.813 +.729 +.771 +.846 +.790 +.818 +Electra +Precision +.587 +.727 +.544 +.619 +.556 +.630 +.516 +.568 +.691 +.936 +.814 +.745 +.974 +.860 +Recall +.758 +.687 +.386 +.610 +.680 +.685 +.373 +.579 +.961 +.570 +.766 +.983 +.656 +.820 +F1-Score +.661 +.706 +.451 +.606 +.612 +.656 +.433 +.567 +.804 +.708 +.756 +.848 +.784 +.816 +XLNet +Precision +.587 +.696 +.523 +.602 +.544 +.589 +.527 +.553 +.713 +.928 +.820 +.772 +.959 +.865 +Recall +.622 +.643 +.495 +.587 +.545 +.614 +.514 +.558 +.953 +.620 +.787 +.968 +.722 +.845 +F1-Score +.604 +.669 +.509 +.594 +.544 +.601 +.520 +.555 +.816 +.743 +.780 +.859 +.824 +.841 +Curie +Precision +.695 +.907 +.010 +.537 +.622 +.887 +.009 +.506 +.763 +.665 +.714 +.635 +.825 +.730 +Recall +.395 +.802 +.375 +.524 +.395 +.804 +.010 +.403 +.576 +.825 +.700 +.975 +.584 +.780 +F1-Score +.504 +.851 +.019 +.458 +.508 +.852 +.019 +.460 +.656 +.736 +.696 +.769 +.684 +.727 +Table 3: Baseline Results for Gender Bias Detection and Classification Tasks. The overall metric refers to Marco average. The +model names and abbreviations refer to § 3.1. Categorical definitions refer to § 2.2. +aa +Metrics +Models +BLEU +METEOR +ROUGE-L +Human Evaluation +Recall +Precision +F1 +Coherence +Gender Bias +*Davinci +.776 +.879 +.203 +.211 +.205 +5.25 +0.96 +Ada +.288 +.429 +.407 +.180 +.250 +5.98 +1.13 +Babbage +.359 +.504 +.716 +.310 +.432 +6.32 +0.69 +Curie +.364 +.506 +.692 +.316 +.434 +6.21 +1.20 +Table 4: Baseline Results for Gender Bias Correction task. Metrics details can be found in Appendix C. * suggests using the +model in zero-shot paradigm and the others refers to fine-tune. +the detection and classification tasks. Baseline re- +sults of detection and classification show that the +classification task is challenging, and there is room +for performance improvement in detecting gender +bias in CORGI-PM, as revealed in Tab. 3. +3.2 +Challenge of Mitigation +Definition. The gender bias mitigation challenge +can be regarded as a natural language generation +task, where the model is asked to generate the cor- +rected version of biased sentences with the human- +annotated ones as references. +Baselines. We test the GPT-3 (Brown et al., 2020) +on CORGI-PM in fine-tune experiment setting +with three different parameter scales, which are +Ada(350M), Babbage(1.3B), and Curie(6.7B), and +Davinci(175B) in zero-shot experiment setting. We +only provide zero-shot results for Davinci because +it is the only released GPT-3 editing model. More +implementation and evaluation details are intro- +duced in Appendix C. +Discussion. We provide both human evaluation +and automated metrics for evaluation. Tab. 4 re- +veals that LMs can learn the annotation pattern of +mitigating gender bias, and the zero-shot editing +model shows competitive performance. The obser- +vation that fine-tuned Babbage outperforms much +larger zero-shot Davinci in the human evaluation, +and ROUGE-L reveals that CORGI-PM has the +potential to be used as strong supervision of the +gender bias mitigation task. We notice that Davinci +tends to apply more conservative edits compared to +fine-tuned models. As a result, the sentences edited +by Davinci keep most of the original sentences and +always only change pronouns and adjectives from +the original sentences, which benefits precision +focusing automatic metrics like BLEU (Papineni +et al., 2002), and METEOR (Agarwal and Lavie, +2007). The performance difference between human +evaluation and automatic metrics reveals the writ- +ing style difference between human and language +models. +4 +Conclusion +We propose CORGI-PM, the first Chinese human- +annotated corpus for both gender bias probing and +mitigation. We also address definitions and evalua- +tion metrics for three challenges based on CORGI- +PM and test the performances of state-of-the-art +language models. Our proposed challenges can +serve as benchmarks for measuring the ability of +language models to detect, classify, and mitigate +textual gender bias. Experiments show that our sen- +tences with fine-grained subclass labels can assist +the language models in gender bias probing, whilst +our parallel human-written debiased data can serve +as strong supervision of the generative language +models. In summary, we imply future work utiliz- +ing CORGI-PM would be benefited the topic of +NLP for gender bias probing and mitigation. +Limitations +There are several major limitations in this research +work. Due to the high requirement of annotators + +for annotating gender-biased sentences and correct- +ing such sentences, we only choose annotators with +higher education, which may lead to potential cog- +nitive bias. In addition, we only conduct limited +implementations and experiments of testing widely- +used Chinese language models’ performance in our +new challenges. More language models and tech- +niques can be further explored in our challenges. +Ethics Statement +We carefully consider the ethical implications dur- +ing the collection process. The collection of our +corpus CORGI-PM sentences only relies on public +available corpora for research purposes. We have +acknowledged the potential usage of our dataset as +well as related privacy issues to the annotators and +received confirmations before the annotation was +initiated. +References +Abhaya Agarwal and Alon Lavie. 2007. Meteor: An +automatic metric for mt evaluation with high levels +of correlation with human judgments. Proceedings +of WMT-08. +Su Lin Blodgett, Solon Barocas, Hal Daumé III, and +Hanna Wallach. 2020. +Language (technology) is +power: A critical survey of “bias” in nlp. In ACL. +Tolga Bolukbasi, Kai-Wei Chang, James Y. 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In +Proceedings of the 19th Chinese National Confer- +ence on Computational Linguistics, pages 31–42. + + ERNIE + BERT + XLNet + ELECTRA + ERNIE + BERT + XLNet + ELECTRA +1 +-0.015 +0.058 +-0.0075 +-0.015 +1 +0.021 +-0.083 +0.058 +0.021 +1 +0.14 +-0.0075 +-0.083 +0.14 +1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 2: Word-level Gender Bias Comparison of Career +Words. +A +Gender Bias Analysis of Chinese +Language Models +A.1 +Evaluation Method and Data Sets +We conduct experiments to explore gender bias con- +tained in widely-used Chinese language models for +research and industrial use. We employ the method +Bolukbasi et al. (2016); Jiao and Luo (2021) pro- +posed to assess gender bias. The gender bias score +for a word is calculated by ⃗w · ( ⃗ +she − ⃗he)based on +its word vector. A positive value means the word +is more relevant to females, while a negative value +means the word is more relevant to males. The +higher the absolute value of the gender bias score, +the more biased the word indicates. +Srivastava et al. propose a big benchmark con- +taining a dataset specifying the existing Chinese ca- +reer words. Zhu and Liu propose AGSS, a manual- +created Chinese word-level adjective list containing +gender bias. To measure gender bias contained in +the language models, we first calculate gender bias +scores of words in the word list provided (Srivas- +tava et al., 2022; Zhu and Liu, 2020) according to +the projection method Bolukbasi et al. (2016); Jiao +and Luo (2021). We compare the career and adjec- +tive word gender bias score vectors to get the ob- +servations of LMs’ influence on word-level learned +gender bias. To make the observations more clear, +we further apply the sign function to the career and +adjective word gender bias score vectors. The sim- +ilarity function used for the heatmaps is Pearson +similarity. +We conduct described comparison of adjectives +between AGSS as a golden standard (Zhu and Liu, +2020), Ernie (Zhang et al., 2019), Chinese Word +Vectors trained by mixed corpus (Qiu et al., 2018), +AGSS + ERNIE + BERT + CWV + XLNet +ELECTRA +AGSS + ERNIE + BERT + CWV + XLNet +ELECTRA +1 +0.22 +-0.023 +0.15 +0.021 0.044 +0.22 +1 +-0.037 0.066 0.012 +0.1 +-0.023 -0.037 +1 +0.074 0.032 -0.0097 +0.15 +0.066 0.074 +1 +-0.065 0.017 +0.021 0.012 0.032 -0.065 +1 +0.0093 +0.044 +0.1 +-0.0097 0.017 0.0093 +1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 3: Word-level Gender Bias Comparison of Adjectives. +CWV denotes the Chinese Word Vectors trained using mixed- +large corpus proposed by Qiu et al.. +AGSS +Mixed-large +PDN +Zhihu_QA +Weibo +Literature +AGSS +Mixed-large +PDN +Zhihu_QA +Weibo +Literature +1 +0.21 +0.15 +-0.093 0.086 +0.21 +0.21 +1 +0.071 +-0.25 -0.043 -0.021 +0.15 +0.071 +1 +-0.041 -0.025 0.046 +-0.093 -0.25 -0.041 +1 +0.12 +-0.037 +0.086 -0.043 -0.025 +0.12 +1 +-0.014 +0.21 +-0.021 0.046 -0.037 -0.014 +1 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 4: Word-level Gender Bias Comparison of Adjectives +of Language Models Pre-trained by Different Corpus. PDN +denotes the People’s Daily News Corpus. +and Chinese-XLNet, Chinese-Bert, and Chinese- +Electra proposed tecui-etal-2020-revisiting to pro- +duce Fig. 3. We conduct described comparison of +career words between Ernie (Zhang et al., 2019), +and Chinese-XLNet, Chinese-Bert, and Chinese- +Electra proposed tecui-etal-2020-revisiting to pro- +duce Fig. 2. The described experiments on career +words is not conducted with the Chinese Word Vec- +tors trained by mixed corpus, because an observing +number of career words are missing in its dictio- +nary. +We don’t provide a golden standard vector (Sri- +vastava et al., 2022) since they didn’t provide a +manual gender bias analysis about the career words. +We also conduct described comparison on adjec- +tives in Chinese Word Vectors pre-trained by dif- +ferent corpus, including Mixed-large corpus, Peo- +ple’s Daily News, Zhihu QA dataset, Weibo, and +Chinese literature dataset to produce Fig. 4 and an- +alyze the learned gender bias difference caused by + +(a) Ch-Ernie-Man-Adj +(b) Ch-Ernie-Woman-Adj +(c) Ch-Ernie-Man-Career +(d) Ch-Ernie-Woman-Career +(e) En-Ernie-Man-Adj +(f) En-Ernie-Woman-Adj +(g) En-Ernie-Man-Career +(h) En-Ernie-Woman-Career +(i) Ch-XLNet-Man-Adj +(j) Ch-XLNet-Woman-Adj +(k) Ch-XLNet-Man-Career +(l) Ch-XLNet-Woman-Career +(m) En-XLNet-Man-Adj +(n) En-XLNet-Woman-Adj +(o) En-XLNet-Man-Career +(p) En-XLNet-Woman-Career +Figure 5: Example Word Cloud Analysis of Ernie and Chinese-XLNet. Ch denotes Chinese. En denotes words’ English +translation. Man and Woman separately denote words with embedding closer to man and woman. Adj denotes adjectives. +Career denotes career words. +using different datasets for pretraining the language +model. +A.2 +Discussion +There exists observing gender bias in the open- +source Chinese language models, especially in +Ernie and Chinese Word Vectors according to +Fig. 3. We hypothesize that the observation is +highly related to the corpus used. Cui et al. claim +that their used corpus is a combination of Chine- +seWiki, and some other universal Chinese datasets, +including encyclopedia, news, and QA dataset. In +sharp contrast, Ernie and Chinese Word Vectors use +corpus, which contains sentences from literature, +forum, and other social media, which may lead to +a gender-biased model. +According to Fig. 4, People’s Daily News, and +Chinese literature corpora contain observing gen- +der bias. The observation indicates that researchers +should be more careful about using literature data +while training a language model. We also hypoth- +esize that this is caused by the literature corpus +and People’s Daily News, which contains more +descriptive expressions. +B +Corpus +B.1 +Word Cloud Analysis +We provide word cloud analysis of Ernie and +Chinese-Electra in the section about adjectives and +career words. More available word cloud analy- +sis will be available in our public repository. The +words are ranked according to the absolute value of +their gender bias score calculated along the method +used by Bolukbasi et al.; Jiao and Luo. There is a +noticeable word-level gender stereotype according +to the word cloud. For example, a man is robust +and a woman is motherly, a man is suitable for +a fitness instructor and a woman is suitable for a +choreographer. We also conduct word cloud anal- +ysis for language models pre-trained by different +corpora. +B.2 +Quality Monitoring and Control +We used a standardized operating method and edu- +cated our annotators to achieve high-quality anno- +tations as follows: +(1). Annotators +We have 6 annotators, which +were all native speakers of Chinese. Annotators + +complacent +emaciated +faithful +stable +handsome +fashionable +Irrogant +lliterate +conscientious +boate +stubborn +ormalserious +vigorous +Interestino +decadent +harsh +truthful +reckless +less +boring +fierce +anxious +procrastination +bold +playful +majestic +heroic +rude +lick +namel +humorous +calm +daring +greedy +S +competent +brave +athletic +slutty +deceitful +attentive +interior swarthy +S +ridiculous +ong-lived +courageous +sturdy +healthy +Tearles +loyal +worldly sloppy +horrible +stern +dull +imbecile +hed +vulgar +unfortunate +lovely +brashshabby +distinguish +lack of virtue +bizarre +pontaneous +illustrious +frank +concentration +proactive +paranoid +lucky +stoic +apable +unruly +capricious +dashing +impatientpedanticunsightly +robustwild +focused +alert +oroad-mindeo +smart and strong +decent +cheerful +self-confident +cautious +rigorouswise and resourcefu +conceited +charming +dexterous +benevolent. +steadfast +casual +coldasice +rouhded +ical +stalwart +lively +obedient +noly +ea +ignoral +disloyal +suspicious +down +peacefu +TO- +berceptive +big-hearted +heartless +nasty +evil +smooth +aloof +abhorrent +dignified +gentle +melancholy +frail +quiet easy-going +soft +nice and charming +polite +timid +ff +motherly +plain +stingy +shcere +sentimental +watery +lazy +spicy +ectionate +clean +OLUS +a +blushing +and +Simpleheadstrong +frivolous +pure +sgood-natureo +disinterested +learned +sensitive +open-minded +bright +money-minded +keen +twisted +Wise +quick-witted +amiable +mean +elegant meek +noble +unpretentious +generous +enlightened +purity +indifferent +innocent +old and spicy +imple +narrow-minded +leisurelyhiheseahdwesterhmediciheahdsurgery +chief executiveofficer +tower crane +operator +judges +tcm chiropractor +flight navigato +operationsmanager +factorymanagerdean +freelancewriter +calligrapher +C +integrativemedicine physician tcm anorectal physician +hadowplayers +mayor safety officer +e +warden +C +military personnel +estate planner +planist +acrobats +headnurse +butcher +town mayormagician +marketing specialist +financier +pilot +otolaryngology +long distance runners +plasterer +director of bureau designer orthotist +cartoonist +producel +dispatcher +art director +sailor +manager +genera +earing +specialists +columnist +economist +waterengineering technician +anthropologist +captain of a plane2 +bankmanager +construction engineering techn +e +prosecutor +businessman +provincial governor +president +curato +dietitian +guitarist +lyricist +orison +guards +archaeologist +police officer +specialist +Iclal +playwright analyst +vice-president +astronaut, +investment banker +el +city party secretary. +sociologist +entrepreneur +mediator +oil and + novelist +technician +gas engineering +psychologist +manager head chef +sprinter +nair stylist +adventureradvisor +author +store manager +footwear designel +humanresourcespecialist +integrative orthopedic surgeoninsurance underwriters +packer +driller +etrigerato +gistician +land engineering technician +S +public health physician +taxpreparer +computerteacher +pet practitioner +agronomist +thology technologist +pharmacist +istant profe +music conductor +dairy processors +pawnbrokers +sound mixer +environmentaldesign +el +higher education teachers +geography teacher +landscaper +relectriclan +O seaman +receiver and dispatcher +elementaryschoolteacher +chinese medicine t +decoration +artist +1O +electrical engineering technician +foreign anguage andliterature.teacher +papermake +tutor +western medicine physiciar +art design +photojournalis +digital +mediaart +copy editors. +grinder +electronicengineeringtechn +funeral service +teller sheet metal worker +internationa +audito +ousihess +domestic helper +cultivator +accountant +leasing salesmar +administrative assistant +physicsteache +taxidermist +gemstone cutter +proofreader +administrativestat +Woodworkel +kindergartenteache +bank personnel +cantin +archivist +child care worker +oomattendant 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All annota- +tors held a bachelor’s degree. Waseem points out +that expert annotators are more cautious and can +improve the corpus quality with a large margin, +which proves the necessity of our training proce- +dure. We also kept the number of male and female +annotators equal. +(2). Gender Equality of Raw Corpus +In the +raw data collection procedure, we keep the num- +ber of man-related keywords and woman-related +keywords equal and make the number of samples +recalled according to different keywords balanced. +As a result, the raw data and the final data should +hold gender equality. +(3). Annotation Procedure +Our annotation +procedure is separated into two stages. In the first +stage, annotators are encouraged to not enter any +samples that they are not certain about. In the +second stage, we have annotators cross-checking +annotations. We did not enter any contradictory +samples. +(4). Inter-annotator Agreement +Given the +domain and purpose of the dataset, we want to +build the dataset as high quality as possible. Af- +ter an initial annotation round with 6 annotators, +we also report inter-annotator agreement in Table +5. to verify annotation reliability, where the IAA +among three annotators on bias classification, de- +tection, and mitigation is 0.802, 0.935, and 0.987, +respectively. +Classification +Detection +Mitigation +IAA +0.802 +0.935 +0.987 +Table 5: Inter-Annotator Agreement (IAA) +C +Implementation Details +For gender bias classification challenge, we +used finetuned Chinese-BERT-wwm, Chinese- +ELECTRA-180g-base, and Chinese-XLNet-base, +(Cui et al., 2020), and the GPT-3 (Curie) in the +in-context paradigm. We first use the train set to +save the multiple labeled examples in a document +with a specific file ID. Then we use the test sets +to perform a classification query on the saved file. +The processing time for the classification of gender +bias is approximately 1 hour. We calculated the +precision, recall, and F1 score to analyze model +performance. +For gender bias detection challenge, we use +the same baseline model set as in the classification +challenge. We test the performance on both "yes" +and "no" detection. The detection tasks also use the +Classification endpoints of GPT3 (Curie), which +requires more time compared to classification as +we use a larger dataset for both training and testing. +For gender bias mitigation challenge, we did +not provide experiment results of finetuning the +largest Davinci (175B) GPT-3 on CORGI-PM be- +cause of the cost and no observing performance +gain comparing Curie and Babbage. For finetune +experiment setting, we follow the tutorial of GPT-3 +official API of the Completion Model and regard +the ground truth edits provided by human annota- +tors as the completion of the original sentences. For +the zero-shot experiment setting, we apply GPT-3 +editing model and set the instructions as "Eliminate +the gender bias contained in the sentence." +For metrics used, on the one hand, we conduct +extensive human evaluations from both gender bias +and coherence aspects on CORGI-PM. For both +gender bias and coherence, we shuffled the correc- +tion results from human annotators and different +models, and asked our annotators to grade the re- +sults using the answer range from 1-not at all to +7-extremely gender biased/extremely fluent without +the information of the source. On the other hand, +we provide the automated metrics result, includ- +ing BLEU (Papineni et al., 2002), ROUGE-L (Lin, +2004), and METEOR (Agarwal and Lavie, 2007) +of the models on CORGI-PM as well. BLEU is the +earliest and most widely-used metric for translation +and NLG tasks. METEOR introduces WordNet and +other external resources to improve the robustness +of the BLEU-based metrics. ROUGE pays more +attention to recall compared to BLEU. +In the gender bias correction task, we fine-tune +Ada, Babbage, and Curie models of GPT3 and test +the performance using the aforementioned metrics. +Fine-tuning on the train set requires 31 minutes +on Ada, 35 minutes on Curie, and 43 minutes on +Babbage. For the generation process, each fine- +tuned model required approximately 30 minutes +to complete. For the zero-show paradigm, we use +"Eliminate the gender bias in the sentence" as the +instruction and use Davinci. Compared with the +fine-tuned model doing the sentence correction, the +zero-shot paradigm requires more time (approxi- + +Original +Ground Truth +Gender-swapped +但是再坚强的你也是个女人吧。 +但是再坚强的你也有脆弱的时候吧。 +但是再坚强的你也是个男人吧。 +(But you are still a woman, even if +you are strong.) +(But even the strongest of you have +moments of vulnerability, right?) +(But you are still a man, even if you +are strong.) +可怕可恨的是，有的女人自己也是这样 +给自己定位的——没有反对，没有抗 +争，有的只是心甘情愿、死心塌地遵照 +执行。 +女人不应该这样给自己定位——没有反 +对，没有抗争，有的只是心甘情愿、死 +心塌地遵照执行。 +可怕可恨的是，有的男人自己也是这样 +给自己定位的——没有反对，没有抗 +争，有的只是心甘情愿、死心塌地遵照 +执行。 +(The scary thing is that some women +themselves are so defined for +themselves - no opposition, no +resistance, there is only a willingness +to follow it to the letter.) +(A woman should not define herself +in this way - there is no opposition, +no resistance, there is only a +willingness to follow it to the letter.) +(The scary thing is that some men +themselves are so defined for +themselves - no opposition, no +resistance, there is only a willingness +to follow it to the letter.) +那个时候，能驾驭金色高跟鞋的女人得 +多摩登啊！ +那个时候，能驾驭金色高跟鞋的人得多 +摩登啊！ +那个时候，能驾驭金色高跟鞋的男人得 +多摩登啊！ +(At that time, the woman who could +handle the gold heels must be very +modern!) +(At that time, how modern must be +the person who can handle the gold +heels!) +(At that time, the man who could +handle the gold heels must be very +modern!) +“鸿雁队”是清一色的“理工男”。 +“鸿雁队”是清一色的“理工生”。 +“鸿雁队”是清一色的“理工女”。 +(The "Hongyan team" team of all +men in STEM.) +(The "Hongyan team" team of all +student in STEM.) +(The "Hongyan team" team of all +women in STEM.) +Figure 6: Case Study of Nonsensical Sentences Created by Gender-swapped Methods. +Original Sentence +Edit Sentence +清洁阿姨一边扫地一边赞扬。 +清洁工一边扫地一边赞扬。 +(The cleaning woman praised while +sweeping the floor.) +(The cleaners praised while sweeping the +floor.) +我，有时文静，有时却调皮得像一个男孩 +。 +我，有时文静，有时调皮。 +(I, sometimes quiet, but sometimes +naughty like a boy.) +(I, sometimes quiet, sometimes naughty.) +在小王眼里，李某高大帅气、温柔体贴， +而且风趣幽默，是一个十分优质的青年男 +性。 +在小王眼里，李某身材高大、外表好看、温 +柔体贴，而且风趣幽默，是一个十分优质的 +青年。 +(In the eyes of Wang, Li is tall and +handsome, gentle and considerate, and +funny, a very high-quality young +male.) +(In the eyes of Wang, Li is tall, good- +looking, caring and gentle, and funny, a +very high-quality young people.) +沙峰起伏，金光灿灿，宛如一座金山，像 +绸缎一样柔软，少女一样娴静。 +沙峰起伏，金光灿灿，宛如一座金山，像绸 +缎一样柔软，宁静。 +(The sandy peaks are undulating and +golden, like a golden mountain, as +soft as silk and as serene as a +maiden.) +(The sandy peaks are undulating and +golden, like a golden mountain, as soft +and serene as silk.) +我想要世界，而世界当时属于男人们。 +我想要世界，而世界当时属于男人们。评: +世界应当属于人们，与男女无关。 +(I want the world, and the world then +belonged to the men.) +(I wanted the world, and the world then +belonged to the men. Comment: The +world should belong to people, not to +men and women.) +哎哟，果然每个追梦男人的背后，都有个 +不世俗的后方！ +哎哟，果然每个追梦男人的背后，都有个不 +世俗的后方！评: 这种感慨是错误的，将男 +女的家庭分工固定化，剥除女性就业的权 +利，应予以鄙弃。 +(Oops, indeed, behind every dream- +chasing man, there is an +unsophisticated back!) +(Oops, indeed, behind every dream- +chasing man, there is an unsophisticated +back! Comment: This is a wrong feeling +that fixes the domestic division of labor +between men and women and strips +women of their employment rights, +which should be despised.) +Change the +Pronoun +Change the +Gender- +specific +Adjectives +Add +Comments +Figure 7: Case Study of Mitigation Annotation Patterns. +mately 1 hour). +D +Case Study +As shown in Fig. 6, gender-swapped methods suffer +from mitigating gender bias expressed by gender- +specific descriptions and inductions, and expressed +gender-stereotyped attitudes, norms and beliefs. As +a result, gender-swapped methods may generate +nonsensical sentences under certain circumstances. +We also use the basic mitigation annotation pat- +terns (Fig. 7). These three major mitigation annota- +tion patterns are not used exclusively in the annota- +tion but optionally in combination. Except for the +three mentioned patterns, we apply several other +linguistic skills, including deleting gender-specific +pronouns and replacing vehicles in gender-related +metaphors, to mitigate the gender bias while keep- +ing semantic information unchanged. + diff --git a/ZNAyT4oBgHgl3EQfifg0/content/tmp_files/load_file.txt b/ZNAyT4oBgHgl3EQfifg0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..391bf0f98acecc3147affcd9131b8cd5ec84daaa --- /dev/null +++ b/ZNAyT4oBgHgl3EQfifg0/content/tmp_files/load_file.txt @@ -0,0 +1,2105 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf,len=2104 +page_content='CORGI-PM : A Chinese Corpus For Gender Bias Probing and Mitigation Ge Zhang1 3 4 ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Yizhi Li2 ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Yaoyao Wu5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Linyuan Zhang 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Chenghua Lin 2 †,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Jiayi Geng7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Shi Wang 3 †,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Jie Fu 1 1 Beijing Academy of Artificial Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' China 2 Department of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The University of Sheffield,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' UK 3 Institute of Computing Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' China 4 University of Michigan Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' USA 5 University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' USA 6 Sichuan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' China 7 McGill University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Canada {yizhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='li, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='lin}@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='uk2, gezhang@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='edu1, wangshi@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='cn3 Abstract As natural language processing (NLP) for gen- der bias becomes a significant interdisciplinary topic, the prevalent data-driven techniques, such as large-scale language models, suffer from data inadequacy and biased corpus, espe- cially for languages with insufficient resources, such as Chinese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' To this end, we propose a Chinese cOrpus foR Gender bIas Probing and Mitigation (CORGI-PM1), which con- tains 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='9k sentences with high-quality labels derived by following an annotation scheme specifically developed for gender bias in the Chinese context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Moreover, we address three challenges for automatic textual gender bias mitigation, which requires the models to de- tect, classify, and mitigate textual gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We also conduct experiments with state-of-the- art language models to provide baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' To our best knowledge, CORGI-PM is the first sentence-level Chinese corpus for gender bias probing and mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 1 Introduction Increasing recognition in consensus is that iden- tifying and preventing toxic gender attitudes and stereotypes is essential for society (Blodgett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Since gender-biased information could be presented and widely propagated in textual format, it is essential to develop automatic methods for detecting and mitigating textual gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Natural language processing (NLP) has been widely used in text-related applications, which have a significant influence on gender bias topics (Costa- jussà, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' On the one hand, large-scale language models (LMs), as a key technique of modern NLP, are proven to learn the subjective gender bias in the training corpus or even amplify it (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2017) On the other hand, it becomes increasingly ∗ The two authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' † Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 1Our code is available at GitHub promising to apply cutting-edge NLP techniques for probing and mitigating gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Language Models Gender-related Vocabulary Polarity Calculation Word Matching Sentence-level Reranking Potentially Biased Corpus Corpus with Biased Vocabulary Raw Corpus Figure 1: Pipeline of Retrieving and Filtering Potentially Bi- ased Sentences from Raw Corpus for Human Annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Building a high-quality text corpus has been one of the key tangents in improving NLP applications for debiasing gender stereotypes in texts (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Some researchers introduce automatic an- notation techniques, such as gender-swapped based methods, to create corpora for gender bias mitiga- tion (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Rudinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' While it is attractive to build a large corpus without heavy labors, automatic gender- swapped based methods highly depend on the qual- ity of base language models and are prone to cre- ating nonsensical sentences (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' To address this issue, some works devote effort to de- veloping human-annotated corpora for gender bias mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' However, these corpora either mainly focus on word- or grammar-level bias (Webster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Zhu and Liu, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Sahai and Sharma, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019), or only concern about sexism-related topics (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Chiril et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2021, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Parikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Moreover, existing works on gender bias exclu- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='00395v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='CL] 1 Jan 2023 sively focus on English (Costa-jussà, 2019), where few datasets exist for other influential languages such as Chinese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' details of generated gen- der bias corpus with nonsensical Chinese sentences can be found in Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We aim to tackle the aforementioned issues by providing a high-quality Chinese human-annotated corpus for contextual- level gender bias probing and mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' To this end, we propose the Chinese cOrpus foR Gender bIas Probing and Mitigation (CORGI-PM) dataset, which consists of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='9k human-annotated sentences, including both gender-biased and non- biased samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For the initial data collection, we propose an automatic method that builds a po- tentially gender-biased sentence set from existing large-scale Chinese corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Inspired by the metric leveraging language models for gender bias score calculation proposed in Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Jiao and Luo (2021), the samples containing words of high gender bias scores are recalled, and then reranked and filtered according to their sentence- level gender-biased probability, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' To ensure the quality of our corpus, the annotation scheme is carefully designed, and an- notators with qualified educational backgrounds are selected to further label and paraphrase the re- trieved sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Additionally, we address three challenges based on CORGI-PM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', gender bias detection, classifi- cation, and mitigation, which provide clear defini- tions and evaluation protocols for NLP tasks in gen- der bias probing and mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' In order to provide referential baselines and benchmarks for our pro- posed challenges, we conduct random data splitting with balanced labels and implement experiments on cutting-edge language models in zero-shot, in- context learning, and fine-tuning paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We discuss the experimental settings and provide result analysis in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The implementation details can be referred to in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' In summary, we provide a well-annotated Chi- nese corpus for gender bias probing and mitiga- tion, along with clearly defined corresponding challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' With a properly designed annotation scheme, CORGI-PM provides a corpus of high quality that assists models in detecting gender bias in texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' More importantly, other than the 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='5k human-annotated non-biased samples, all the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2k biased sentences in our corpus are further labeled with gender bias subclasses and companies with parallel bias-free versions provided by the annota- Sample Quantity Category Train Valid Test Biased AC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='90k 235 237 DI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='70k 334 337 ANB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='47k 306 309 Non-biased 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='4k 516 526 Overall 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1k 1391 1409 Table 1: Overall Statistics of the CORGI-PM Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The notations, AC, DI, and ANB represent specific bias labels described in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Our codes and dataset will be released for the benefit of the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 2 Data Collection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1 Sample Filtering We propose an automatic processing method to recall, rerank, and filter annotation candidates from raw corpora using a two-stage filtering from word- level to sentence-level, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The Chinese sentence samples are mainly screened out from the SlguSet (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2021) and the CCL corpus (Weidong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' To recall gender-biased words or retrieve candi- date sentences with gender bias scores, we com- pare the target word/sentence representations with the seed direction, which can be calculated by the subtraction between the word embeddings of she and he (Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Jiao and Luo, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We leverage different Chinese LMs includ- ing ERNIE (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019), CBert (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2020), and Chinese word vectors (Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2018) to acquire the word-level and sentence-level rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For word-level filtering, we use the mentioned metric to build a vocabulary of high bias scores and recall sentences containing such words from the raw corpora with exact matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We compute gender bias scores of the crawled sen- tences and group them by the gender bias keywords acquired in the previous stage for sentence-level fil- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The final sentences for annotation are then selected according to a specific global threshold gender bias score and an in-group threshold rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The word-level filtering process presented as word clouds can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2 Annotation Scheme The annotation scheme is designed for gender bias probing and mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For gender bias probing, the annotators are required to provide the follow- ing information given a sentence: whether gender bias exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' if so, how the bias is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For gender bias mitigation, the corrected non-biased version of the biased sentences is also required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We Linguistic Non-biased Biased Corrected Biased Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Train Valid Test Train Valid Test Train Valid Test Word 724k 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='9k 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='7k 228k 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='8k 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='3k 265k 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1k 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='0k Dictionary 574k 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='4k 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1k 167k 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='4k 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='4k 191k 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='9k 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='5k Character 1,156k 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1k 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1k 358k 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2k 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='4k 417k 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='8k 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='9k Sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Length 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='952 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='397 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='473 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='837 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='087 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='214 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='839 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='853 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='939 Table 2: Linguistic Characteristics of the Corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Word, Dictionary, and Character separately denote the total Chinese word number, total unique Chinese word number, and total character number of the specific categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The sentence lengths are defined as the number of containing characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' further describe the annotation scheme details in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Existence and Categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The annotators are required to annotate whether the sentence is gender-biased (B) or non-biased (N) in contextual-level or word-level, and further clar- ify how the bias is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Given that our raw data is collected using gender-related keywords or from gender-related corpus, the samples annotated without gender bias are useful human-annotated negative samples for detecting gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' To addi- tionally provide information about gender bias cate- gorization, we classify gender bias types into three subtypes : (1) Gender Stereotyped activity and career choices (AC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (2) Gender Stereotyped de- scriptions and inductions (DI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' and (3) Expressed gender-stereotyped attitudes, norms and beliefs (ANB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The classification standard is inspired by (King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2021) and further summed up into the mentioned subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Bias Mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Annotators are also required to mitigate the gender bias of selected sentences while keeping the original semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We also ask our annotators to diversify the expres- sions if applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The major revision patterns can be summarized as follows: (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Replace the gender-specific pronouns with neutral pronouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Replace the gender-specific adjectives with neutral descriptions with similar semantics defini- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Add additional comments to neutralize the sentences which cannot be directly mitigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='3 Corpus Analysis In this section, we report the linguistic statistics of CORGI-PM as Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We design a balanced split to create the valid and test set considering the negative-positive ratio and bias subclass proportion in the global distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' As revealed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 22, we observe two major differences compared the de- biased samples with the original ones: longer and more diverse expressions (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' sentence length and vocabulary size of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We hypothesize that it 2We use the Jieba to parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' is due to human annotators’ intention to keep the semantic information unchanged and the sentence coherent while mitigating gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' They may use more conjunctions and longer descriptions com- pared to some gender-biased inherent expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' More details for quality managing and control can be referred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 3 Gender Bias Mitigation Challenges To provide a clear definition for automatic textual gender bias probing and mitigation tasks, we pro- pose corresponding challenges and standardize the evaluation protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We address two tasks, detec- tion, and classification, for gender bias probing and formalize the gender mitigation challenge as a text mitigation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1 Challenges of Detection and Classification We regard both the gender bias detection and clas- sification challenges as supervised classification tasks and evaluate them with metrics of consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The gender bias detection challenge can be regarded as a binary classification task, where the model is required to predict the prob- ability that a given sentence contains gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' As described in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2, biased samples are further categorized into one or more kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Therefore, we can address the gender classification challenge as a multi-label classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The precision, re- call, and F1-score are selected as the main metrics in these two challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Class-wise metrics and macro average summarized evaluation are required through both valid and test sets to show the perfor- mance of language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We finetune Chinese language mod- els from three representative different pretrained paradigms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', Chinese BERT, Electra, and XL- Net Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (2020), for both the detection and classification tasks by adding an additional dense prediction layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 3 We also provide GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2020) curie’s few-shot performance for both 3Pretrained models can be found at theHFL Anthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Model Metrics Classification (Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') Classification (Test) Detection (Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') Detection (Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') AC DI ANB Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' AC DI ANB Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' N B Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' N B Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' BERT Precision .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='609 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='729 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='533 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='624 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='556 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='615 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='521 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='564 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='699 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='950 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='824 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='742 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='980 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='861 Recall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='594 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='665 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='543 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='601 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='493 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='652 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='585 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='577 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='971 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='591 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='781 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='985 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='662 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='823 F1-Score .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='602 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='695 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='538 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='522 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='633 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='551 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='567 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='813 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='729 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='771 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='846 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='790 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='818 Electra Precision .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='587 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='727 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='544 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='619 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='556 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='630 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='516 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='568 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='691 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='936 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='814 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='745 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='974 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='860 Recall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='758 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='687 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='386 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='610 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='680 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='685 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='373 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='579 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='961 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='570 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='766 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='983 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='656 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='820 F1-Score .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='661 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='706 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='451 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='606 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='656 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='433 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='567 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='804 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='708 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='756 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='848 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='784 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='816 XLNet Precision .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='587 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='696 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='523 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='602 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='544 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='589 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='527 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='553 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='713 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='928 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='820 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='772 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='959 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='865 Recall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='622 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='643 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='495 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='587 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='545 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='614 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='514 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='558 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='953 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='787 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='968 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='722 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='845 F1-Score .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='604 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='669 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='509 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='594 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='544 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='601 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='520 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='555 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='816 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='743 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='780 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='859 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='824 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='841 Curie Precision .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='695 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='907 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='537 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='622 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='887 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='009 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='506 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='763 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='665 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='714 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='635 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='825 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='730 Recall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='395 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='802 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='375 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='524 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='395 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='804 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='403 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='576 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='825 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='700 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='975 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='584 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='780 F1-Score .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='504 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='851 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='019 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='458 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='508 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='852 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='019 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='460 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='656 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='736 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='696 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='769 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='684 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='727 Table 3: Baseline Results for Gender Bias Detection and Classification Tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The overall metric refers to Marco average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The model names and abbreviations refer to § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Categorical definitions refer to § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' aa Metrics Models BLEU METEOR ROUGE-L Human Evaluation Recall Precision F1 Coherence Gender Bias Davinci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='776 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='879 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='203 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='211 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='205 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='96 Ada .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='288 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='429 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='407 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='180 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='250 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='13 Babbage .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='359 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='504 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='716 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='310 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='432 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='69 Curie .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='364 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='506 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='692 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='316 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='434 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='20 Table 4: Baseline Results for Gender Bias Correction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Metrics details can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' * suggests using the model in zero-shot paradigm and the others refers to fine-tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' the detection and classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Baseline re- sults of detection and classification show that the classification task is challenging, and there is room for performance improvement in detecting gender bias in CORGI-PM, as revealed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2 Challenge of Mitigation Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The gender bias mitigation challenge can be regarded as a natural language generation task, where the model is asked to generate the cor- rected version of biased sentences with the human- annotated ones as references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We test the GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2020) on CORGI-PM in fine-tune experiment setting with three different parameter scales, which are Ada(350M), Babbage(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='3B), and Curie(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='7B), and Davinci(175B) in zero-shot experiment setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We only provide zero-shot results for Davinci because it is the only released GPT-3 editing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' More implementation and evaluation details are intro- duced in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We provide both human evaluation and automated metrics for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 4 re- veals that LMs can learn the annotation pattern of mitigating gender bias, and the zero-shot editing model shows competitive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The obser- vation that fine-tuned Babbage outperforms much larger zero-shot Davinci in the human evaluation, and ROUGE-L reveals that CORGI-PM has the potential to be used as strong supervision of the gender bias mitigation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We notice that Davinci tends to apply more conservative edits compared to fine-tuned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' As a result, the sentences edited by Davinci keep most of the original sentences and always only change pronouns and adjectives from the original sentences, which benefits precision focusing automatic metrics like BLEU (Papineni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2002), and METEOR (Agarwal and Lavie, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The performance difference between human evaluation and automatic metrics reveals the writ- ing style difference between human and language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 4 Conclusion We propose CORGI-PM, the first Chinese human- annotated corpus for both gender bias probing and mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We also address definitions and evalua- tion metrics for three challenges based on CORGI- PM and test the performances of state-of-the-art language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Our proposed challenges can serve as benchmarks for measuring the ability of language models to detect, classify, and mitigate textual gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Experiments show that our sen- tences with fine-grained subclass labels can assist the language models in gender bias probing, whilst our parallel human-written debiased data can serve as strong supervision of the generative language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' In summary, we imply future work utiliz- ing CORGI-PM would be benefited the topic of NLP for gender bias probing and mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Limitations There are several major limitations in this research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Due to the high requirement of annotators for annotating gender-biased sentences and correct- ing such sentences, we only choose annotators with higher education, which may lead to potential cog- nitive bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' In addition, we only conduct limited implementations and experiments of testing widely- used Chinese language models’ performance in our new challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' More language models and tech- niques can be further explored in our challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Ethics Statement We carefully consider the ethical implications dur- ing the collection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The collection of our corpus CORGI-PM sentences only relies on public available corpora for research purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We have acknowledged the potential usage of our dataset as well as related privacy issues to the annotators and received confirmations before the annotation was initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' References Abhaya Agarwal and Alon Lavie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 2007.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' ERNIE BERT XLNet ELECTRA ERNIE BERT XLNet ELECTRA 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='015 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='021 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='14 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='0 Figure 2: Word-level Gender Bias Comparison of Career Words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' A Gender Bias Analysis of Chinese Language Models A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1 Evaluation Method and Data Sets We conduct experiments to explore gender bias con- tained in widely-used Chinese language models for research and industrial use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We employ the method Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Jiao and Luo (2021) pro- posed to assess gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The gender bias score for a word is calculated by ⃗w · ( ⃗ she − ⃗he)based on its word vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' A positive value means the word is more relevant to females, while a negative value means the word is more relevant to males.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The higher the absolute value of the gender bias score, the more biased the word indicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' propose a big benchmark con- taining a dataset specifying the existing Chinese ca- reer words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Zhu and Liu propose AGSS, a manual- created Chinese word-level adjective list containing gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' To measure gender bias contained in the language models, we first calculate gender bias scores of words in the word list provided (Srivas- tava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Zhu and Liu, 2020) according to the projection method Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Jiao and Luo (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We compare the career and adjec- tive word gender bias score vectors to get the ob- servations of LMs’ influence on word-level learned gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' To make the observations more clear, we further apply the sign function to the career and adjective word gender bias score vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The sim- ilarity function used for the heatmaps is Pearson similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We conduct described comparison of adjectives between AGSS as a golden standard (Zhu and Liu, 2020), Ernie (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019), Chinese Word Vectors trained by mixed corpus (Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2018), AGSS ERNIE BERT CWV XLNet ELECTRA AGSS ERNIE BERT CWV XLNet ELECTRA 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='023 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='0 Figure 3: Word-level Gender Bias Comparison of Adjectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' CWV denotes the Chinese Word Vectors trained using mixed- large corpus proposed by Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='0 Figure 4: Word-level Gender Bias Comparison of Adjectives of Language Models Pre-trained by Different Corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' PDN denotes the People’s Daily News Corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' and Chinese-XLNet, Chinese-Bert, and Chinese- Electra proposed tecui-etal-2020-revisiting to pro- duce Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We conduct described comparison of career words between Ernie (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019), and Chinese-XLNet, Chinese-Bert, and Chinese- Electra proposed tecui-etal-2020-revisiting to pro- duce Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The described experiments on career words is not conducted with the Chinese Word Vec- tors trained by mixed corpus, because an observing number of career words are missing in its dictio- nary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We don’t provide a golden standard vector (Sri- vastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2022) since they didn’t provide a manual gender bias analysis about the career words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We also conduct described comparison on adjec- tives in Chinese Word Vectors pre-trained by dif- ferent corpus, including Mixed-large corpus, Peo- ple’s Daily News, Zhihu QA dataset, Weibo, and Chinese literature dataset to produce Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 4 and an- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='alyze the learned gender bias difference caused by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(a) Ch-Ernie-Man-Adj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(b) Ch-Ernie-Woman-Adj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(c) Ch-Ernie-Man-Career ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(d) Ch-Ernie-Woman-Career ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(e) En-Ernie-Man-Adj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(f) En-Ernie-Woman-Adj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(g) En-Ernie-Man-Career ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(h) En-Ernie-Woman-Career ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(i) Ch-XLNet-Man-Adj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(j) Ch-XLNet-Woman-Adj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(k) Ch-XLNet-Man-Career ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(l) Ch-XLNet-Woman-Career ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(m) En-XLNet-Man-Adj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(n) En-XLNet-Woman-Adj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(o) En-XLNet-Man-Career ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='(p) En-XLNet-Woman-Career ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='Figure 5: Example Word Cloud Analysis of Ernie and Chinese-XLNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Ch denotes Chinese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' En denotes words’ English translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Man and Woman separately denote words with embedding closer to man and woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Adj denotes adjectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Career denotes career words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' using different datasets for pretraining the language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2 Discussion There exists observing gender bias in the open- source Chinese language models, especially in Ernie and Chinese Word Vectors according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We hypothesize that the observation is highly related to the corpus used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' claim that their used corpus is a combination of Chine- seWiki, and some other universal Chinese datasets, including encyclopedia, news, and QA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' In sharp contrast, Ernie and Chinese Word Vectors use corpus, which contains sentences from literature, forum, and other social media, which may lead to a gender-biased model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 4, People’s Daily News, and Chinese literature corpora contain observing gen- der bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The observation indicates that researchers should be more careful about using literature data while training a language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We also hypoth- esize that this is caused by the literature corpus and People’s Daily News, which contains more descriptive expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' B Corpus B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='1 Word Cloud Analysis We provide word cloud analysis of Ernie and Chinese-Electra in the section about adjectives and career words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' More available word cloud analy- sis will be available in our public repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The words are ranked according to the absolute value of their gender bias score calculated along the method used by Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Jiao and Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' There is a noticeable word-level gender stereotype according to the word cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For example, a man is robust and a woman is motherly, a man is suitable for a fitness instructor and a woman is suitable for a choreographer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We also conduct word cloud anal- ysis for language models pre-trained by different corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='2 Quality Monitoring and Control We used a standardized operating method and edu- cated our annotators to achieve high-quality anno- tations as follows: (1).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='钣金工 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='舞美设计 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='疾病控制医师文字编辑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='老师 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='卫生检疫人员were only qualified to do the annotation if they ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='went through several societal (King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019) and computer science research works (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2018) about gender bias before the annotation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' All annota- tors held a bachelor’s degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Waseem points out that expert annotators are more cautious and can improve the corpus quality with a large margin, which proves the necessity of our training proce- dure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We also kept the number of male and female annotators equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Gender Equality of Raw Corpus In the raw data collection procedure, we keep the num- ber of man-related keywords and woman-related keywords equal and make the number of samples recalled according to different keywords balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' As a result, the raw data and the final data should hold gender equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Annotation Procedure Our annotation procedure is separated into two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' In the first stage, annotators are encouraged to not enter any samples that they are not certain about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' In the second stage, we have annotators cross-checking annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We did not enter any contradictory samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Inter-annotator Agreement Given the domain and purpose of the dataset, we want to build the dataset as high quality as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Af- ter an initial annotation round with 6 annotators, we also report inter-annotator agreement in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' to verify annotation reliability, where the IAA among three annotators on bias classification, de- tection, and mitigation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='802, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='935, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='987, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Classification Detection Mitigation IAA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='987 Table 5: Inter-Annotator Agreement (IAA) C Implementation Details For gender bias classification challenge, we used finetuned Chinese-BERT-wwm, Chinese- ELECTRA-180g-base, and Chinese-XLNet-base, (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2020), and the GPT-3 (Curie) in the in-context paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We first use the train set to save the multiple labeled examples in a document with a specific file ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Then we use the test sets to perform a classification query on the saved file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The processing time for the classification of gender bias is approximately 1 hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We calculated the precision, recall, and F1 score to analyze model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For gender bias detection challenge, we use the same baseline model set as in the classification challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We test the performance on both "yes" and "no" detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' The detection tasks also use the Classification endpoints of GPT3 (Curie), which requires more time compared to classification as we use a larger dataset for both training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For gender bias mitigation challenge, we did not provide experiment results of finetuning the largest Davinci (175B) GPT-3 on CORGI-PM be- cause of the cost and no observing performance gain comparing Curie and Babbage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For finetune experiment setting, we follow the tutorial of GPT-3 official API of the Completion Model and regard the ground truth edits provided by human annota- tors as the completion of the original sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For the zero-shot experiment setting, we apply GPT-3 editing model and set the instructions as "Eliminate the gender bias contained in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='" For metrics used, on the one hand, we conduct extensive human evaluations from both gender bias and coherence aspects on CORGI-PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For both gender bias and coherence, we shuffled the correc- tion results from human annotators and different models, and asked our annotators to grade the re- sults using the answer range from 1-not at all to 7-extremely gender biased/extremely fluent without the information of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' On the other hand, we provide the automated metrics result, includ- ing BLEU (Papineni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=', 2002), ROUGE-L (Lin, 2004), and METEOR (Agarwal and Lavie, 2007) of the models on CORGI-PM as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' BLEU is the earliest and most widely-used metric for translation and NLG tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' METEOR introduces WordNet and other external resources to improve the robustness of the BLEU-based metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' ROUGE pays more attention to recall compared to BLEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' In the gender bias correction task, we fine-tune Ada, Babbage, and Curie models of GPT3 and test the performance using the aforementioned metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Fine-tuning on the train set requires 31 minutes on Ada, 35 minutes on Curie, and 43 minutes on Babbage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For the generation process, each fine- tuned model required approximately 30 minutes to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' For the zero-show paradigm, we use "Eliminate the gender bias in the sentence" as the instruction and use Davinci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Compared with the fine-tuned model doing the sentence correction, the zero-shot paradigm requires more time (approxi- Original Ground Truth Gender-swapped 但是再坚强的你也是个女人吧。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 但是再坚强的你也有脆弱的时候吧。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 但是再坚强的你也是个男人吧。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (But you are still a woman, even if you are strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (But even the strongest of you have moments of vulnerability, right?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (But you are still a man, even if you are strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') 可怕可恨的是，有的女人自己也是这样 给自己定位的——没有反对，没有抗 争，有的只是心甘情愿、死心塌地遵照 执行。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 女人不应该这样给自己定位——没有反 对，没有抗争，有的只是心甘情愿、死 心塌地遵照执行。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 可怕可恨的是，有的男人自己也是这样 给自己定位的——没有反对，没有抗 争，有的只是心甘情愿、死心塌地遵照 执行。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (The scary thing is that some women themselves are so defined for themselves - no opposition, no resistance, there is only a willingness to follow it to the letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (A woman should not define herself in this way - there is no opposition, no resistance, there is only a willingness to follow it to the letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (The scary thing is that some men themselves are so defined for themselves - no opposition, no resistance, there is only a willingness to follow it to the letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') 那个时候，能驾驭金色高跟鞋的女人得 多摩登啊！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 那个时候，能驾驭金色高跟鞋的人得多 摩登啊！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 那个时候，能驾驭金色高跟鞋的男人得 多摩登啊！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (At that time, the woman who could handle the gold heels must be very modern!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (At that time, how modern must be the person who can handle the gold heels!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (At that time, the man who could handle the gold heels must be very modern!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') “鸿雁队”是清一色的“理工男”。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' “鸿雁队”是清一色的“理工生”。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' “鸿雁队”是清一色的“理工女”。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (The "Hongyan team" team of all men in STEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (The "Hongyan team" team of all student in STEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (The "Hongyan team" team of all women in STEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') Figure 6: Case Study of Nonsensical Sentences Created by Gender-swapped Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Original Sentence Edit Sentence 清洁阿姨一边扫地一边赞扬。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 清洁工一边扫地一边赞扬。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (The cleaning woman praised while sweeping the floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (The cleaners praised while sweeping the floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') 我，有时文静，有时却调皮得像一个男孩 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 我，有时文静，有时调皮。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (I, sometimes quiet, but sometimes naughty like a boy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (I, sometimes quiet, sometimes naughty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') 在小王眼里，李某高大帅气、温柔体贴， 而且风趣幽默，是一个十分优质的青年男 性。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 在小王眼里，李某身材高大、外表好看、温 柔体贴，而且风趣幽默，是一个十分优质的 青年。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (In the eyes of Wang, Li is tall and handsome, gentle and considerate, and funny, a very high-quality young male.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (In the eyes of Wang, Li is tall, good- looking, caring and gentle, and funny, a very high-quality young people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') 沙峰起伏，金光灿灿，宛如一座金山，像 绸缎一样柔软，少女一样娴静。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 沙峰起伏，金光灿灿，宛如一座金山，像绸 缎一样柔软，宁静。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (The sandy peaks are undulating and golden, like a golden mountain, as soft as silk and as serene as a maiden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (The sandy peaks are undulating and golden, like a golden mountain, as soft and serene as silk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') 我想要世界，而世界当时属于男人们。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 我想要世界，而世界当时属于男人们。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='评: 世界应当属于人们，与男女无关。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (I want the world, and the world then belonged to the men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (I wanted the world, and the world then belonged to the men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Comment: The world should belong to people, not to men and women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') 哎哟，果然每个追梦男人的背后，都有个 不世俗的后方！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 哎哟，果然每个追梦男人的背后，都有个不 世俗的后方！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content='评: 这种感慨是错误的，将男 女的家庭分工固定化，剥除女性就业的权 利，应予以鄙弃。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' (Oops, indeed, behind every dream- chasing man, there is an unsophisticated back!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') (Oops, indeed, behind every dream- chasing man, there is an unsophisticated back!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Comment: This is a wrong feeling that fixes the domestic division of labor between men and women and strips women of their employment rights, which should be despised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=') Change the Pronoun Change the Gender- specific Adjectives Add Comments Figure 7: Case Study of Mitigation Annotation Patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' mately 1 hour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' D Case Study As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 6, gender-swapped methods suffer from mitigating gender bias expressed by gender- specific descriptions and inductions, and expressed gender-stereotyped attitudes, norms and beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' As a result, gender-swapped methods may generate nonsensical sentences under certain circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' We also use the basic mitigation annotation pat- terns (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' These three major mitigation annota- tion patterns are not used exclusively in the annota- tion but optionally in combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} +page_content=' Except for the three mentioned patterns, we apply several other linguistic skills, including deleting gender-specific pronouns and replacing vehicles in gender-related metaphors, to mitigate the gender bias while keep- ing semantic information unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfifg0/content/2301.00395v1.pdf'} diff --git a/b9FAT4oBgHgl3EQfXx3V/content/tmp_files/2301.08536v1.pdf.txt b/b9FAT4oBgHgl3EQfXx3V/content/tmp_files/2301.08536v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f5ffcfcab79c63368bb6acade79bf38905ccf20 --- /dev/null +++ b/b9FAT4oBgHgl3EQfXx3V/content/tmp_files/2301.08536v1.pdf.txt @@ -0,0 +1,2994 @@ +RASTI 000, 1–20 (2022) +Preprint 23 January 2023 +Compiled using RASTI LATEX style file v3.0 +Overcoming Separation Between Counterparts Due to Unknown +Proper Motions in Catalogue Cross-Matching +Tom J. Wilson1★ +ID +1School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +To perform precise and accurate photometric catalogue cross-matches – assigning counterparts +between two separate datasets – we need to describe all possible sources of uncertainty in object +position. With ever-increasing time baselines between observations, like 2MASS in 2001 and +the next generation of surveys, such as the Vera C. Rubin Observatory’s LSST, Euclid, and +the Nancy Grace Roman telescope, it is crucial that we can robustly describe and model +the effects of stellar motions on source positions in photometric catalogues. While Gaia has +revolutionised astronomy with its high-precision astrometry, it will only provide motions for +≈10% of LSST sources; additionally, LSST itself will not be able to provide high-quality +motion information for sources below its single-visit depth, and other surveys may measure +no motions at all. This leaves large numbers of objects with potentially significant positional +drifts that may incorrectly lead matching algorithms to deem two detections too far separated +on the sky to be counterparts. +To overcome this, in this paper we describe a model for the statistical distribution of +on-sky motions of sources of given sky coordinates and brightness, allowing for the cross- +match process to take into account this extra potential separation between Galactic sources. +We further detail how to fold these probabilistic proper motions into Bayesian cross-matching +frameworks, such as those of Wilson & Naylor. This will vastly improve the recovery of +e.g. very red objects across optical-infrared matches, and decrease the false match rate of +photometric catalogue counterpart assignment. +Key words: Algorithms – methods: statistical – catalogues – astrometry – proper motions – +Galaxy: kinematics and dynamics +1 +INTRODUCTION +Counterpart assignment, the merging of bandpass detections in two +(or more) datasets, enables a wide range of value-added science, +and is therefore a crucial aspect of many areas of astronomical re- +search. Fundamentally, we require the ability to answer the question +‘are these two detections observations of two different objects, or +two observations or the same astrophysical object?’ Thus, to pro- +vide accurate and precise cross-matches between two photometric +catalogues, we require a complete description of all sources of sep- +aration between detections of a single astrophysical object. +Unfortunately for astronomers, there are many reasons for the +same source, detected by two different telescopes in different parts +of the world at different times, to have recorded positions that are +not perfectly aligned with one another. The first, and frequently +assumed only, contribution is that from the act of measuring the +position of the source on the detector image as part of the catalogue +creation process. This ‘centroid’ uncertainty is related to the size +★ Email: t.j.wilson@exeter.ac.uk; onoddil@pm.me +of the telescope and the wavelength of the observation, as well +as the atmospheric seeing, if applicable – all of which affect the +telescope ‘point spread function’ (PSF), as well as the signal-to- +noise ratio (SNR) of the detection, related to its brightness (e.g. King +1983). Wilson & Naylor (2017) highlighted an additional source of +positional shift that can affect detections in crowded fields: sources, +too close together on the sky to be resolved by the telescope, can +appear as a single observation, leading the fainter source to influence +the position of the (assumed singular) brighter object (sometimes +referred to as ‘classical confusion’; see also e.g. Hogg 2001). +Here we consider an extra source of apparent separation be- +tween detections: that of the physical motion of the source across +the sky. If the observations to be combined are sufficiently sepa- +rated in time, the ‘proper motion’ of sources introduces a drift in +the separation between consecutive measurements of the objects’ +locations. Thus for pairs of observations with significant baselines, +the proper motion-induced separations can become significant for +large enough numbers of objects that, if we failed to consider these +motions, we would decide the objects were too far apart to be coun- +terpart detections of one object, and fail to assign them properly. +© 2022 The Authors +arXiv:2301.08536v1 [astro-ph.SR] 20 Jan 2023 + +2 +Tom J. Wilson +For probabilistic cross-matching algorithms this problem of +object drift is further compounded. It is not only objects with sig- +nificant proper motion that suffer, those few objects with motions +large enough to render them completely incompatible with the hy- +pothesis that the two detections are counterparts. All objects, even +those with relatively small motions, are affected. Any motion on +the same length scale as the astrometric precisions will impact the +derived match confidence, and potentially render quoted match or +non-match probabilities meaningless. This issue of your chosen +model completely encapsulating the information contained within +your data (or not) is often referred to as ‘model (mis)specification’. +Therefore, the effect of source motion must be accounted for, even +if not so extreme as to completely move an object beyond its prior +position. If it is not taken into account, users of any resulting cross- +match tables may not be able to put trust in the quoted match +likelihoods and be able to take reliable, high-confidence cuts of the +merged datasets. +This effect is particularly important for the upcoming Vera +C. Rubin Observatory’s Legacy Survey of Space and Time (LSST; +Ivezić et al. 2019), for a few key reasons. First, it will operate from +∼2025-2035, and thus have a two or three-decade baseline to the +numerous surveys that operated during the 2000s and 2010s, such +as 2MASS (Skrutskie et al. 2006) or SDSS (e.g. York et al. 2000). +And second, it will lack measured proper motions for almost all of +its sources for a large fraction of its survey lifetime. In part this is +because the survey will require a multi-year baseline before reliable +proper motions can be derived, but more simply because most ob- +jects within the full LSST catalogue will be below the completeness +limit of the single-visit images. Even in this specific case, with Ru- +bin’s high-fidelity time-series capabilities, proper motions will only +ever be available for objects that appear in multiple images, which +sets the proper motion magnitude limit much higher than that of +inclusion in the full coadd catalogue. Worse still, the sheer number +density of objects in the full LSST catalogue mean that up to 10 +LSST sources will be potential counterparts to every single oppos- +ing catalogue object, and the ‘re-shuffle’ of objects, even of order +the precision of the measured positions, may lead to false matches +being returned for a sizeable fraction of the catalogues. Thus, the +survey will be especially susceptible to this match misspecification +due to proper motion drift, primarily at faint magnitudes. Addition- +ally, other upcoming missions such as Euclid (Laureijs et al. 2011) +and the Nancy Grace Roman Space Telescope (Green et al. 2012) +will likely lack the multi-epoch capabilities that Rubin and LSST +offer, but still suffer the effects of decade-long time baselines back +to previous generations of deep surveys, such as SDSS or VISTA +(e.g. VHS, McMahon et al. 2013). +As time goes on, and we accumulate increasing numbers of +surveys we wish to combine to maximum scientific return, we will +increasingly no longer be able to ignore even relatively small lev- +els of apparent on-sky motion. One obvious solution is to use the +individual proper motions available through datasets such as the +Gaia (Gaia Collaboration et al. 2016) mission. These positions – +combined with the rate-of-change of position from the proper mo- +tions – can be ‘fast-forwarded’ through time, allowing for sources +to be placed in the epoch of the opposing catalogue, removing on- +sky drift as a factor in considering the separation between sources. +However, this is impractical for surveys such as LSST for a couple +of reasons. +First, and simplest, is that proper motions are not available for +the entire Gaia catalogue. Something like 20% of sources in the +early Data Release 3 (eDR3; Gaia Collaboration et al. 2021; Lin- +degren et al. 2021) do not have the five- or six-parameter solutions +necessary to include proper motions, and a not insignificant frac- +tion of those that have quoted proper motions have uncertainties +that render the quoted values useless for any meaningful position +projection. Second, this would result in needing to run two Gaia- +to-other-catalogue matches, merge the most likely of those matches +in turn, and then run an internal Gaia-Gaia look up to get the in- +ner join of the two catalogues. This would significantly affect the +quality of the resulting matched datasets, with probabilistic cross- +matching processes not being able to provide the proper probability +of sources in the two ‘other catalogue’ datasets matching. Third, +and most crucial, is the dynamic range consideration. Gaia is, for +all its superb data, a relatively bright survey – at least by LSST +standards. It also lacks coverage against longer wavelength surveys, +where Galactic extinction is less oppressive. LSST will, with its +∼7 magnitude fainter completeness limit, include at least an order +of magnitude more stars (Ivezić et al. 2019), and VISTA, as an +example infrared (IR) catalogue of consideration as an ancillary +dataset to extend LSST information with, will have little overlap +with Gaia due to differing wavelength coverage. Other surveys with +deeper completeness limits, such as Euclid, Roman, and SDSS, will +also suffer significant numbers of matches beyond the Gaia com- +pleteness limit, with neither offering reliable proper motions at 21st +magnitude or fainter. Thus, even if we did decide to peg Gaia as +our gold standard, this would leave perhaps 9 out of every 10 LSST +Galactic sources without a proper motion match. Those LSST ob- +jects, and many others in other catalogues, would be in need of a +separate, and worse, cross-match once we had handled those few +objects with a Gaia proper motion. +If we cannot trust matches between observations with signifi- +cant time between observations, and we cannot necessarily use an- +cillary datasets with measured proper motions, how can we recover +robust catalogue counterpart assignments through cross-matching? +Naively, we might think that we can ‘re-center’ the distribution of +offsets, by subtracting some mean separation between all of our +counterpart pairings to account for the drift of our objects. How- +ever, different average proper motions across the dynamic range of +the two catalogues would cause further systematics, affecting the +distribution of counterpart separations. One may also think to use a +Galactic model that calculates stellar velocities, and hence provides +proper motions as viewed from Earth, for example the Besançon +model (Robin et al. 2003). However, as discussed in more detail +in Section 4.8, there are various reasons that these proper motions +do not provide robust enough statistics for the determination of a +statistical separation drift between two potential counterpart stars +during a probabilistic cross-match process. +Thus, in this paper we put forward a model to build a statistical +distribution of proper motions of sources, based on their Galactic +motions. Modelling all sources of motion a star orbiting the Galactic +center is subject to – its bulk circular orbit, and any ‘random’ scatter +of sources from e.g. stellar cluster interactions – combined with the +Sun’s motion, we are able to build a picture of the apparent motion +of the given object. To do so, we begin with the velocity of the +star as it orbits around the Galactic center. The conversion from +velocity – in units like km s−1 – to proper motion – in units like +arcsec yr−1 – is, roughly speaking, an inverse relation with distance. +As we detail later, instead of using distance directly, we intend to use +the brightness of an object as its more readily available surrogate, +accepting that this is only an approximation. +Hence, combining the apparent motions of a wide range of +objects at the same sky position and brightness we obtain all proper +motions such a source might have – faint M dwarfs close by would +have larger apparent motions than very intrinsically bright super- +RASTI 000, 1–20 (2022) + +Overcoming Separation Between Counterparts Due to Unknown Proper Motions +3 +giants, but all ‘types’ of object contribute to the spread of motion +drifts a source of this particular brightness could have. We are not +particularly interested in a precise reconstruction of Galactic orbital +dynamics – not being overly concerned with the details of spiral +arm dynamics, or the specifics of the Milky Way Bar shape, for ex- +ample. The use of the proper motions here is to ‘spread’ the motion +drift, the separation between potential counterparts, allowing for +the recovery, and increasing the reliability of match probability, of +these objects within a Bayesian cross-matching framework. There- +fore, the exact shape is less important than its central location and +width; so long as these match reality to a fraction of the precision +of the objects’ positions and the underlying distribution width to a +factor 1.5 − 2, the model has served its purpose. The probability of +two stars being counterpart ranges over many orders of magnitude, +and hence the resulting probability density functions (PDFs) we +derived to model the statistical proper motions having widths, and +overall PDF heights, correct to a factor two is sufficient to improve +counterpart recovery. We also desire computational simplicity, and +thus speed, over an overly prescriptive or detailed exact model of +the Galaxy, as this model must fit within a wider counterpart assign- +ment framework and be able to be run on-the-fly for some arbitrary +sets of sky positions and brightnesses. +Once we have constructed our distribution of theoretical proper +motions, we can then consider their effect on our potential cross- +match pairings. Here we can, in a similar way to how we would han- +dle known Gaia proper motions, translate one object’s position into +the epoch of the second catalogue observations, and consider the +additional separation caused by the motion of the object. We must +then test all modelled proper motions, with appropriate weighting, +ultimately accounting for these potential additional time-based sep- +arations in answering the question of whether these two detections +are two physical sky objects or one source viewed twice in time. +1.1 +Paper Layout +This paper is split into two parts. First, we detail the construction of +a simple analytic model for the statistical distribution of potential +proper motions of a source of a given magnitude and sky coordi- +nates. In Section 2 we detail the constituent parts necessary to build +the model of proper motions. We describe the process of building +the distributions in Section 3, while Section 4 discusses the preci- +sion and accuracy of the model at various Galactic sightlines and +brightnesses. Here we will return to Gaia, using its high-precision +stellar proper motions across many Galactic sightlines to evaluate +and corroborate our model distributions. +The second part of this work describes how to include this +unknown proper motion distribution – or any distribution of proper +motions, theoretical or poorly constrained yet detected – in the +cross-matching process. We discuss the mathematical framework +necessary for including proper motion drift in the evaluation of +the separation between potential counterpart detections in Section +5. We also touch upon how to use these statistical distributions of +proper motion drift as a discriminator between stars and galaxies. +Concluding remarks are given in Section 7. +In Appendix A we outline the various coordinate systems and +derive the transformation matrices used throughout this work, while +in Appendix B we detail the inclusion of the proper motion PDFs +within a probabilistic cross-match astrometric separation likelihood +framework. +2 +CONSTRUCTING THE PROPER MOTIONS +We need to build a model to describe the observed motions of +sources across the sky. There are, essentially, three components that +matter: first, the ‘peculiar’ motion of the Sun itself; second, the +expected velocity of a source moving with the Galactic rotation; +and third, the random velocities of the Galactic sources; these will +be discussed individually. First, we must consider how we will build +this model. +As the model involves consideration of the Galactic rotation +(and we will see later that source random motion is location depen- +dent), we will require a description of the position of the source in +the Galaxy. The first two components are easy: sky coordinate in +Galactic coordinates (converting Equatorial 𝛼 and 𝛿 by rotation to +longitude 𝑙 and latitude 𝑏, if necessary). The only other component +we would need is a distance, or parallax; however, if we have paral- +lax we likely have a unique proper motion, as these are generally fit +for simultaneously, and hence we use the next best proxy: magni- +tude. This will blend several ‘types’ of source together (e.g. dwarfs +and giants of the same brightness are at different distances). We +will see that while this might introduce extra scatter in the proper +motion drifts, our models match Gaia proper motions at magnitude +cuts well – and indeed account for the fact that we do not know the +type of any individual source from its photometry a priori! +However, we do need some distance metric, and hence we turn +to the TRILEGAL simulations (Girardi et al. 2005) to provide a +theoretical magnitude-distance relation. Thus, while our catalogue +sources have their proper motions built as a function of sky coordi- +nates and photometric brightness, our model is coordinate/distance +based. In the following sections we describe how we formulate a +description of the observed proper motion of a set of sources based +on their given parameters. +2.1 +Solar Peculiar Motion +The first component in the Galactic motion is the unique velocity +of the Sun, relative to the local standard of rest (LSR). The Sun’s +motion through the Galaxy will induce a ‘secular’ parallax effect +(i.e. distance-dependent, albeit non-periodic, as a trigonometric par- +allax would be) in the apparent movement of all other sources in +the sky, with opposite sign. Hence we need to know the Sun’s mo- +tion, relative to this ‘zero point’ motion, the LSR, defined in the +Heliocentric Cartesian coordinate frame, (𝑈, 𝑉, 𝑊) – velocities +corresponding to the (𝑥, 𝑦, 𝑧) coordinate system. Here we use the +values of Schönrich et al. (2010): +𝑈⊙ = 11.1 km s−1 +𝑉⊙ = 12.2 km s−1 +𝑊⊙ = 7.3 km s−1. +(1) +2.2 +Galactic Rotation +The main component of our model that will dictate the Galaxy-wide +observed motions of sources is that of the Galactic rotation, and +the stellar streaming. Descriptions of this motion go back to Oort +(1927), with the Oort constants describing the motion of stars on +closed orbits around the Galaxy. However, through modern kine- +matic surveys, obtaining the three-dimensional velocities and in- +dependent distances to a host of well-characterized objects, it is +possible to directly measure the rotation curve of the Milky Way. +RASTI 000, 1–20 (2022) + +4 +Tom J. Wilson +Thus, we can derive the average tangential velocity of sources or- +biting the Galactic center at a given radius, here following Model 3 +of Mróz et al. (2019). +Obtaining the rotational velocity Θ at a given Galactocentric +radius 𝑅𝑐, we can transform this Galactocentric Cylindrical az- +imuthal velocity into a Galactocentric Cartesian coordinate frame +and subtract the Solar peculiar and LSR motion, obtaining 𝑈1, 𝑉1, +and 𝑊1 (Mróz et al. 2019, equations 5-7). To do so, we use the +transformation +T𝑡 = +����� +� +𝑅2 +𝑐+𝑅2 +⊙−𝑑2 +ip +2 𝑅𝑐 𝑅⊙ +𝑑ip +𝑅𝑐 sin(𝑙) +0 +− 𝑑ip +𝑅𝑐 sin(𝑙) +𝑅2 +𝑐+𝑅2 +⊙−𝑑2 +ip +2 𝑅𝑐 𝑅⊙ +0 +0 +0 +1 +����� +� +; +(2) +here 𝑅⊙ is the Solar Galactocentric Cylindrical radius, 𝑑ip is the +in-plane distance from the Sun to the particular location, and 𝑙 is +Galactic longitude – see Appendix A1.1 for details. Deviating from +Mróz et al. (2019), we set (𝑈𝑠, 𝑉𝑠, 𝑊𝑠), the non-circular motion of +the source, all to zero, and hence have +𝑈1 = Θ(𝑅𝑐) × +𝑑ip +𝑅𝑐 +sin(𝑙) − 𝑈⊙ +𝑉1 = Θ(𝑅𝑐) × +𝑅2𝑐 + 𝑅2 +⊙ − 𝑑2 +ip +2 𝑅𝑐 𝑅⊙ +− 𝑉⊙ − Θ⊙ +𝑊1 = −𝑊⊙. +(3) +Once we have the Galactocentric Cartesian components of the rota- +tional velocity, relative to the Sun, we can transpose into the in-plane +Heliocentric radial and tangential velocities, +𝑣𝑑 = 𝑈1 cos(𝑙) + 𝑉1 sin(𝑙) +𝑣𝑙 = −𝑈1 sin(𝑙) + 𝑉1 cos(𝑙), +(4) +along with 𝑣𝑧 = 𝑊1, since the two axes are still in alignment. Finally, +once we have appropriate Heliocentric Cylindrical velocities, we can +construct our latitudinal velocity as +𝑣𝑏 = 𝑣𝑧 cos(𝑏) − 𝑣𝑑 sin(𝑏) +(5) +and relate the Heliocentric longitudinal and latitudinal velocities to +their proper motions through +𝜇𝑙∗ ≡ 𝜇𝑙 cos(𝑏) = 𝑘 × 𝜋𝑣𝑙, +(6) +𝜇𝑏 = 𝑘 × 𝜋𝑣𝑏, +(7) +where 𝜋 is the parallax of the source. As our distances come from +Galactic models, we assume they are not subject to any observational +bias or uncertainty, and simply treat 𝜋−1 = 𝑑. The factor 𝑘 describes +the translation from units of km s−1 kpc−1 to mas yr−1, and is given +by 𝑘 = 0.2108 mas yr−1 km−1 s kpc. +In practice, the TRILEGAL simulations do not provide either +distance or parallax, but provide its distance modulus. Hence, to +obtain a (three-dimensional) distance 𝑑 in kpc, we invert the absolute +magnitude equation: +𝑑 = 10−3 × 100.2(𝑚−𝑀)+1 +(8) +where 𝑚 − 𝑀 is the distance modulus. +2.3 +Asymmetric Drift Velocity +The above equations describe the average Galactic rotation velocity +around the center of the Galaxy. However, objects have other sources +of velocity that impact their observed proper motions, and this leads +to a deviation away from the expected velocity, and thus proper +motion. This is termed the asymmetric drift velocity, and essentially +controls how much of the theoretical velocity a source should have +is taken by other, random motions. Thus, we must include this +component in the velocities. +We model three Galactic components (see Sections 2.4.1-2.4.3 +for more details of their construction) in our simulations: the Galac- +tic thin and thick discs, and a single Galactic (outer) halo. Each +of these is given their own asymmetric drift, as a measure of the +levels to which their motions are different from ‘pure’ streaming +motion. We assume the thin disc of the Galaxy has an azimuthal +asymmetric drift velocity of 10 km s−1 (Robin et al. 2003). As Mróz +et al. (2019) used Classical Cepheids in the derivation of their ro- +tation curve, we also assume the rotation curve is valid for the thin +disc and already folds in any drift velocity, and therefore just need +to consider the relative drift velocities of the thick disc and the +halo. We use a thick disc drift velocity of 49 km s−1 (Pasetto et al. +2012a), and give the halo a drift velocity of 240 km s−1 (Golubov +et al. 2013), to essentially counteract the motion of the LSR, mod- +elling the halo as stationary relative to the Galaxy. We therefore use +𝑣𝑎,𝜙 = {0 , 39 , 230} km s−1 for the relative thin disc, thick disc, +and halo drift velocities, respectively. +For a given location in the Galaxy, our decomposition of the +drift velocity, from Galactocentric Cylindrical coordinates into He- +liocentric Cylindrical coordinates, is given by the transformation +𝒗′drift = T𝑐 𝒗drift, +(9) +with +T𝑐 = +����� +� +𝑅2 +𝑐+𝑑2 +ip−𝑅2 +⊙ +2𝑅𝑐𝑑ip +𝑅⊙ +𝑅𝑐 sin(𝑙) +0 +𝑅⊙ +𝑅𝑐 sin(𝑙) +− +𝑅2 +𝑐+𝑑2 +ip−𝑅2 +⊙ +2𝑅𝑐𝑑ip +0 +0 +0 +1 +����� +� +(10) +and +𝒗drift = �� +� +0 +𝑣𝑎,𝜙 +0 +�� +� +, +(11) +with 𝑣𝑎,𝜙 taking on any one of the three given drift velocities above, +depending on which component of the Galaxy is being considered. +For details on the derivation of this transformation (rotation and +mirror) matrix, see Appendix A1.2. +2.4 +Velocity Dispersion +The asymmetric drift velocity suggests that some of the motion +that ought to be used by a given source in its rotation around the +Galaxy is being used otherwise, in a random component. Hence, +a collection of sources in a particular part of the Galaxy will have +some spread of their velocities around some mean value. Thus, to +be able to model our collection of sources in the Galaxy we require +a description of the dispersion of the velocities. +Each component of the Galaxy modelled – thin and thick discs, +and halo – have their own prescription of velocity dispersion. In +addition, when simulating sources, we do not initially know to +which component to assign a given simulated object. Therefore we +simply generate a set of proper motion realisations for each of the +three components, weighted according to their a priori density at +that location. Hence, in the following sections we also describe +the formulation of each component’s density profile, which are +simply re-normalised by the sum of their densities to provide a prior +probability, used as the weight for the proper motion distribution. +RASTI 000, 1–20 (2022) + +Overcoming Separation Between Counterparts Due to Unknown Proper Motions +5 +We currently do not consider the Bulge, a common component +of Galactic simulation models such as TRILEGAL, in our proper +motion model. We chose to ignore this additional component for this +initial, exploratory model, focussing on relatively bright sources, +with detected Gaia proper motions to compare and verify our model +against. This should mean they are sufficiently far from the Galactic +center to not be influenced by the Bulge, avoiding the complexities +that the inner region of the Galaxy impose on velocities of orbiting +stars. However, with the upcoming LSST survey and the need to +model much fainter, more distant objects in the next few years, +we will investigate a robust Galactic Bulge/Bar model for inclusion +within this proper motion framework. We simply conclude, for now, +that it would be easy to add additional components, and we could +model a simple Bulge component after e.g. Jackson et al. (2002). +Once a given Galactic component has its dispersion vector – +its covariance matrix – in the Heliocentric Cylindrical coordinate +system, then a realisation is drawn from a multivariate normal, given +by +𝒗noisy,𝑖 ∼ N (𝒗 − 𝒗′drift,𝑖, 𝚺′𝑖) +(12) +where 𝑖 ∈ {thin, thick, halo} and 𝚺′𝑖 is the Cylindrical frame rotated +covariance matrix of the 𝑖th component. +2.4.1 +Thin Disc +The thin disc is modelled as an exponentially decaying density +profile with given radial and vertical scale heights, as per Jurić et al. +(2008) and Ivezić et al. (2008): +𝜌(𝑅𝑐, 𝑧) = Γ exp +� +− 𝑅𝑐 − 𝑅⊙ +𝑙thin +− 𝑧 + 𝑧⊙ +ℎthin +� +, +(13) +with 𝑅⊙ = 8.09 kpc (Mróz et al. 2019), 𝑧⊙ = 25 pc (Jurić et al. +2008); Γ is an irrelevant normalisation constant, used simply to +explicitly cancel in the re-normalisation from density to weighting. +The radial and vertical scale lengths we use are the bias-corrected +values calculated by Jurić et al.: 𝑙thin = 2.6 kpc, and ℎthin = 0.3 kpc. +The dispersion vector for the thin disc is based on observations +of RAVE stars from Pasetto et al. (2012b). However, the nature of the +observations limit their calculation of covariances to approximately +1 kpc from the Sun, and we need to extrapolate these dispersions +out to perhaps five times that distance. Thus we turn to Amendt & +Cuddeford (1991) for relations between the various (co-)variances +in the dispersion vector. +First, we assume that the variance in the vertical direction, 𝜎2𝑧𝑧, +scales with radial distance in the mid-plane of the Galaxy: +𝜎2 +𝑧𝑧 (𝑅𝑐, 𝑧 = 0) = 𝜎2 +𝑧𝑧(0, 0) exp +� +− 𝑅𝑐 +𝑙thin +� += 𝜎2 +𝑧𝑧(𝑅′⊙, 0) exp +� +− 𝑅𝑐 − 𝑅′⊙ +𝑙thin +� +. +(14) +As discussed by Amendt & Cuddeford, this scaling relation is also +sometimes assumed for 𝜎2 +𝑅𝑐𝑅𝑐, but a second, valid formalism can +be used where the rotation curve of the Galaxy is flat – which it +should be safe to assume given the very small gradient from Mróz +et al. (2019) for most of the Galaxy. This formalism is based on a +constant Toomre (1964) local stability parameter, and gives +𝜎2 +𝑟𝑟 ≡ 𝜎2 +𝑅𝑐𝑅𝑐 ∝ 𝑅2 +𝑐 exp +� +−2 𝑅𝑐 +ℎthin +� +. +(15) +Hence we use1 +𝜎2 +𝑟𝑟 (𝑅𝑐, 0) = 𝜎2 +𝑟𝑟 (𝑅′⊙, 0) +� 𝑅𝑐 +𝑅′⊙ +�2 +exp +� +−2 𝑅𝑐 − 𝑅′⊙ +ℎthin +� +. +(16) +For the vertical extrapolation, we assume a local Taylor expan- +sion to first order (i.e. we extrapolate linearly to above and below +the plane, from 𝑧 = 0). We limit this extrapolation to the inner one +kiloparsec of the plane, and assume a constant dispersion beyond +that, and hence: +𝜎2 +𝑧𝑧(𝑅𝑐, 𝑧) ≃ 𝜎2 +𝑧𝑧(𝑅𝑐, 0) + min (1 kpc, |𝑧|) 𝜕𝜎2𝑧𝑧 (𝑅𝑐, 0) +𝜕|𝑧| +(17) +𝜎2 +𝑟𝑟 (𝑅𝑐, 𝑧) ≃ 𝜎2 +𝑟𝑟 (𝑅𝑐, 0) + min (1 kpc, |𝑧|) 𝜕𝜎2𝑟𝑟 (𝑅𝑐, 0) +𝜕|𝑧| +. +(18) +Using the Pasetto et al. data in the range 8.2 kpc ≤ 𝑅𝑐 ≤ +8.8 kpc, −0.5 kpc ≤ 𝑧 ≤ 0.5 kpc, we find +𝜎2 +𝑧𝑧(𝑅′⊙, 0) = 243.71 km2 s−2, +𝜕𝜎2𝑧𝑧(𝑅𝑐, 0) +𝜕|𝑧| += 306.84 km2 s−2 kpc−1, +𝜎2 +𝑟𝑟 (𝑅′⊙, 0) = 715.93 km2 s−2, +𝜕𝜎2𝑟𝑟 (𝑅𝑐, 0) +𝜕|𝑧| += 1236.97 km2 s−2 kpc−1. +(19) +Additionally, to be consistent with the data as derived by Pasetto +et al., we use 𝑅′⊙ = 8.5 kpc – note that 𝑅′⊙ ≠ 𝑅⊙ – for extrapolating +the dispersions. Here we have assumed their quoted location of the +Sun ‘in the range 𝑅 ∈ ]8.4, 8.6] kpc’ implies2 an assumed default +location in the middle of the bin. +We assume, following Amendt & Cuddeford (1991) and Val- +lenari et al. (2006), that +𝜎2 +𝑟𝑧 (𝑅𝑐, 𝑧) ≃ 𝜎2 +𝑟𝑧(𝑅𝑐, 0) + 𝑧 𝜕𝜎2𝑟𝑧 (𝑅𝑐, 0) +𝜕𝑧 +(20) +where the first term on the right hand side vanishes by symmetry at +𝑧 = 0, and the derivative is given by +𝜕𝜎2𝑟𝑧 (𝑅𝑐, 0) +𝜕𝑧 += 𝜆(𝑅) 𝜎2𝑟𝑟 (𝑅𝑐, 0) − 𝜎2𝑧𝑧(𝑅𝑐, 0) +𝑅𝑐 +. +(21) +Given no information on the radial dependence of 𝜆, we fix it +to the local value of 𝜆 = 0.6 (Amendt & Cuddeford 1991). In +cases where the linear extrapolation would result in a correlation +� +𝜌𝑟𝑧 ≡ +𝜎2 +𝑟𝑧 +𝜎𝑟𝑟 𝜎𝑧𝑧 +� +larger in absolute value than one, we force the +correlation back to either +1 or -1. +Following Vallenari et al. (2006), this is the only off-diagonal +term we consider for the thin disc covariance matrix. Finally, again +following the prescription of Amendt & Cuddeford, we assume that +the azimuthal and radial dispersions are related by a constant, and +hence use +𝜎2 +𝜙𝜙 = +−𝐵 +𝐴 − 𝐵 𝜎2 +𝑟𝑟, +(22) +with 𝐴 and 𝐵 the Oort (1927) constants, for all 𝑅𝑐 and 𝑧. We use the +Olling & Dehnen (2003) Oort constant values (their table 5, figure +1 Using 𝜎2𝑟𝑟 for the Galactocentric Cylindrical frame radial dispersion +component, to avoid the slightly clunky notation 𝜎2 +𝑅𝑐 𝑅𝑐 . +2 Inclusive of 8.6 kpc but exclusive of 8.4 kpc, equivalent to (8.4, 8.6]. +RASTI 000, 1–20 (2022) + +6 +Tom J. Wilson +6), as a function of intrinsic colour 𝐵 − 𝑉. Here we interpolate 𝐴 +and 𝐵 as a linear function of (𝐵 − 𝑉)0, fitting: +𝐴 = 1.94553 × (𝐵 − 𝑉)0 + 11.33138 +𝐵 = −2.63360 × (𝐵 − 𝑉)0 − 13.60611 +(23) +as shown in Figure 1 (left-hand panel). As we are using TRILEGAL +simulations, we require a conversion from available TRILEGAL +parameters to (𝐵 − 𝑉)0; for this we use the dwarf colour sequence +of Pecaut & Mamajek (2013). We fit a two-step function to the +intrinsic B-V colour as a function of effective temperature (with 𝑇 +in units of Kelvin), +(𝐵 − 𝑉)0 = +����� +����� +− 0.40739 + 5.07836 × +exp(−0.27083 × 𝑇/1000K) +𝑇 < 10000 K +− 0.35093 + 0.69012 × +exp(−0.08179 × 𝑇/1000K) +𝑇 ≥ 10000 K +(24) +as shown in Figure 1, right-hand panel. +Once we have all of the terms, we rotate the covariance ma- +trix from its Galactocentric cylindrical coordinate frame into the +Heliocentric coordinate system by +𝚺′ = T𝑐 𝚺 T𝑇 +𝑐 = T𝑐 +��� +� +𝜎2𝑟𝑟 +0 +𝜎2𝑟𝑧 +0 +𝜎2 +𝜙𝜙 +0 +𝜎2𝑟𝑧 +0 +𝜎2𝑧𝑧 +��� +� +T𝑇 +𝑐 +(25) +using the rotation matrix as defined in equation 10. +2.4.2 +Thick Disc +The formalism for the thick disc is very similar to that of the thin +disc. We also use the exponential decay model for the density profile, +albeit with different scale lengths: +𝜌(𝑅𝑐, 𝑧) = Γ 𝑓thick exp +� +− 𝑅𝑐 − 𝑅⊙ +𝑙thick +− 𝑧 + 𝑧⊙ +ℎthick +� +, +(26) +again following the Jurić et al. (2008) and Ivezić et al. (2008) +formalism, with 𝑅⊙ = 8.09 kpc and 𝑧⊙ = 25 pc again, and 𝑙thick = +3.6 kpc, and ℎthick = 0.9 kpc. In addition, the parameter 𝑓thick = +0.13 sets the relative densities of the thin and thick discs, and Γ +again is an arbitrary normalisation constant. +The thick disc dispersion vector uses the data from Pasetto +et al. (2012a), again following the same radial scaling relations for +the thin disc: +𝜎2 +𝑟𝑟 (𝑅𝑐, 0) = 𝜎2 +𝑟𝑟 (𝑅′⊙, 0) +� 𝑅𝑐 +𝑅′⊙ +�2 +exp +� +−2 𝑅𝑐 − 𝑅′⊙ +ℎthick +� +(27) +𝜎2 +𝜙𝜙(𝑅𝑐, 0) = 𝜎2 +𝜙𝜙(𝑅′⊙, 0) +� 𝑅𝑐 +𝑅′⊙ +�2 +exp +� +−2 𝑅𝑐 − 𝑅′⊙ +ℎthick +� +(28) +𝜎2 +𝑧𝑧(𝑅𝑐, 0) = 𝜎2 +𝑧𝑧(𝑅′⊙, 0) exp +� +− 𝑅𝑐 − 𝑅′⊙ +ℎthick +� +(29) +where, once again, we use 𝑅′⊙ = 8.5 kpc from Pasetto et al. (2012b), +assuming the two papers were jointly analysed and hence have the +same 𝑅′⊙, although neither paper in the series quote a specific +value. This time, we do not describe any vertical dependency of +the dispersions. Finally, we take the diagonal terms as presented +by Pasetto et al. (2012a) within or without the Solar circle as the +values approximately at (𝑅′⊙, 0), as given by their tables 3 and 4 +respectively, but ignore the off-diagonal terms, which are all within +≈ 1.5𝜎 of zero. +Exactly the same as with the thin disc, we rotate the Galac- +tocentric Cylindrical reference frame into Heliocentric Cylindrical +coordinates by +𝚺′ = T𝑐 𝚺 T𝑇 +𝑐 = T𝑐 +��� +� +𝜎2𝑟𝑟 +0 +0 +0 +𝜎2 +𝜙𝜙 +0 +0 +0 +𝜎2𝑧𝑧 +��� +� +T𝑇 +𝑐 +(30) +again using the cylindrical rotation matrix of equation 10. +2.4.3 +Halo +The halo Galactic component follows the density profile +𝜌(𝑅𝑐, 𝑧) = Γ 𝑓ℎ +����� +� +𝑅⊙ +√︂ +𝑅2𝑐 + +� +𝑧 +𝑞 +�2 +����� +� +𝑛 +, +(31) +again using the Jurić et al. (2008) and Ivezić et al. (2008) formalism, +with 𝑓ℎ = 0.0051, 𝑞 = 0.64, 𝑛 = 2.77. Γ is a normalising constant +once again. This parameterization of an inverse power law leads, +at 𝑅𝑐 = 0, 𝑧 = 0, to an infinite halo density, and hence unphysical +normalising weighting in the Galactic model. We therefore truncate +the halo density within the solar circle, 𝑅⊙, fixing it at its value at +𝑅𝑐 = 𝑅⊙ at smaller radii. This ought to be possible because the old +Galactic halo should be negligible in relative density by the solar +circle. +The dispersion vector for the halo is derived from King et al. +(2015), given in spherical coordinates. We take the full covariance +matrix from the closest radial bin from the ‘Equally Populated Bins’ +in their table 3 for a given set of (𝑅𝑠, 𝜙, 𝜃) parameters for a given +source, with the exception of their 𝑅𝑠 = 12 kpc bin. This bin gives +a covariance matrix that is not positive semi-definite, and hence we +ignore the off-diagonal terms for that individual bin. These have no +scaling applied to them and are taken exactly as quoted. +To rotate into the Heliocentric Cylindrical reference frame, we +use +𝚺′ = R𝑠𝑐 𝚺 R𝑇 +𝑠𝑐 +(32) +where3 +𝚺 = +��� +� +𝜎2𝑟𝑟 +Σ𝑟 𝜙 +Σ𝑟 𝜃 +Σ𝑟 𝜙 +𝜎2 +𝜙𝜙 +Σ𝜙𝜃 +Σ𝑟 𝜃 +Σ𝜙𝜃 +𝜎2 +𝜃 𝜃 +��� +� +, +(33) +R𝑠𝑐 = T𝑐R𝑠, +(34) +R𝑠 = �� +� +cos(𝛽) +0 +−𝑑/𝑅𝑠 sin(𝑏) +0 +1 +0 +𝑑/𝑅𝑠 sin(𝑏) +0 +cos(𝛽) +�� +� +, +(35) +and where Σ𝑟 𝜃 ≡ 𝜎2 +𝑟 𝜃, following the King et al. notation. R𝑠 +describes the rotation from Galactocentric Spherical coordinates to +Galactocentric Cylindrical coordinates, with T𝑐, as before, defining +the rotation from Galactocentric Cylindrical to Heliocentric Cylin- +drical coordinates. 𝛽 is defined as the angle between the spherical +radial vector and the Galactic plane (𝑏 = 0◦), with 𝑑 the three- +dimensional distance to the source in question, and 𝑅𝑠 the three- +dimensional Galactocentric distance to the star. For more details on +the derivation of this transformation matrix, see Appendix A1.3. +3 Once again, 𝑟 has been used instead of 𝑅𝑠 for notation’s sake, analogous +to the thin and thick disc notations. +RASTI 000, 1–20 (2022) + +Overcoming Separation Between Counterparts Due to Unknown Proper Motions +7 +0 +10 +20 +30 +40 +Teff / 103 K +0 +1 +2 +(B - V)0 / mag +0.0 +0.5 +1.0 +(B - V)0 / mag +−10 +0 +10 +20 +Oort Constant / km s−1 kpc−1 +A +B +Figure 1. Relationships used to derive the dependencies of 𝐴 and 𝐵 on intrinsic colour. Left: linear relationships between intrinsic B-V and Oort constants, +using the Oort constants as derived by Olling & Dehnen (2003). Right: a two-piece fit between effective temperature and intrinsic B-V, using the empirical +colour sequence of Pecaut & Mamajek (2013). +3 +CREATING A PROPER MOTION DISTRIBUTION +Now that we have described the model for simulating the velocity of +a source at a given position in the Galaxy, we can create a theoretical +distribution of sources. In a small sky coordinate window (in our +tests limiting ourselves to a few square degrees in the Galactic +plane, and relatively small polar cap latitude windows), we run +a TRILEGAL simulation in the center of the defined region. We +simulate either 1.5 million sources down to Gaia 𝐺 = 25, or as many +as we are allowed within 10 square degrees, the maximum limit of +the public simulation API. Distances for these simulated sources +are derived from their absolute distance modulus, and – with no +positional information in the simulated dataset – we randomly place +the sources within the rectangle defining the coordinate window. +For a given small magnitude range of sources, each source then +has its proper motions calculated as though it were from each of the +three Galactic components in turn, with some number of realisations +(𝑁 ≈ 1000) of the multivariate dispersion drawn. We then calculate +a weighted histogram of proper motions. For each source, 𝑗 = +1, 2, ..., 𝑀, where 𝑀 is the number of simulated stars (and thus +distances), the three Galactic components at the given Galactic +longitude, latitude, and distance have their respective weights 𝑤𝑖 𝑗 +(𝑖 ∈ {1, 2, 3}, or 𝑖 ∈ {thin, thick, halo}) calculated. The weighted +histogram is therefore built with each derived proper motion being +given weight 𝑤𝑖 𝑗/𝑁 (𝑁 the number of derived Galactic velocities +for the 𝑗th source, in each of the three components), for each of +the 3 × 𝑁 × 𝑀 derived proper motions, across all 𝑀 objects. The +histogram (which will contain 𝑀 weighted counts across all bins) +is then converted to a PDF. +For the purposes of visualisation and testing, we extract all of +the proper motions of Gaia eDR3 sources with flux SNRs greater +than five in the same coordinate window and magnitude range de- +fined for the simulated proper motions. Finally, one additional step +is taken, solely for the purposes of distribution comparison: we +convolve the model with the median uncertainty of the Gaia proper +motions in the dataset for this magnitude cut and sightline. This +allows for the inclusion of non-negligible Gaussian uncertainties in +our comparison of our generated model to the Gaia data. This step +was purely for visualisation purposes, and is not part of the model +itself. +4 +ASSESSING THE ACCURACY AND PRECISION OF +THE PROPER MOTION MODEL +4.1 +Overall Model Shape +The simulated proper motions are good across all sightlines and +brightnesses; some examples are shown in Figures 2-4. We get +good agreement in the mean proper motion, and shape of the dis- +tributions, of bulk source motions. For most sightline-brightness +combinations the agreement is quantitative, while sometimes the +shapes are merely broadly in agreement. Disagreement in modal +proper motion drift is likely largely caused by our Galactic rotation +model not capturing the fine detail of Galactic potentials or inac- +curacies in our asymmetric drift velocity, while distribution width +issues can mostly be explained by the extrapolation of the velocity +dispersion vector. However, we stress that the model’s simplicity +is one of its strengths in the context of inclusion within a larger +cross-match process, and that these minor differences are more than +acceptable for the purpose of improving Bayesian match likelihoods. +These distributions are intended to reflect a wide range of potential +positional shifts through time, rather than model any one specific +proper motion. Hence so long as the rough widths – to within some- +thing like a factor two, which we achieve – and mean offsets – good +to high accuracy using the Galactic rotation curve – are modelled +to reasonable accuracy, our distributions are good enough for our +work, and as intended. +Figure 5 shows some reduced statistics for the entire set of +sightline-brightness combinations we tested in the Galactic plane +– 𝐺 = {12, 15, 18, 19.5}, 𝑙 in the range from 0◦ to 345◦ in 15 +degree intervals, and 𝑏 = {−50◦, −30◦, 0◦, 25◦, 40◦}, as well as +the Galactic north and south poles at −90◦ ≤ 𝑏 ≤ −80◦, −80◦ ≤ +𝑏 ≤ −70◦, 70◦ ≤ 𝑏 ≤ 80◦, and 80◦ ≤ 𝑏 ≤ 90◦. Overall, we find +that the widths of the Gaia data are approximately 80% that of our +model (i.e. our model is too wide by 25%) across all positions and +brightnesses. There is a roughly 10% spread in relative widths – +middle column, Figure 5, cf. Figure 2, bottom right panel, where +our red model has a slightly wider wing than the histogram of the +black Gaia data. We also see evidence for overly narrow simulated +proper motion distributions (Gaia-to-model ratios larger than one) +along various sightlines, in approximately 8% of cases – but, again, +get extremely good agreement along others. These slightly-too-wide +RASTI 000, 1–20 (2022) + +8 +Tom J. Wilson +−0.2 +0.0 +0.2 +∆l / arcsecond [10yr baseline] +0 +5 +10 +PDF / arcsecond−1 +G = 12.0 +l = 90.0 +b = 0.0 +−0.2 +−0.1 +0.0 +∆b / arcsecond [10yr baseline] +0 +10 +20 +30 +PDF / arcsecond−1 +−0.1 +0.0 +0.1 +∆l / arcsecond [10yr baseline] +0 +5 +10 +15 +PDF / arcsecond−1 +G = 15.0 +−0.05 +0.00 +∆b / arcsecond [10yr baseline] +0 +10 +20 +30 +40 +PDF / arcsecond−1 +−0.10 +−0.05 +0.00 +0.05 +∆l / arcsecond [10yr baseline] +0 +5 +10 +15 +20 +PDF / arcsecond−1 +G = 18.0 +−0.050 −0.025 +0.000 +0.025 +∆b / arcsecond [10yr baseline] +0 +20 +40 +PDF / arcsecond−1 +−0.10 +−0.05 +0.00 +∆l / arcsecond [10yr baseline] +0 +10 +20 +PDF / arcsecond−1 +G = 19.5 +−0.050 +−0.025 +0.000 +0.025 +∆b / arcsecond [10yr baseline] +0 +20 +40 +PDF / arcsecond−1 +Figure 2. Distributions of proper motions (Galactic longitude, left-hand +columns, and latitude, right-hand columns) for sources at 𝑙 = 90◦, 𝑏 = 0◦, +for 𝐺 = 12, 𝐺 = 15, 𝐺 = 18, and 𝐺 = 19.5 (each respective row). Proper +motions have been converted from a per-year drift to decadal positional +change. Gaia proper motions are shown in the black histogram, with sim- +ulated distributions of proper motions in the red solid lines. Errorbars in +the corner of each subplot show the typical uncertainty of each individual +Gaia proper motion, while the plot labels show the Galactic longitude and +latitude, and 𝐺 magnitude, of the subset of sources. +distributions are likely related to our modelled radial and vertical +dependencies of the thin disc dispersion vector, as the thin disc is the +dominant term at the distances our Gaia data probe. Additionally, as +can be seen in the top row of e.g. Figure 2, low-number statistics of +brighter Gaia stars could be interpreted as lower standard deviations, +as the ‘real’ distributions fail to probe the wings of the simulated +distributions to high precision. +Relative to the standard deviation of the distributions, our bi- +ases – the mean motion offsets – are within ∼ 10% two-thirds of the +time, and almost always within 15 − 20%, as shown by right-hand +column, Figure 5. Absolutely, on a decade baseline, we find most of +our mean motion offsets are within approximately 0.01 arcseconds +(or 0.025 arcseconds on a 25-year baseline, as maybe be important +for LSST), well within the ‘centroid’ precision of most photometric +catalogues – left-hand column, Figure 5. These results – even where +qualitative (e.g. Figure 4, bottom-left panel) as opposed to quantita- +0.0 +0.1 +0.2 +∆l / arcsecond [10yr baseline] +0 +5 +10 +15 +PDF / arcsecond−1 +G = 12.0 +l = 180.0 +b = 0.0 +−0.1 +0.0 +0.1 +∆b / arcsecond [10yr baseline] +0 +5 +10 +15 +20 +PDF / arcsecond−1 +0.0 +0.1 +0.2 +∆l / arcsecond [10yr baseline] +0 +5 +10 +15 +20 +PDF / arcsecond−1 +G = 15.0 +−0.10 +−0.05 +0.00 +0.05 +∆b / arcsecond [10yr baseline] +0 +10 +20 +30 +PDF / arcsecond−1 +0.00 +0.05 +0.10 +∆l / arcsecond [10yr baseline] +0 +10 +20 +30 +PDF / arcsecond−1 +G = 18.0 +−0.05 +0.00 +∆b / arcsecond [10yr baseline] +0 +10 +20 +30 +40 +PDF / arcsecond−1 +0.00 +0.05 +0.10 +∆l / arcsecond [10yr baseline] +0 +10 +20 +30 +PDF / arcsecond−1 +G = 19.5 +−0.05 +0.00 +∆b / arcsecond [10yr baseline] +0 +10 +20 +30 +PDF / arcsecond−1 +Figure 3. Distributions of proper motions for 𝑙 = 180◦, 𝑏 = 0◦. Lines and +symbols have the same meaning as in Figure 2. +tively (e.g. Figure 4, top-left panel) good fits – are satisfactory, and +we therefore have chosen not to over-explore the residuals, as the +subtleties of the spiral arm structure of the Milky Way are outside +of the scope of this work. +4.2 +Galactic Poles +All previous examples shown (Figures 2-4) were limited in Galac- +tic latitude to |𝑏|≤ 50◦, exploring primarily the proper motions +of sources roughly in the Galactic plane. However, we must also +verify that our model is good at high absolute Galactic latitudes, +where we are viewing sources orbiting around the Galactic cen- +ter ‘above’ us. As shown in Figure 6, we get good agreement +for the Galactic longitudinal and latitudinal proper motions, af- +ter removing objects with parallax 𝜋 < 0.05 mas (𝑑 ≳ 20 kpc) or +𝜋/𝜎𝜋 < 2. Here, close to 𝑏 = 90◦, our equations for the average +rotational velocities (equations 3-7) simplify somewhat. First, look- +ing straight up out of the Galactic plane, we have 𝑑ip ≈ 0; we also +have 𝑅𝑐 ≈ 𝑅⊙, and hence (𝑅2𝑐 + 𝑅2 +⊙ − 𝑑2 +ip)/(2 𝑅𝑐 𝑅⊙) ≈ 1, and can +assume Θ(𝑅𝑐) = Θ⊙, giving 𝑈1 = −𝑈⊙, 𝑉1 = −𝑉⊙. This further +gives 𝑣𝑑 = −𝑈⊙ cos(𝑙) −𝑉⊙ sin(𝑙), 𝑣𝑙 = 𝑈⊙ sin(𝑙) −𝑉⊙ cos(𝑙), and +RASTI 000, 1–20 (2022) + +Overcoming Separation Between Counterparts Due to Unknown Proper Motions +9 +−0.5 +0.0 +∆l / arcsecond [10yr baseline] +0 +2 +4 +PDF / arcsecond−1 +G = 12.0 +l = 255.0 +b = 25.0 +−0.4 +−0.2 +0.0 +∆b / arcsecond [10yr baseline] +0 +2 +4 +6 +PDF / arcsecond−1 +−0.2 +0.0 +0.2 +∆l / arcsecond [10yr baseline] +0 +2 +4 +6 +PDF / arcsecond−1 +G = 15.0 +−0.2 +−0.1 +0.0 +∆b / arcsecond [10yr baseline] +0 +5 +10 +PDF / arcsecond−1 +−0.2 +0.0 +∆l / arcsecond [10yr baseline] +0.0 +2.5 +5.0 +7.5 +10.0 +PDF / arcsecond−1 +G = 18.0 +−0.2 +−0.1 +0.0 +∆b / arcsecond [10yr baseline] +0 +5 +10 +15 +PDF / arcsecond−1 +−0.2 +0.0 +∆l / arcsecond [10yr baseline] +0 +2 +4 +6 +8 +PDF / arcsecond−1 +G = 19.5 +−0.2 +−0.1 +0.0 +∆b / arcsecond [10yr baseline] +0 +5 +10 +15 +PDF / arcsecond−1 +Figure 4. Distributions of proper motions for 𝑙 = 255◦, 𝑏 = 25◦. Lines and +symbols have the same meaning as in Figure 2. +hence, along with a simplified 𝑣𝑏 = −𝑣𝑑, +𝜇𝑙∗ = 𝑘 +𝑑 [𝑈⊙ sin(𝑙) − 𝑉⊙ cos(𝑙)] , +(36) +𝜇𝑏 = 𝑘 +𝑑 [𝑈⊙ cos(𝑙) + 𝑉⊙ sin(𝑙)] . +(37) +This renders the orbital motions effectively just those of the Sun, +with our dispersion vector giving good shape agreement to the Gaia +data. Overall, we see good agreement in the shape of our model and +the Gaia proper motions. +4.3 +Widths of Distributions vs. Position and Proper Motion +Precisions +At this point it is worth briefly considering if, or at what bright- +nesses, this additional information is necessary. Figures 2-6 show a +representative sample of Galactic sightlines and the various widths +of the distributions of potential proper motions in each sightline- +magnitude combination. Overall, the widths of these distributions +are approximately 0.2 − 0.5 arcsecond drifts over a 10 yr baseline +(20 − 50 mas yr−1) at the bright end of our tests (𝐺 = 12), reduc- +ing to 0.1 − 0.3 arcsecond drifts in 10 years (10 − 30 mas yr−1) at +𝐺 = 19.5. +The first parameter we should compare the proper motions to +is the precision on an individual position. If one or both of the +positions in a given cross-match were highly uncertain, factors 10 +or higher than the proper motion drift, this would dominate over the +extra positional spread caused by the potential proper motion of the +source. Its inclusion would then not contribute to the determination +of potential counterparts. However, for a decade-long baseline, the +spread of separations induced by unknown proper motion is at least +a factor two or three higher than typical astrometric precisions, +with even small motions over long enough baselines moving objects +several astrometric precisions apart. For Gaia astrometric precisions +are vastly higher; while 80% of its sources will also have incredibly +high precision proper motions even the remaining sources will have +coordinate positions significantly higher than the unknown proper +motion distributions. A more typical ground-based survey, LSST +should have at worse 0.07 arcsecond precision on each individual +visit at 𝑟 = 24 (Ivezić et al. 2019). While this is a factor ≈ 3 times +smaller than the widths of our proper motion distributions on decade +baselines, the real power of LSST lies in its repeated observations. +Depending on whether the object is in the full ‘Wide-Fast-Deep’ +(WFD) survey or in the Galactic Plane footprint, it will either be +observed approximately 800 or 150 times across LSST’s full survey +lifetime (Bianco et al. 2022). Hence the statistical precision on a co- +added detection at 𝑟 = 24 is 0.003−0.006 arcseconds depending on +the exact number of visits. Even including ≈ 0.01 arcsec systematic +precision (Ivezić et al. 2019) this is far below the widths of our +models for proper motion drift. While those objects will likely +also have proper motions after LSST DR3-4, 𝑟 = 26.5 coadded +detections will have statistical astrometric precisions a factor +√ +10 +higher, 0.008 − 0.02 arcseconds, still a factor 10 or more below our +10-year baseline drift spread. It will be therefore important to take +these long-baseline drifts into account for faint LSST objects. For +current-generation surveys such as SDSS, its very faintest sources +have statistical positional uncertainties comparable to the tightest +of our proper motion distribution widths (≈ 0.2 arcsec) so 𝑟 = +24 objects in SDSS may only see limited gains matching across +a 10-year timespan. Of course, the drifts increase linearly with +time, and so a 15-year baseline (2015-2030, for example, in the +case of SDSS-LSST) increases the potential proper motion drifts +to a larger impact than astrometric precision. Additionally, in the +context of crowded field Bayesian cross-matching, even a ‘one- +sigma’ positional movement will be enough to significantly disrupt +match likelihoods. +It is also useful to ask if proper motion precisions are ever com- +parable to the width of potential unknown proper motion. Again, +for Gaia this is not the case due to its extremely high precision and +repeated observations of all objects. At 𝐺 = 20 the median precision +on its proper motions are of order 1 mas yr−1 (Gaia Collaboration +et al. 2021). For LSST, quoted proper motion uncertainties are also +of order 1 mas yr−1 (Ivezić et al. 2019) – but these assume ≈ 800 +visits. Hence the stellar proper motion precisions for Galactic Plane +objects will be a factor ≈ 3 higher due to the reduced number of +observations within the same timeframe. However, even 5 mas yr−1 +is 0.05 arcsec over a 10-year timeframe and therefore a smaller, +but sizeable, fraction than the unknown proper motion distribution +widths. Right at the detection limit of proper motions with LSST +it may be the case that it is preferable to not use the detected-but- +unconstrained proper motions, though. SDSS has typical limiting +proper motion precisions of 5 mas yr−1 by 𝑟 ≈ 20. Hence, like +LSST, where constrained individual SDSS proper motions brighter +than about 20th magnitude are likely preferable to unknown proper +motions, but below this limit and the single-visit detection limit of +RASTI 000, 1–20 (2022) + +10 +Tom J. Wilson +−0.03 +−0.02 +−0.01 +0.00 +0.01 +0.02 +0.03 +Gaia − Model Average / arcsecond [10yr baseline] +0 +25 +50 +75 +100 +125 +150 +N +Mean +Median +Mode +0.4 +0.6 +0.8 +1.0 +1.2 +Ratio of Gaia-to-Model Distribution Widths +0 +20 +40 +60 +80 +100 +N +StDev +90th−10th +84th−16th +75th−25th +−0.4 +−0.2 +0.0 +0.2 +0.4 +Gaia − Model Median / Gaia StDev +0 +10 +20 +30 +40 +50 +60 +70 +N +−0.03 +−0.02 +−0.01 +0.00 +0.01 +0.02 +0.03 +Gaia − Model Average / arcsecond [10yr baseline] +0 +25 +50 +75 +100 +125 +150 +N +Mean +Median +Mode +0.4 +0.6 +0.8 +1.0 +1.2 +Ratio of Gaia-to-Model Distribution Widths +0 +20 +40 +60 +80 +N +StDev +90th−10th +84th−16th +75th−25th +−0.4 +−0.2 +0.0 +0.2 +0.4 +Gaia − Model Median / Gaia StDev +0 +10 +20 +30 +40 +50 +60 +N +Figure 5. Comparison between Gaia proper motions and those of our model across all sightlines and brightnesses in the Galactic plane, in Galactic longitude +(top row) and latitude (bottom row). Left: Data-to-model average proper motion drift offsets. Middle: Ratio of data and model proper motion drift distribution +widths. Right: Ratio of the median proper motion drift offset between Gaia data and the model distribution, normalised by the standard deviation of the Gaia +proper motions. +𝑟 = 22 unknown proper motions may be the more precise constraint. +In the IR, CatWISE (Eisenhardt et al. 2020) has proper motion un- +certainties of 20 mas yr−1 at 𝑊1 ≈ 15 (Marocco et al. 2021) and the +VVV survey (Smith et al. 2018) cites uncertainties of 10 mas yr−1 +around 𝐾𝑠 ≈ 16; below these brightnesses the precisions on in- +dividual proper motions become comparable to or larger than the +widths of typical unknown proper motion distributions. +4.4 +Missing Galactic Components +As discussed in Section 2.4, we do not currently include a full +prescription for the Galaxy. In particular, we do not model the +Galactic Bulge (or Bar). This may have an effect at very low Galactic +longitudes and latitudes. The Gaia data show a broader, almost +flat distribution of longitudinal proper motions, where our simpler +Galactic model, using the thin disc as the dominant term, has a bi- +modal distribution of two narrower peaks, as shown in Figure 7, left +hand panel. It can also be seen in the data (Figure 7, right hand panel) +that there is a slightly too narrow distribution of latitudinal proper +motions, as compared to the Gaia data. This likely comes back to +the minor effects of radial dependencies of the 𝜎2𝑟𝑟 term, either +following a Gaussian- or Rayleigh-like distribution (as discussed in +Section 2.4.1). However, it could also be the case that our radial +and vertical dispersion scalings are failing at these smaller Galactic +radii, as a significant fraction of Gaia sources ought to be sufficiently +far removed from the Galactic center to be Bar or Bulge objects. +Once again, we deem these minor issues beyond the scope of this +preliminary work – the combined bi-modal longitudinal distribution +almost entirely covers the distribution of Gaia motion drifts, to +within better than a factor 1.5 or so, which is our goal. However, we +highlight the issue that the very inner few degrees of the Galactic +center may suffer systematic proper motion effects due to the nature +of the Galactic Bulge and Bar complexities. +We also have not modelled the Magellanic Clouds, and in- +deed during testing found that several of our test fields are heavily +‘contaminated’ by sitting on the Small Magellanic Cloud (SMC) +and Large Magellanic Cloud (LMC). These extra terms, as with +the Bulge, would be easy to implement; provided a relative num- +ber density of sources, with some positional distribution, and bulk +and dispersal proper motion – assuming the Magellanic Clouds are +orbiting internally, and around the Milky way – the proper motions +can be modelled in much the same way with the Galactic discs and +halo. For now, we also simply urge the reader to take care when sim- +ulating sources centered on the Magellanic Clouds (SMC 𝑙 ∼ 300◦, +𝑏 ∼ −45◦; LMC 𝑙 ∼ 280◦, 𝑏 ∼ −35◦). +4.5 +Missing Binarity Perturbation +Our model for motions of objects in the plane of the sky assumes +all sources are single stars, subject solely to the Galactic potential. +However, half of objects are in some form of higher-order system +(e.g. Raghavan et al. 2010) and should therefore be subject to ad- +ditional on-sky motion. It is therefore reasonable to ask whether +the non-inclusion of this effect, of unresolved binary objects, would +have any impact on our derived proper motions. +If the binary were equal mass, any orbital motion of the two +sources around their common barycentre would completely cancel +by symmetry, and show no impact on the photocentre and proper +motion of the blended sources. On the other hand, if the objects +were very unequal in mass then both the barycentre and photo- +centre of the pair will be dominated by the larger, brighter main +source, and effectively reduce to a singular object for our purposes. +RASTI 000, 1–20 (2022) + +Overcoming Separation Between Counterparts Due to Unknown Proper Motions +11 +−0.5 +0.0 +0.5 +∆l / arcsecond [10yr baseline] +0 +1 +2 +PDF / arcsecond−1 +G = 12.0 +l = 180.0 +b = -75.0 +−0.5 +0.0 +0.5 +∆b / arcsecond [10yr baseline] +0 +1 +2 +PDF / arcsecond−1 +−0.5 +0.0 +0.5 +∆l / arcsecond [10yr baseline] +0 +1 +2 +3 +PDF / arcsecond−1 +G = 15.0 +−0.5 +0.0 +0.5 +∆b / arcsecond [10yr baseline] +0 +1 +2 +3 +PDF / arcsecond−1 +−0.25 +0.00 +0.25 +∆l / arcsecond [10yr baseline] +0 +1 +2 +3 +PDF / arcsecond−1 +G = 18.0 +−0.25 +0.00 +0.25 +∆b / arcsecond [10yr baseline] +0 +1 +2 +3 +PDF / arcsecond−1 +−0.25 +0.00 +0.25 +∆l / arcsecond [10yr baseline] +0 +1 +2 +3 +PDF / arcsecond−1 +G = 19.5 +−0.25 +0.00 +0.25 +∆b / arcsecond [10yr baseline] +0 +1 +2 +3 +PDF / arcsecond−1 +Figure 6. Distributions of proper motions for 𝑙 = 180◦, 𝑏 = −75◦. Lines and +symbols have the same meaning as in Figure 2. Additionally, Gaia objects +were filtered for parallaxes consistent with zero to remove extragalactic +contamination. +−0.10 +−0.05 +0.00 +∆l / arcsecond [10yr baseline] +0 +5 +10 +15 +20 +PDF / arcsecond−1 +G = 18.0 +l = 0.0 +b = 0.0 +−0.05 +0.00 +0.05 +∆b / arcsecond [10yr baseline] +0 +5 +10 +15 +20 +PDF / arcsecond−1 +−0.10 +−0.05 +0.00 +∆l / arcsecond [10yr baseline] +0.0 +2.5 +5.0 +7.5 +10.0 +PDF / arcsecond−1 +G = 19.5 +−0.05 +0.00 +0.05 +∆b / arcsecond [10yr baseline] +0 +10 +20 +PDF / arcsecond−1 +Figure 7. Distributions of proper motions for 𝑙 = 0◦, 𝑏 = 0◦. Lines and +symbols have the same meaning as in Figure 2. +An object of approximately half the mass of the primary, however, +contributes very little in luminosity but significantly in astrometric +effects. Placing such an object on a worse-case orbit of approxi- +mately 10 AU would give an orbital period around 25 years, and a +half-phase orbit on our key decade-long time interval between pho- +tometric catalogue ‘generations’. In that time, the primary object +would travel halfway around the orbit, appearing to move a total +of ≈ 6.5AU, twice its orbital distance from the barycentre. At a +typical distance of roughly 1 kpc this is 6.5 mas or 0.54 mas yr−1. +Such a perturbation is well below the of order 10 mas yr−1 widths +to the proper motion distributions observed for faint objects in our +model. If the object were significantly closer – say 100 pc instead – +the motion effects would be 10 times higher, and comparable to the +model widths. In those cases the object would be much brighter, and +likely have an individually measured proper motion or be known to +be a multiple system through other means. We therefore believe the +non-inclusion of higher-order systems is justifiable at the resolution +we are aiming to achieve. +4.6 +Random Positions of Sources +As noted in Section 3, the TRILEGAL simulations we use to con- +struct our models of proper motions do not provide individual posi- +tions for simulated sources. To overcome this, we simply uniformly +distributed sources within the rectangular area we sampled our Gaia +proper motions in. This effect may explain some small disagree- +ments between our simulated and Gaia proper motion distributions, +as we are therefore not properly modelling any clustering, extinction +effects, or other non-uniformity and correlations in the distances and +positions of Galactic sources. +However, the effect on each individual proper motion should +be relatively small, as the regions in question were mostly limited +to several degrees in extent, and cos(𝑥 + 5◦) − cos(𝑥) ≲ 0.08, +sin(𝑥 + 5◦) − sin(𝑥) ≲ 0.08 over the entire Galactic longitude. +Thus our values for, e.g. the decomposition of 𝑈⊙, or Θ, within +our proper motion equations, are of order 8% wrong at their most +extreme, in the case of a simulated patch of sky five degrees wide. As +an ensemble, however, this assumption should be a reasonable one, +and the ‘incorrectness’ should average out, with uniformity of source +distribution acceptable for small enough patches of sky, providing a +statistical distribution of variations of velocity decomposition across +the whole region. +4.7 +Gaia Proper Motion Uncertainty +To compare our ensemble proper motion distribution with the distri- +bution of Gaia proper motions, we included the Gaia measurement +uncertainty in our theoretical distribution of drifts. The Gaia data +have individual uncertainties but to smooth the model with the +uncertainty we had to select a single average value (the error bar +included in the corners of sub-plots in e.g. Figure 2). A small part +of the discrepancies between model and data in our analysis could +therefore stem from this simplifying assumption, with no bearing +on the model itself. If the Gaia data have a particularly broad dis- +tribution of measurement uncertainties, as they tend to at fainter +magnitudes, our single uncertainty value would not produce an +uncertainty-convolved motion drift distribution that reflected that +of the Gaia data. Testing more complex treatments of Gaia uncer- +tainty distributions in the comparison between model and data, we +found that more fully describing the non-singular value of mea- +surement precision did produce theoretical drift distributions that +RASTI 000, 1–20 (2022) + +12 +Tom J. Wilson +−0.50 +−0.25 +0.00 +∆l / arcsecond [10yr baseline] +0 +5 +10 +PDF / arcsecond−1 +G = 12.0 +l = 270.0 +b = 0.0 +−0.2 +0.0 +∆b / arcsecond [10yr baseline] +0 +5 +10 +15 +20 +PDF / arcsecond−1 +−0.1 +0.0 +∆l / arcsecond [10yr baseline] +0 +5 +10 +15 +20 +PDF / arcsecond−1 +G = 19.5 +−0.05 +0.00 +∆b / arcsecond [10yr baseline] +0 +10 +20 +30 +40 +PDF / arcsecond−1 +Figure 8. Distributions of proper motions for 𝑙 = 270◦, 𝑏 = 0◦. Lines and +symbols have the same meaning as in Figure 2. In addition, the dashed blue +line shows simulated Besançon proper motions. +better matched the expected data, but not completely. We therefore +still find a few sightlines with slight differences in central proper +motion or distribution width, but perhaps 30% of these tensions are +explainable by the different measurement precisions of faint Gaia +proper motions. Ultimately, however, as mentioned in Section 3, we +do not include this measurement uncertainty in the final model, just +performing the convolution to compare to the Gaia distributions +more accurately. +4.8 +Comparison with the Besançon Model +Throughout this work we have used the TRILEGAL simulations +to provide a set of theoretical distances for sources of a particular +Galactic sightline and magnitude. We could, of course, use any +model of the Milky Way to achieve this, such as the Besançon +model (Robin et al. 2003, 2012, 2014, 2017; Czekaj et al. 2014; +Bienaymé et al. 2015). With the Besançon models, however, we +receive simulated proper motions for the objects returned in our +query, unlike with TRILEGAL. We can use these simulated proper +motions to further verify the robustness of our proper motion model; +but this then raises the question of why we simply do not use these +simulated proper motions for use in our cross-matches. We will +address both of these issues in the next two sections. +4.8.1 +Verifying the Accuracy of Our Model with Besançon +With simulated Besançon proper motions, we can compare our +model’s statistical distribution of proper motions with those of the +Galactic model. Shown in Figure 8 are distributions of proper mo- +tions at 𝑙 = 270◦, 𝑏 = 0◦ for Gaia, our simple model for stellar +velocities, and Besançon proper motions. Overall, at fainter mag- +nitudes (bottom row), both models are in agreement with the Gaia +data, with our distribution a slightly better match in Galactic latitude +than the Besançon model. +However, there are some sightlines within the Galaxy where +our model has some mismatches to the Gaia data – an example +sightline demonstrating this effect is shown in Figure 9. Here, at +faint magnitudes (𝐺 = 19.5), the Besançon model better reproduces +the Galactic longitude proper motion distribution seen with Gaia, +−0.2 +0.0 +∆l / arcsecond [10yr baseline] +0 +5 +10 +15 +20 +PDF / arcsecond−1 +G = 12.0 +l = 60.0 +b = 0.0 +−0.2 +−0.1 +0.0 +∆b / arcsecond [10yr baseline] +0 +10 +20 +30 +PDF / arcsecond−1 +−0.10 +−0.05 +0.00 +∆l / arcsecond [10yr baseline] +0 +10 +20 +PDF / arcsecond−1 +G = 19.5 +−0.050 +−0.025 +0.000 +0.025 +∆b / arcsecond [10yr baseline] +0 +20 +40 +PDF / arcsecond−1 +Figure 9. Distributions of proper motions for 𝑙 = 60◦, 𝑏 = 0◦. Lines and +symbols have the same meaning as in Figure 8. +where our model shows a slight bias, and a distribution slightly +too broad. Neither model can reproduce the Galactic latitude Gaia +proper motions, and both look very similar in their over-broad dis- +tribution. +At bright magnitudes (𝐺 = 12), however, we can see that our +distribution (red solid lines) much better matches the Gaia data +points than the Besançon simulation (blue dashed lines). In almost +all cases, 𝑙 = 60◦ and 𝑙 = 270◦ in Figures 8 and 9, but more +generally across multiple sightlines, the Besançon models are too +sharp in distribution, and fail to match the Gaia data as well as our +model for proper motion. We discuss this magnitude-dependence +of the Besançon model fits further in Section 4.8.2. +Here we conclude that our model matches the Besançon models +very well, as it does the Gaia data, and see cases where both our +model and the Besançon model fail to match the Gaia data perfectly. +4.8.2 +Why Not Just Use the Besançon Proper Motions? +We have used TRILEGAL simulations to construct our Galactic +model throughout this work, but we could have used any Galactic +model. If we had used the Besançon model, we would also have +been provided with simulated proper motions for the objects we use +for their distances in constructing our proper motions. It is therefore +reasonable to ask why we would go to the effort of using another +model, if we already had a set of proper motions from which to +construct a PDF of unknown proper motions. +First, as our model is broken up into separate smaller sub- +models, as opposed to being wrapped in a full Galaxy model, our +magnitude-to-distance relation is flexible. As mentioned, we have +been using TRILEGAL simulations to get our potential distribution +of distances of sources of a given magnitude, but we could use any +Galactic model. Indeed, we do not need to use a model at all; if we +instead had a known distribution of tip of the red-giant branch stars, +or some other class of standard candle, we would immediately know +the distance of our sources from their brightnesses. We therefore +do not necessarily need to rely on fully resolved Galactic models +to provide proper motions or distances with our simple model. On +the other hand, our options become slightly more limited if we wish +to use a full Galactic model to obtain simulated proper motions in +one pass, as opposed to generating more ‘static’ distributions of +RASTI 000, 1–20 (2022) + +Overcoming Separation Between Counterparts Due to Unknown Proper Motions +13 +object brightnesses (or distances), and using other functionality to +continue on to create our final proper motion distributions, as we +do here. +The second consideration is that of dimensionality; each Be- +sançon source is provided with a simulated proper motion – but +only one. Our model uses the simulated distance for each source +once, but draws 𝑁 simulated velocities – and hence 𝑁 simulated +proper motions – for each source. We therefore much more com- +pletely sample the 3-D velocity space than any one simulation from +the Besançon Galactic model will. This effect can be seen in the +𝐺 = 12 panels of Figures 8 and 9, where our model (red solid lines) +much better agrees with the Gaia proper motions, where limited +sample size means the Besançon model is not fully populating the +velocity dispersion dimensions. At 𝐺 = 19.5, number counts have +increased by a factor 100, and the velocity dispersion, having an +inverse-distance component (and fainter stars being further away, +on average), has reduced in size. This reduces the effect of the lack +of realisations; the Besançon models therefore agree much better at +these fainter magnitudes than they do at bright ones. +By using each source only for its distance, as opposed to using it +to sample the 4-D distance-velocity dimensionality, we much more +accurately sample from the full potential proper motion distribution +at bright magnitudes. This is crucial for bright sources, being closer +to the Sun on average, which have larger proper motions (cf. the 𝑥 +axis ranges on the top and bottom rows of Figures 8 and 9). It is here +where our constructed model has the edge on the proper motions +constructed from large-scale Galactic simulations, although brighter +objects are, of course, more likely to have a robustly detected proper +motion from other sources. +5 +INCLUDING PROPER MOTIONS IN PROBABILISTIC +CATALOGUE CROSS-MATCHING +No matter how you construct your proper motion distributions, it +is still important to consider them in a match between two photo- +metric catalogues of differing epochs. The Astrometric Uncertainty +Function (AUF; Wilson & Naylor 2017; Wilson & Naylor 2018b) is +the description of the belief as to a true position of the source, given +its measured position. This is typically assumed to be a Gaussian, +which describes the most obvious term affecting the measured posi- +tions of sources in photometric catalogues, and hence the separation +between two potential counterparts: the noise-based centroiding of +the individual objects during the catalogue creation process. The +function representing the likelihood of two sources having a given +separation under the assumption that they are counterparts to one +another – two detections of the same physical object – is given by +𝐺(Δ𝑥, Δ𝑦) = (ℎ𝛾 ∗ ℎ𝜙)(Δ𝑥, Δ𝑦) +(38) +(Wilson & Naylor 2018a). Here Δ𝑥, Δ𝑦 are the two-dimensional sky +offsets (e.g. right ascension and declination, or Galactic longitude +and latitude), ℎ𝛾 and ℎ𝜙 the AUFs of the sources from the two +catalogues respectively, and ( 𝑓 ∗ 𝑔)(𝑥, 𝑦) denotes the convolution +of two arbitrary functions 𝑓 and 𝑔 evaluated at 𝑥 and 𝑦. +As discussed by Wilson & Naylor (2018b), the AUF ℎ can be +extended with any additional terms, ℎ𝛾 = ℎ𝛾,1∗ℎ𝛾,2∗ℎ𝛾,3 etc. Here +the additional ℎ𝛾,𝑖 components, after the first noise-based centroid +term, describe extra potential movement away from the ‘true’ sky +position of the source in the limit of infinite precision. These could +include, for example, stochastic processes such as the perturbation +of objects due to hidden contaminants affecting the center-of-light +of sources, or systematic effects like offsets of the coordinate frame +of the catalogue from a common reference frame, such as the ICRS. +Whatever the effects, the point is that each source is considered +individually, and has all of its ℎ𝛾,𝑖 components applied to it on an +isolated, per-source basis. Proper motion, however, does not work +like this; the effect of proper motion drift works on offsets between +two positions, as opposed to affecting the absolute position of one +source. +Thus the proper motion drift must be applied to 𝐺, giving, +effectively 𝐺′ = 𝐺 ∗ ℎ′pm (see Appendix B1 for details). ℎ′pm +should be calculated in the sense of mapping from oldest to youngest +epoch, in units of distance; thus for Δ𝑡 > 0 we have, crudely, +Δ𝛿 = 𝜇𝛿 × Δ𝑡. Mapping from most recent to older data would +have a negative Δ𝑡, but the proper motion would have to be of +the opposite sign as well (being a ‘rewind’ of the motion), and +thus the sign of Δ𝛿 would be the same. If a source has a purely +positive proper motion distribution, such that all 𝜇𝑙∗ > 0 for this +simulated source, then we would expect a source observed in the year +J2000 to have a smaller Galactic longitude than a source observed +at J2015, for example. This convolution can be performed as any +other convolution done to calculate 𝐺 by the convolution of all ℎ +components – e.g. either numerically, or through expression as a +mixture of analytically convolvable models. +We also highlight here that while Sections 2-4 detail a method +for the construction of a distribution of unknown proper motions, +ℎ′pm can be constructed through any available means. For exam- +ple, Kerekes et al. (2010) construct sets of data-driven proper mo- +tion distributions for the purpose of improving cross-matches, using +available proper motions to construct priors for weighting the search +for unknown proper motions between potential source counterparts. +On the other hand, the faint end of a photometric catalogue will sys- +tematically have worse precision on its measurements (see Section +4.3), and at some point will have detected the proper motion of an +object but be unable to constrain it with high precision. In these +cases, ℎ′pm could very well be constructed as a Gaussian PDF with +mean and covariance matrix that of the best-fit and uncertainty of +the proper motion. +5.1 +Star-Galaxy Separation +Our model for proper motions assumes the source in question is a +star – objects orbiting the Galactic center in some fashion. However, +for an all-sky catalogue cross-match we will also, at high Galactic +latitudes, be matching a considerable number of galaxies. We there- +fore need to model the two cases. First, that the sources being +matched are stars, and hence have the statistically modelled un- +known proper motion distribution, with which we wish to ‘blur’ out +our potential match separations. Second, they are galaxies, which +have zero proper motion, being altogether too distant to have visibly +moved anywhere. We therefore have a slightly different probability +of match (sources being ‘counterparts’, under hypothesis 𝑐) given +separation 𝑑, now also conditioned on the ‘type of source’ hypoth- +esis, which we will denote as 𝑝(𝑐|𝑑, S) and 𝑝(𝑐|𝑑, G) for a ‘star’ +and ‘galaxy’ pairing respectively. +When matching, we are generally only concerned with the +overall probability of the two sources having a given sky separation +under the hypothesis of their being matched, 𝑝(𝑑|𝑐) – this term is +denoted 𝐺 by Wilson & Naylor (2018a). Note that this differs from +𝑝(𝑐|𝑑), the probability of the two sources being counterparts given +their sky separation, Wilson & Naylor (2018a)’s 𝑔. 𝑝(𝑑|𝑐) we can +RASTI 000, 1–20 (2022) + +14 +Tom J. Wilson +obtain by the marginalisation over the two hypotheses: +𝑝(𝑑|𝑐) = 𝑝(𝑑, S|𝑐) + 𝑝(𝑑, G|𝑐) += 𝑝(𝑑|𝑐, S)𝑃(S|𝑐) + 𝑝(𝑑|𝑐, G)𝑃(G|𝑐) +(39) +where 𝑃(S|𝑐) is the prior probability that these counterparts (with +given sky positions, brightnesses, etc.) are stars (or galaxies, in the +opposite case). We work under the assumption there is no third +type of object – crudely labelling objects as ‘in the Milky Way’ or +‘outside the Milky Way’ – and thus 𝑃(S|𝑐) + 𝑃(G|𝑐) = 1. +However, we can also ask a related but separate question: ‘what +is the probability that these two detections are of a star, given that +they are counterparts with a given separation?’, which looks like +𝑃(S|𝑑, 𝑐) = 𝑝(𝑑|𝑐, S)𝑃(S|𝑐) +𝑝(𝑑|𝑐) +. +(40) +Here we have the likelihood of the separation given the hypoth- +esis that the sources are counterparts and stars, multiplied by the +prior chance of the sources being stars given they are counterparts, +normalised by the overall chance of either a galaxy or star pair hav- +ing this particular detection offset. To calculate both 𝑃(S|𝑑, 𝑐) and +𝑝(𝑑|𝑐) we therefore need both prior and likelihood terms. +Calculating the likelihood terms 𝑝(𝑑|𝑐, S) and 𝑝(𝑑|𝑐, G) is +relatively straightforward, simply being the convolution of the re- +spective AUFs (containing all relevant AUF components for the +two catalogues) of the sources in question. For 𝑝(𝑑|𝑐, G) this does +not include any proper motion terms, as the ‘proper motion model’ +for galaxies is a static one – mathematically, this is equivalent to +the convolution of 𝐺 and a delta function at zero proper motion, +with 𝑓 ∗ 𝛿 = 𝑓 – and hence 𝑝(𝑑|𝑐, G) = 𝐺. For 𝑝(𝑑|𝑐, S), how- +ever, we wish to include the motion of Galactic sources, and hence +subsequently convolve by the ℎ′pm PDF, describing the potential +additional on-sky movement due to the epoch difference between +the two sets of observations; 𝑝(𝑑|𝑐, S) = 𝐺′ ≡ 𝐺 ∗ ℎ′pm. +Thus, the likelihood for our new question is easy to calculate; +we are therefore left with the derivation of the prior, 𝑃(S|𝑐). The +‘conditioned on the fact that the sources are counterparts’ aspect +of the prior is tricky to implement in practice. We wish to know, +analogous to Wilson & Naylor (2018a)’s derivation of photomet- +ric likelihoods, the distribution of stars and galaxies as a function +of the two bandpasses in question – e.g. 𝑟 and 𝐽, for a match be- +tween optical and infrared data. Thus, 𝑃(S|𝑐) is really ‘what is +the probability that these two sources are stars given that they are +counterparts with magnitude limits (or dynamic ranges) in their re- +spective bandpasses?’, 𝑃(S|𝑐, 𝑚lim,r, 𝑚lim,J). Due to the nature of +the simulated objects – being derived from one-sided distributions, +a function of just a single magnitude in one bandpass in one of the +two catalogues – we are unable to create two-dimensional relation- +ships between stars and galaxies in the construction of these priors. +In fact, our likelihoods, 𝑝(𝑑|𝑐, S), should implicitly assume coun- +terparts for sources, but are built from the full distribution of sources +of just a single magnitude. Here we could have, for example, a case +where sources of 𝐽 = 17 either have optical brightnesses 𝑟 = 18 or +𝑟 = 25 (being two classes of objects at differing distances, say); this +distance distribution is blurred into a bimodal proper motion distri- +bution in the IR, but one class of object is rejected if we consider +the dynamic range of the optical data for an example 𝑚lim,𝑟 = 20. +At present, the explicit dependency on the two-sided, +magnitude-magnitude relationship between sources in our two cat- +alogues is beyond the scope of this work, due to the nature of the +outputs available from most Galactic simulations being limited to a +particular set of bandpasses for a specific catalogue. We therefore +simply note here that for now, the construction of these models is +one-sided – in contrast to the cross-matching algorithms of Wilson +& Naylor (2018a), taking into account both catalogues symmet- +rically, in both AUF-based astrometry and photometry. We thus +sidestep this dependency by constructing our priors on star and +galaxy counts on single magnitude source counts, effectively cre- +ating 𝑃(S) and 𝑃(G), removing the dependency on 𝑐 within the +priors. We can still, however, account for the dynamic range of each +bandpass within its given catalogue on a per-filter basis, and hence +implicitly use 𝑃(S|𝑚lim) in a practical implementation. +With this minor practical dependency removed, we conclude +that with the inclusion of a distribution of unknown proper motions +for Galaxy-based stars, it is possible to discriminate between stars +and galaxies in photometric catalogues. The equations +𝑃(S|𝑑, 𝑐) = 𝑝(𝑑|𝑐, S)𝑃(S) +𝑝(𝑑|𝑐) +(41) +and +𝑃(G|𝑑, 𝑐) = 𝑝(𝑑|𝑐, G)𝑃(G) +𝑝(𝑑|𝑐) +(42) +allow for the drift of Galactic sources with time, recovering them +as non-static sources. This is the most certain question that can +be answered; stars, as shown in e.g. Figure 3, can have a very +high probability of small proper motions in certain sightlines in the +Galaxy. Thus, zero proper motion does not necessarily mean galaxy; +but a combination of delta-function likelihood for Galactic proper +motion and imbalanced priors at high Galactic latitudes mean that +zero proper motion objects will bias towards being extragalactic. +On the other hand, if a source has a proper motion distribution +which is significantly non-zero, as is the case for Galactic longitude +proper motions at 𝑙 = 270◦, 𝑏 = 0◦ (Figure 8), then we should see +a breaking of this degeneracy. The offset between the two sources +being considered as potential counterparts should now be able to +tell whether the sources are further apart than their respective AUFs +would suggest – at which point they are very likely detections of +a star – or if they have an offset compatible with their AUFs – at +which point they are very likely a galaxy. +5.2 +Inclusion of Proper Motions in the Non-Match +Hypothesis +We also note that we should also consider the proper motions within +the context of non-matches, but it is easy to see that this results +in a trivial case, effectively ignoring the proper motions. For the +counter hypothesis of ‘these sources are unrelated to one another, +and separate detections of two physical sky objects’, each source can +have its own proper motion, based on its own statistical distribution +of potential motions. In these cases, we need to compare to the +hypothesis that these sources are not related to one another given +the separation between them. This involves the double, but separate, +marginalisation over all possible unknown locations and proper +motions, for both objects. Ultimately, as the integrals are separate +the proper motions do not affect the end result – see Appendix +B2. This is obvious intuitively: the distribution of separations of +unrelated, randomly placed objects is independent of the unknown +motion history of those objects. +6 +WHEN ARE UNKNOWN PROPER MOTION +DISTRIBUTIONS NEEDED? +The theoretical framework for accounting for unknown proper mo- +tions presented here is relatively indifferent to the type of surveys +RASTI 000, 1–20 (2022) + +Overcoming Separation Between Counterparts Due to Unknown Proper Motions +15 +being matched and brightnesses at which it is used. However, prac- +tically it is more useful in some situations than others. Hence, we +summarise here some key surveys, magnitude ranges, and science +cases for which the inclusion of statistical proper motions may be +most crucial. +The main criterion for considering whether the inclusion of +unknown proper motion distributions is important or not is the +surveys being matched. +• The model is most useful outside of Gaia dynamic ranges, as +such high-precision individual proper motions may be too impor- +tant to ignore at brighter magnitudes. The main downside here is +that Gaia is quite a bright survey relative to the next generation +of photometric catalogues, and therefore large numbers of objects +won’t be detected in Gaia at all. +• In the case of LSST, it will also offer proper motions down +to perhaps 𝑟 = 24 (Ivezić et al. 2019) but cannot offer proper +motions for objects not detectable in its single-visit images, and thus +those objects will have to rely solely on statistical proper motions +to avoid risking underestimating match probability or unnecessary +false match rates. +• More generally, any science done in the Northern Hemisphere, +where LSST has no coverage, will be unable to take advantage of +the dataset – for its increased proper motion coverage or otherwise. +These cases will more likely require the falling back on unknown +proper motions when outside of Gaia or SDSS proper motion dy- +namic ranges (𝐺 ≈ 20 and 𝑟 ≈ 21 respectively). +• Where proper motions are not important to the science case, +and any motion drift is a nuisance parameter, it may also be prefer- +able to avoid relying on matching to an intermediate catalogue that +contains proper motions. When trying to match catalogue 𝐴 to cata- +logue 𝐵, we may not want to perform separate LSST-𝐴 and LSST-𝐵 +(or Gaia-𝐴 and Gaia-𝐵, depending on your source of individual +proper motions) matches, then join across common LSST (Gaia) +objects to obtain a final cross-match. In these cases, where the abil- +ity to select high-quality matches using the added-value information +from a probabilistic cross-match algorithm is important, reliance on +proper motion distributions may suffice to gain in other areas. +• With the key exceptions of CatWISE, albeit with order-of- +magnitude larger uncertainties than LSST or Gaia, and, but with +much less sky coverage, VVV, most IR surveys are single-epoch, +and matching longer wavelength surveys to one another therefore +relies far more than optical catalogues on unknown proper motion +distributions. +In terms of science cases, the main areas that benefit from in- +cluding unknown proper motions are those in which proper motions +are crucial but lacking by other methods. +• Nearby faint objects, which LSST especially will find signifi- +cant numbers of, will have appreciable on-sky motions that may not +be derived as part of the survey’s dataset construction due to their +faint fluxes. +• Red objects will suffer a bias in current- and future-generation +surveys such as LSST, Euclid, and Roman where they will system- +atically be less likely to have measured proper motions. +• Very faint transient progenitors will also suffer from a lack +of known proper motions, and potentially may require matching +back to a number of long-time-baseline surveys to probe progenitor +characteristics. +• For LSST, Galactic Plane science will systematically be af- +fected due to the much lower number of visits currently planned +than in the main WFD survey (Bianco et al. 2022). Current simu- +lated LSST precisions (e.g. Ivezić et al. 2019, table 3) assume WFD +cadences and hence numbers of observations, but reduced visit +count will lead to worse proper motion accuracies and precisions +by factors of a few. +Finally, care should be taken when attempting to extrapolate +reasonably uncertain, but ‘detected’ proper motions. In these cases +it may be more advantageous to not use the best-fit value, but +marginalise over all potential proper motions based on the likely +more robustly determined position and brightness. Alternatively, +the best-fit proper motion can be used, but ‘blurred’ out with the +detection’s precision, representing the proper motion offset PDF +ℎ′pm as a Gaussian with given mean proper motion and one-sigma +uncertainty. +7 +CONCLUSION +We described a model of the bulk motion of a random set of sources +through the Galaxy. The model uses the rotation curve of the Galaxy, +the Solar motion, and a prescription for the random motion of +sources due to e.g. their interaction history to create a statistical +distribution of potential proper motions of a source at a particular +set of sky coordinates and brightness. We compared this model to +Gaia sources in various sightlines across the Galactic plane – in the +mid-plane and out of plane – in different magnitude regimes, and +to the proper motions provided by the Besançon Galactic model, to +verify its robustness and accuracy. +Overall we find that our model matches the observed proper +motions with a high degree of both accuracy and precision, and +hence believe that our model is an acceptable description of the +statistical proper motions of sources. This will be invaluable when +matching the next generation of deep photometric surveys to other +datasets, in the regime where Gaia cannot provide individual proper +motions for sources. Without the inclusion of unknown proper mo- +tions we could be subject to a source separation bias that will impact +the number of cross-matches reported between two such catalogues. +This will be particularly crucial in the coming years in light of the +revolution in Galactic studies that the Rubin Observatory’s LSST +will bring, where – with its long time baseline back to previous +brighter infrared surveys – this effect has the potential to dominate +a systematic search for classes of sources such as faint, red objects. +We have made a Python version of the model described in this +paper available through the macauff GitHub codebase4. +ACKNOWLEDGEMENTS +TJW thanks the reviewers for their useful comments and sugges- +tions, which much improved the manuscript. TJW would also like to +thank Sergey Koposov for useful conversations and suggestions that +improved the accuracy of the model, and Tim Naylor for his helpful +discussions and proofreading assistance throughout this work. This +work has been supported by STFC funding for UK participation +in LSST, through grant ST/S 006117/1. This work has made use +of Python (Van Rossum & Drake 2009), and the SciPy (Virtanen +et al. 2020), NumPy (Harris et al. 2020), Astropy (Astropy Collab- +oration et al. 2013; Astropy Collaboration et al. 2018), astroquery +4 At this URL. +RASTI 000, 1–20 (2022) + +16 +Tom J. Wilson +(Ginsburg et al. 2019), Matplotlib (Hunter 2007), and F2PY (Pe- +terson 2009) Python modules, as well as NASA’s Astrophysics +Data System. +This work has made use of data from the European Space +Agency (ESA) mission Gaia (https://www.cosmos.esa.int/ +gaia), processed by the Gaia Data Processing and Analy- +sis Consortium (DPAC, https://www.cosmos.esa.int/web/ +gaia/dpac/consortium). Funding for the DPAC has been pro- +vided by national institutions, in particular the institutions partici- +pating in the Gaia Multilateral Agreement. +DATA AVAILABILITY +The datasets used in this manuscript were derived from sources in +the public domain, from the Gaia archive (https://gea.esac. +esa.int/archive/), TRILEGAL (http://stev.oapd.inaf. +it/cgi-bin/trilegal_1.7), and Besançon (https://model. +obs-besancon.fr/modele_home.php). +REFERENCES +Amendt P., Cuddeford P., 1991, ApJ, 368, 79 +Astropy Collaboration et al., 2013, A&A, 558, A33 +Astropy Collaboration et al., 2018, AJ, 156, 123 +Bianco F. B., et al., 2022, ApJS, 258, 1 +Bienaymé O., Robin A. C., Famaey B., 2015, A&A, 581, A123 +Budavári T., Szalay A. S., 2008, ApJ, 679, 301 +Czekaj M. A., Robin A. C., Figueras F., Luri X., Haywood M., 2014, A&A, +564, A102 +Eisenhardt P. R. 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We use several coordinate +systems: Heliocentric Cartesian space (𝑥, 𝑦, 𝑧); the observable, He- +liocentric Spherical coordinate space (𝑑, 𝑙, 𝑏); Heliocentric Cylin- +drical coordinates (ˆ𝑣𝑑, ˆ𝑣𝑙, ˆ𝑣𝑧); the Galactocentric Cylindrical co- +ordinate system (𝑅𝑐, 𝜙, 𝑧); the Galactocentric Spherical coordinate +system (𝑅𝑠, 𝜙, 𝜃); and the Galactocentric Cartesian coordinate sys- +tem (𝑋, 𝑌, 𝑍). +The Heliocentric Cylindrical coordinate system is defined as +the radial and tangential velocity components of the in-plane stellar +motions, as measured from the Sun in (and orthogonal to) the direc- +tion towards the source, as well as the orthogonal, vertical compo- +nent of the motion. Its transformation from Heliocentric Spherical +coordinates is a simple rotation from (𝑑, 𝑏) through the angle 𝑏 to +(ˆ𝑣𝑑, ˆ𝑣𝑧), albeit with the caveat that the direction of rotation varies +with the sign of 𝑏; its transformation from Galactocentric Cartesian +coordinates is a rotation through longitudinal angle 𝑙. +The Heliocentric Cartesian coordinate system can be obtained +from the Heliocentric Spherical coordinates, the observables, dis- +tance 𝑑, and Galactic coordinates 𝑙 and 𝑏, with +𝑥 = 𝑑 cos(𝑙) cos(𝑏) +(A1) +𝑦 = 𝑑 sin(𝑙) cos(𝑏) +(A2) +𝑧 = 𝑑 sin(𝑏), +(A3) +where we have used the ‘right-handed’ system that defines 𝑥 as +pointing towards the Galactic center from the Sun, to 𝑙 = 0◦; 𝑦 +towards 𝑙 = 90◦; and 𝑧 towards 𝑏 = +90◦. +The Galactocentric Cartesian coordinates are a simple shift of +zero-point, relative to the Heliocentric coordinates: +𝑋 = 𝑥 − 𝑅⊙ +(A4) +𝑌 = 𝑦 +(A5) +𝑍 = 𝑧 + 𝑧⊙ +(A6) +with a shift of the origin up by ≃ 8kpc and down ≃ 25pc in the 𝑋 +and 𝑍 directions (e.g. Jurić et al. 2008). +For the Galactocentric non-Cartesian coordinate systems, we +have to define new angles, as well as two additional radii. The radii +are fairly straightforward, being based simply on the Galactocentric +Cartesian coordinates. First, the Galactocentric Cylindrical radius, +being defined as the in-plane radius, is given by +𝑅𝑐 = +√︁ +𝑋2 + 𝑌2 +(A7) +RASTI 000, 1–20 (2022) + +Overcoming Separation Between Counterparts Due to Unknown Proper Motions +17 +and the Galactocentric Spherical radius by +𝑅𝑠 = +√︁ +𝑋2 + 𝑌2 + 𝑍2. +(A8) +The angle 𝜙, used in both Galactocentric Cylindrical and +Spherical coordinate systems, is defined as the angle around the +Galaxy – if viewed top-down, from the Galactic North Pole – from +the line running from the Sun through the Galactic center. This an- +gle, however, is defined as clockwise for the Galactocentric Cylin- +drical coordinates (and during the derivation of the Heliocentric +Spherical proper motions; see Section 2.3), but counter-clockwise +for the Galactocentric Spherical coordinate system. Equivalently, +this counter-clockwise angle can be considered as being measured +in the 𝑋 𝑌 plane, from the 𝑋 axis towards the 𝑌 axis (or −𝑌 axis, for +a clockwise defined 𝜙), analagous to how 𝑙 is defined as the angle +from the 𝑥 axis towards the 𝑦 axis. Finally, when in Galactocentric +Spherical coordinates, we could calculate 𝜃 by +𝜃 = arccos(𝑍/𝑅𝑠), +(A9) +where 𝜃 is the co-latitude, the angle as measured from the Cartesian +𝑍-axis, which differs from the system defining the Galactic latitude +𝑏, measured from the (𝑥, 𝑦) plane. We will find that we never need +to consider 𝜙 or 𝜃 themselves, as they will entirely be used to de- +fine rotation. The rotation matrices to convert from Galactocentric +Spherical to Galactocentric Cylindrical coordinates, or Galactocen- +tric Cylindrical to Heliocentric Cylindrical coordinates, along with +the conversion from Galactocentric Cylindrical to Galactocentric +Cartesian coordinates, are derived in Appendix A1. +A1 +Rotating Covariance Matrices +In this section we briefly outline the transformation, reflection, +and rotation matrices used to convert between four coordinate sys- +tems: the Galactocentric and Heliocentric Cylindrical, and Galac- +tocentric Cartesian and Spherical frames. First, we need to convert +from Galactocentric Cylindrical to Galactocentric Cartesian coor- +dinates, following the rotation curve-based methodology of Mróz +et al. (2019). Additionally, to work entirely in Sun-based radial, +tangential, and vertical velocity space, we need to rotate the co- +variance matrices calculated from Pasetto et al., and King et al., in +Galactocentric Cylindrical and Spherical coordinates respectively, +to ˆ𝑣𝑑 − ˆ𝑣𝑙 − ˆ𝑣𝑧 space. +A1.1 +Galactocentric Cylindrical to Galactocentric Cartesian +Rotation +First we need to calculate the rotation matrix describing the change +from Galactocentric Cylindrical to Galactocentric Cartesian coordi- +nates. Starting with Figure A1, we first consider the case of 𝑙 ≤ 180◦ +(left-hand schematics), a counter-clockwise rotation from 𝑅 through +angle 𝜔 to 𝑈. As these are left-handed cartesian coordinate systems, +this is a negative rotation, and hence the rotation matrix is +TCCW = +� cos(𝜔) +sin(𝜔) +− sin(𝜔) +cos(𝜔) +� +. +(A10) +We therefore need to calculate sin(𝜔) and cos(𝜔). sin(𝜔) can be +derived using the law of sines, and is given by +sin(𝜔) = 𝑑 +𝑅 sin(𝑙), +(A11) +while the cosine can be calculated from its corresponding law, +cos(𝜔) = +𝑅2 + 𝑅2 +0 − 𝑑2 +2 𝑅 𝑅0 +. +(A12) +Sun +GC +R +R0 +d +l +ω +Sun +GC +R +R0 +d +l +ω +360∘ − l +V +U +ϕ +R +ω +V +U +ϕ +R +ω +Figure A1. Schematic showing the transformation from 𝑅 − 𝜙 Galactocen- +tric Cylindrical coordinates to Galactocentric Cartesian 𝑈 − 𝑉 coordinate +system. +In the 𝑙 ≥ 180◦ case, right-hand side of Figure A1, we now have +a clockwise rotation, which in our left-handed coordinate system is +a positive rotation, +TCW = +�cos(𝜔) +− sin(𝜔) +sin(𝜔) +cos(𝜔) +� +. +(A13) +We can, as before, calculate the sine and cosine of 𝜔: +sin(𝜔) = 𝑑 +𝑅 sin(360◦ − 𝑙) = − 𝑑 +𝑅 sin(𝑙), +(A14) +cos(𝜔) = +𝑅2 + 𝑅2 +0 − 𝑑2 +2 𝑅 𝑅0 +. +(A15) +We can therefore now see that the changing from positive to +negative rotation in T, which changes the sign of sin(𝜔) in the +rotation matrix, is correlated with a change of sign of sin(𝜔). Thus +we can simplify our matrices, giving us +T𝑡 = �� +� +𝑅2+𝑅2 +0−𝑑2 +2 𝑅 𝑅0 +𝑑 +𝑅 sin(𝑙) +− 𝑑 +𝑅 sin(𝑙) +𝑅2+𝑅2 +0−𝑑2 +2 𝑅 𝑅0 +�� +� +. +(A16) +Expanding to the full three-dimensions of our problem, we note +that the third axis is unchanged by the rotation within the plane of +the Galaxy, and therefore the final axis has a trivial transformation, +giving +T𝑡 = +���� +� +𝑅2+𝑅2 +0−𝑑2 +2 𝑅 𝑅0 +𝑑 +𝑅 sin(𝑙) +0 +− 𝑑 +𝑅 sin(𝑙) +𝑅2+𝑅2 +0−𝑑2 +2 𝑅 𝑅0 +0 +0 +0 +1 +���� +� +. +(A17) +A1.2 +Galactocentric Cylindrical to Heliocentric Cylindrical +Rotation +Here we calculate the Pasetto et al. rotation from the Galactic center- +based cylindrical frame on to one centered on the Sun. Consider the +RASTI 000, 1–20 (2022) + +18 +Tom J. Wilson +Sun +GC +R +R0 +d +l +θ +̂vl +R +ϕ +̂vd +α +Sun +GC +R +R0 +d +l +θ +̂vl +R +ϕ +̂vd +α +360∘ − l +Figure A2. Schematic showing the transformation from 𝑅 − 𝜙 Galactocen- +tric coordinates to a Heliocentric ˆ𝑣𝑑 − ˆ𝑣𝑙 coordinate system. +left-hand panel of Figure A2; to rotate from 𝑅 − 𝜙 coordinates to +ˆ𝑣𝑑 − ˆ𝑣𝑙 is a negative (clockwise) rotation – working in the more +traditional right-handed coordinate system – through 𝛼, as well as a +mirroring around the 𝑅 axis (i.e. a flip of the 𝜙 axis on to the 𝑣𝑙 axis, +after rotation). Hence a rotation-then-mirror transformation matrix +would look like +TCW = +�1 +0 +0 +−1 +� � cos(𝛼) +sin(𝛼) +− sin(𝛼) +cos(𝛼) +� += +�cos(𝛼) +sin(𝛼) +sin(𝛼) +− cos(𝛼) +� +. +(A18) +As can be seen in Figure A2, 𝛼 = 𝜃, and hence +cos(𝛼) = cos(𝜃) = +𝑅2 + 𝑑2 − 𝑅2 +0 +2𝑅𝑑 +, +(A19) +sin(𝛼) = sin(𝜃) = 𝑅0 +𝑅 sin(𝑙). +(A20) +In the right-hand case of Figure A2, where 𝑙 ≥ 180◦, we now +have a counter-clockwise, positive rotation from 𝑅 through ˆ𝑣𝑑, but +still have a mirror reflection. This simply changes the sign of sin(𝛼) +in the rotation matrix, and hence +TCCW = +�1 +0 +0 +−1 +� �cos(𝛼) +− sin(𝛼) +sin(𝛼) +cos(𝛼) +� += +� cos(𝛼) +− sin(𝛼) +− sin(𝛼) +− cos(𝛼) +� +. +(A21) +Again, we can consider the inner triangle of 𝑅0−𝑑−𝑅 and calculate +angles for 𝛼 using 𝜃: +cos(𝛼) = cos(𝜃) = +𝑅2 + 𝑑2 − 𝑅2 +0 +2𝑅𝑑 +, +(A22) +sin(𝛼) = sin(𝜃) = 𝑅0 +𝑅 sin(360◦ − 𝑙) = − 𝑅0 +𝑅 sin(𝑙). +(A23) +Similar to Appendix A1.1, we can see that no matter the di- +rection of the rotation – i.e. if 𝑙 ≤ 180◦ or 𝑙 ≥ 180◦ – the sign +R +z +θ +ρ +β +R +z +θ +ρ +β +Sun +GC +Sun +GC +ρ +R0 +d +R0 +b +β +d +ρ +b +β +Figure A3. Schematic showing the rotation from 𝜌−𝜃 Galactocentric Spher- +ical coordinates to a Galactocentric Cylindrical 𝑅 − 𝑧 coordinate system. +of sin(𝛼) cancels with the sign within the sin(𝛼) elements of the +transformation matrix, and hence +TCCW = TCW = �� +� +𝑅2+𝑑2−𝑅2 +0 +2𝑅𝑑 +𝑅0 +𝑅 sin(𝑙) +𝑅0 +𝑅 sin(𝑙) +− +𝑅2+𝑑2−𝑅2 +0 +2𝑅𝑑 +�� +� +. +(A24) +We can also now explicitly include the third axis, the vertical +coordinate in our three-dimensional cylindrical reference frame, a +trivial continued alignment of the 𝑧 axis with our 𝑣𝑧 axis, giving +the final transformation matrix as +T𝑐 = +���� +� +𝑅2+𝑑2−𝑅2 +0 +2𝑅𝑑 +𝑅0 +𝑅 sin(𝑙) +0 +𝑅0 +𝑅 sin(𝑙) +− +𝑅2+𝑑2−𝑅2 +0 +2𝑅𝑑 +0 +0 +0 +1 +���� +� +. +(A25) +A1.3 +Galactocentric Spherical to Galactocentric Cylindrical +Rotation +Finally, we consider the King et al. rotation from a spherical refer- +ence frame into a cylindrical one, albeit still centered on the Galactic +center. To do this, we consider the frame goes from 𝜌 − 𝜙 − 𝜃 to +𝑟 −𝜙−𝑧; here, (𝜌, 𝜃) and (𝑅, 𝑧) are both in a right-handed cartesian +coordinate systems. We therefore define our rotation matrices in the +opposite sense to Section A1.1, +TCW = +� cos(𝛽) +sin(𝛽) +− sin(𝛽) +cos(𝛽) +� +, +TCCW = +�cos(𝛽) +− sin(𝛽) +sin(𝛽) +cos(𝛽) +� +. +(A26) +The upper case of Figure A3 shows the rotation necessary for 𝑏 ≥ +0◦, with the left hand side showing the clockwise rotation through +𝛽, and the right hand side showing a schematic of the various known +distances and angles. Here, considering a negative – clockwise in +a right-handed frame – rotation through 𝛽, we can calculate sin(𝛽) +RASTI 000, 1–20 (2022) + +Overcoming Separation Between Counterparts Due to Unknown Proper Motions +19 +and cos(𝛽) as +sin(𝛽) = 𝑑 +𝜌 sin(𝑏), +(A27) +where 𝜌2 = 𝑅2 +0 + 𝑑2 − 2𝑅0𝑑 cos(𝑏), and +cos(𝛽) = +𝑅2 +0 + 𝜌2 − 𝑑2 +2𝑅0𝜌 +. +(A28) +For the lower case of Figure A3, 𝑏 < 0◦, with a positive rotation, +cos(𝛽) = (𝑅2 +0 + 𝜌2 − 𝑑2)/(2𝑅0𝜌), as previously, as the triangle +is unchanged, just mirrored. sin(𝛽) is a little more complicated to +derive, however, as the triangle in Figure A3 uses 𝑏 as its modulus +value, but it is negative in value. Using |𝑏| explicitly, the law of +sines gives +sin(𝛽) = 𝑑 +𝜌 sin(|𝑏|), +(A29) +as previously. However, if we use, as we will in practice, 𝑏′ = −|𝑏|, +we get sin(𝑏′) = − sin(|𝑏|), and hence sin(𝛽) = −𝑑/𝜌 sin(𝑏′). +Once again, we find – as with Appendices A1.1 and A1.2 – +that the sign of sin(𝛽) cancels with the sign of the term within +the rotation matrices. Thus, for either orientation – positive and +negative Galactic latitude – the rotation matrix from Galactocentric +spherical to Galactocentric cylindrical coordinates (from the (𝜌, 𝜃) +to (𝑟, 𝑧) plane) is given by +R𝑠 = +� +cos(𝛽) +𝑑/𝜌 sin(𝑏) +−𝑑/𝜌 sin(𝑏) +cos(𝛽) +� +, +(A30) +with cos(𝛽) still defined consistently as before. +While we are using 𝜙 to represent the two azimuthal angles, +they are defined in the opposite sense (see Section A). We therefore +need to reflect the 𝜙 axis through the (𝑟, 𝑧) plane, after the rotation +has occurred, given by +R𝜙,reflect = �� +� +1 +0 +0 +0 +−1 +0 +0 +0 +1 +�� +� +. +(A31) +Thus, our full three-dimensional transformation matrix is given by +R𝑠 = �� +� +cos(𝛽) +0 +𝑑/𝜌 sin(𝑏) +0 +−1 +0 +−𝑑/𝜌 sin(𝑏) +0 +cos(𝛽) +�� +� +, +(A32) +APPENDIX B: CONVOLUTION MATHEMATICS FOR +COUNTERPART AND NON-COUNTERPART +HYPOTHESES +B1 +Counterpart Likelihood Including Proper Motion +In this Appendix we detail the derivation of the inclusion of the +proper motion PDF in the hypothesis that two objects are one astro- +physical object given their separation. Starting from a similar place +to Wilson & Naylor (2018a)’s equation 14, we have +𝐺′ = ++∞ +∬ +−∞ +𝑝(Δ𝑢, Δ𝑣) ++∞ +∬ +−∞ +ℎ𝛾(𝑥0 − 𝑥𝛾, 𝑦0 − 𝑦𝛾)× +ℎ𝜙(𝑥𝜙 − 𝑥0 − Δ𝑢, 𝑦𝜙 − 𝑦0 − Δ𝑣) d𝑥0 d𝑦0 dΔ𝑢 dΔ𝑣. +(B1) +Here we have the simultaneous marginalisation over an unknown +common position – dropping the prior, 𝑝(𝑥0, 𝑦0) for being uniform +and independent of unknown position (and proper motion), as per +Wilson & Naylor (2018a) – and a marginalisation over the PDF of all +unknown proper motions drifts 𝑝 (here representing proper motions +in the two orthogonal sky directions with 𝑢 and 𝑣). Substituting +Δ𝑥 = 𝑥𝜙 − 𝑥𝛾 and Δ𝑦 = 𝑦𝜙 − 𝑦𝛾 into ℎ𝛾 we get +𝐺′ = ++∞ +∬ +−∞ +𝑝(Δ𝑢, Δ𝑣) ++∞ +∬ +−∞ +ℎ𝛾(𝑥0 − 𝑥𝜙 + Δ𝑥, 𝑦0 − 𝑦𝜙 + Δ𝑦)× +ℎ𝜙(𝑥𝜙 − 𝑥0 − Δ𝑢, 𝑦𝜙 − 𝑦0 − Δ𝑣) d𝑥0 d𝑦0 dΔ𝑢 dΔ𝑣. +(B2) +Now we change variables from 𝑥0 and 𝑦0 to 𝑥 and 𝑦 via 𝑥 = 𝑥𝜙−𝑥0− +Δ𝑢, 𝑦 = 𝑦𝜙 − 𝑦0 −Δ𝑣. This rearranges such that 𝑥0 −𝑥𝜙 = −Δ𝑢 −𝑥, +𝑦0 − 𝑦𝜙 = −Δ𝑣 − 𝑦, and thus +𝐺′ = ++∞ +∬ +−∞ +𝑝(Δ𝑢, Δ𝑣) ++∞ +∬ +−∞ +ℎ𝛾(Δ𝑥 − Δ𝑢 − 𝑥, Δ𝑦 − Δ𝑣 − 𝑦)× +ℎ𝜙(𝑥, 𝑦) d𝑥 d𝑦 dΔ𝑢 dΔ𝑣. +(B3) +As per Wilson & Naylor (2018a), we note that the inner integral is +the definition of a convolution, and thus setting +𝐺(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) ≡ (ℎ𝛾 ∗ ℎ𝜙)(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) += ++∞ +∬ +−∞ +ℎ𝛾(Δ𝑥 − Δ𝑢 − 𝑥, Δ𝑦 − Δ𝑣 − 𝑦)ℎ𝜙(𝑥, 𝑦) d𝑥 d𝑦, +(B4) +we have +𝐺′ = ++∞ +∬ +−∞ +𝑝(Δ𝑢, Δ𝑣)𝐺(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) dΔ𝑢 dΔ𝑣. +(B5) +Now it is clear that this is itself a convolution, of 𝑝 and 𝐺, and hence +we can now write +𝐺′(Δ𝑥, Δ𝑦) ≡ (𝑝 ∗ 𝐺)(Δ𝑥, Δ𝑦) += ++∞ +∬ +−∞ +𝑝(Δ𝑢, Δ𝑣)𝐺(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) dΔ𝑢 dΔ𝑣. +(B6) +We note that our equation B1 is of similar form to equation 5 +of Kerekes et al. (2010), with the interchange of integrals. Here we +have chosen to construct a semi-analytic simulated model for the +construction of the distribution of unknown proper motions, while +Kerekes et al. built theirs from survey data. However, as discussed +in Section 5, we can substitute such a data-driven distribution of +proper motions within our matches, using any valid distribution as +𝑝(Δ𝑢, Δ𝑣) (or ℎ′pm). +Finally, consistent with Wilson & Naylor (2018a)’s original +derivation, we explicitly remind the reader that the AUFs ℎ must be +defined such that ℎ(𝑥, 𝑦) = ℎ(−𝑥, −𝑦). +B2 +Unrelated Object Likelihood Including Proper Motion +For the case where the sources are unrelated to one another, we have +a slightly different equation to that of equation B1; something more +RASTI 000, 1–20 (2022) + +20 +Tom J. Wilson +like equation 10 of Budavári & Szalay (2008), +𝐺′ = ++∞ +∬ +−∞ +𝑝(Δ𝑢, Δ𝑣) +� +∞ +∬ +−∞ +ℎ𝛾(𝑥0 − 𝑥𝛾 − Δ𝑢, 𝑦0 − 𝑦𝛾 − Δ𝑣) +d𝑥0 d𝑦0 +� +dΔ𝑢 dΔ𝑣 × ++∞ +∬ +−∞ +𝑝(Δ𝑢, Δ𝑣) +� +∞ +∬ +−∞ +ℎ𝜙(𝑥0 − 𝑥𝜙 − Δ𝑢, 𝑦0 − 𝑦𝜙 − Δ𝑣) +d𝑥0 d𝑦0 +� +dΔ𝑢 dΔ𝑣. +(B7) +Here, as with equation B1, we have explicitly assumed that 𝑝(𝑥0, 𝑦0) +is independent of both unknown position and proper motion, and +thus can be removed as a factor from the equation. As ℎ𝛾 and +ℎ𝜙 are normalised PDFs, the inner integral is trivially integrable +to unity; but with 𝑝, the PDF of unknown proper motions, also +normalised, the outer integral then also evaluates to unity. Thus +we have the trivial case, for unrelated objects, that 𝐺 = 1 and +𝐺′ = 1. In these cases, as with the counterpart hypothesis having +a prior 𝑝(𝑥0, 𝑦0) = 𝑁𝑐 as per Wilson & Naylor (2018a), we can +say that the equivalent priors in the ‘unrelated’ hypothesis case +are 𝑝(𝑥0, 𝑦0) = 𝑁 𝑓 , Wilson & Naylor (2018a)’s ‘field’ source +density. Hence, for the hypothesis of two sources being unrelated +to one another and two detections of different sky objects, the PDF +describing the likelihood of the objects having some separation +is independent of proper motion, just as it is independent of the +respective sources’ AUFs. +This paper has been typeset from a TEX/LATEX file prepared by the author. +RASTI 000, 1–20 (2022) + diff --git a/b9FAT4oBgHgl3EQfXx3V/content/tmp_files/load_file.txt b/b9FAT4oBgHgl3EQfXx3V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa253f45607ea967cce99113be194c02f2cd24ab --- /dev/null +++ b/b9FAT4oBgHgl3EQfXx3V/content/tmp_files/load_file.txt @@ -0,0 +1,1251 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf,len=1250 +page_content='RASTI 000, 1–20 (2022) Preprint 23 January 2023 Compiled using RASTI LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 Overcoming Separation Between Counterparts Due to Unknown Proper Motions in Catalogue Cross-Matching Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson1★ ID 1School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' in original form ZZZ ABSTRACT To perform precise and accurate photometric catalogue cross-matches – assigning counterparts between two separate datasets – we need to describe all possible sources of uncertainty in object position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' With ever-increasing time baselines between observations, like 2MASS in 2001 and the next generation of surveys, such as the Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Rubin Observatory’s LSST, Euclid, and the Nancy Grace Roman telescope, it is crucial that we can robustly describe and model the effects of stellar motions on source positions in photometric catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' While Gaia has revolutionised astronomy with its high-precision astrometry, it will only provide motions for ≈10% of LSST sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' additionally, LSST itself will not be able to provide high-quality motion information for sources below its single-visit depth, and other surveys may measure no motions at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This leaves large numbers of objects with potentially significant positional drifts that may incorrectly lead matching algorithms to deem two detections too far separated on the sky to be counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To overcome this, in this paper we describe a model for the statistical distribution of on-sky motions of sources of given sky coordinates and brightness, allowing for the cross- match process to take into account this extra potential separation between Galactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We further detail how to fold these probabilistic proper motions into Bayesian cross-matching frameworks, such as those of Wilson & Naylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This will vastly improve the recovery of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' very red objects across optical-infrared matches, and decrease the false match rate of photometric catalogue counterpart assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Key words: Algorithms – methods: statistical – catalogues – astrometry – proper motions – Galaxy: kinematics and dynamics 1 INTRODUCTION Counterpart assignment, the merging of bandpass detections in two (or more) datasets, enables a wide range of value-added science, and is therefore a crucial aspect of many areas of astronomical re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Fundamentally, we require the ability to answer the question ‘are these two detections observations of two different objects, or two observations or the same astrophysical object?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, to pro- vide accurate and precise cross-matches between two photometric catalogues, we require a complete description of all sources of sep- aration between detections of a single astrophysical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Unfortunately for astronomers, there are many reasons for the same source, detected by two different telescopes in different parts of the world at different times, to have recorded positions that are not perfectly aligned with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The first, and frequently assumed only, contribution is that from the act of measuring the position of the source on the detector image as part of the catalogue creation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This ‘centroid’ uncertainty is related to the size ★ Email: t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='wilson@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' onoddil@pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='me of the telescope and the wavelength of the observation, as well as the atmospheric seeing, if applicable – all of which affect the telescope ‘point spread function’ (PSF), as well as the signal-to- noise ratio (SNR) of the detection, related to its brightness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' King 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson & Naylor (2017) highlighted an additional source of positional shift that can affect detections in crowded fields: sources, too close together on the sky to be resolved by the telescope, can appear as a single observation, leading the fainter source to influence the position of the (assumed singular) brighter object (sometimes referred to as ‘classical confusion’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' see also e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hogg 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we consider an extra source of apparent separation be- tween detections: that of the physical motion of the source across the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If the observations to be combined are sufficiently sepa- rated in time, the ‘proper motion’ of sources introduces a drift in the separation between consecutive measurements of the objects’ locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus for pairs of observations with significant baselines, the proper motion-induced separations can become significant for large enough numbers of objects that, if we failed to consider these motions, we would decide the objects were too far apart to be coun- terpart detections of one object, and fail to assign them properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='08536v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='SR] 20 Jan 2023 2 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson For probabilistic cross-matching algorithms this problem of object drift is further compounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is not only objects with sig- nificant proper motion that suffer, those few objects with motions large enough to render them completely incompatible with the hy- pothesis that the two detections are counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' All objects, even those with relatively small motions, are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Any motion on the same length scale as the astrometric precisions will impact the derived match confidence, and potentially render quoted match or non-match probabilities meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This issue of your chosen model completely encapsulating the information contained within your data (or not) is often referred to as ‘model (mis)specification’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Therefore, the effect of source motion must be accounted for, even if not so extreme as to completely move an object beyond its prior position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If it is not taken into account, users of any resulting cross- match tables may not be able to put trust in the quoted match likelihoods and be able to take reliable, high-confidence cuts of the merged datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This effect is particularly important for the upcoming Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Rubin Observatory’s Legacy Survey of Space and Time (LSST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019), for a few key reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, it will operate from ∼2025-2035, and thus have a two or three-decade baseline to the numerous surveys that operated during the 2000s and 2010s, such as 2MASS (Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2006) or SDSS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' And second, it will lack measured proper motions for almost all of its sources for a large fraction of its survey lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In part this is because the survey will require a multi-year baseline before reliable proper motions can be derived, but more simply because most ob- jects within the full LSST catalogue will be below the completeness limit of the single-visit images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Even in this specific case, with Ru- bin’s high-fidelity time-series capabilities, proper motions will only ever be available for objects that appear in multiple images, which sets the proper motion magnitude limit much higher than that of inclusion in the full coadd catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Worse still, the sheer number density of objects in the full LSST catalogue mean that up to 10 LSST sources will be potential counterparts to every single oppos- ing catalogue object, and the ‘re-shuffle’ of objects, even of order the precision of the measured positions, may lead to false matches being returned for a sizeable fraction of the catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, the survey will be especially susceptible to this match misspecification due to proper motion drift, primarily at faint magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Addition- ally, other upcoming missions such as Euclid (Laureijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2011) and the Nancy Grace Roman Space Telescope (Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2012) will likely lack the multi-epoch capabilities that Rubin and LSST offer, but still suffer the effects of decade-long time baselines back to previous generations of deep surveys, such as SDSS or VISTA (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' VHS, McMahon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As time goes on, and we accumulate increasing numbers of surveys we wish to combine to maximum scientific return, we will increasingly no longer be able to ignore even relatively small lev- els of apparent on-sky motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' One obvious solution is to use the individual proper motions available through datasets such as the Gaia (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2016) mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These positions – combined with the rate-of-change of position from the proper mo- tions – can be ‘fast-forwarded’ through time, allowing for sources to be placed in the epoch of the opposing catalogue, removing on- sky drift as a factor in considering the separation between sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, this is impractical for surveys such as LSST for a couple of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, and simplest, is that proper motions are not available for the entire Gaia catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Something like 20% of sources in the early Data Release 3 (eDR3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lin- degren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2021) do not have the five- or six-parameter solutions necessary to include proper motions, and a not insignificant frac- tion of those that have quoted proper motions have uncertainties that render the quoted values useless for any meaningful position projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Second, this would result in needing to run two Gaia- to-other-catalogue matches, merge the most likely of those matches in turn, and then run an internal Gaia-Gaia look up to get the in- ner join of the two catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This would significantly affect the quality of the resulting matched datasets, with probabilistic cross- matching processes not being able to provide the proper probability of sources in the two ‘other catalogue’ datasets matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Third, and most crucial, is the dynamic range consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Gaia is, for all its superb data, a relatively bright survey – at least by LSST standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It also lacks coverage against longer wavelength surveys, where Galactic extinction is less oppressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' LSST will, with its ∼7 magnitude fainter completeness limit, include at least an order of magnitude more stars (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019), and VISTA, as an example infrared (IR) catalogue of consideration as an ancillary dataset to extend LSST information with, will have little overlap with Gaia due to differing wavelength coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Other surveys with deeper completeness limits, such as Euclid, Roman, and SDSS, will also suffer significant numbers of matches beyond the Gaia com- pleteness limit, with neither offering reliable proper motions at 21st magnitude or fainter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, even if we did decide to peg Gaia as our gold standard, this would leave perhaps 9 out of every 10 LSST Galactic sources without a proper motion match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Those LSST ob- jects, and many others in other catalogues, would be in need of a separate, and worse, cross-match once we had handled those few objects with a Gaia proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If we cannot trust matches between observations with signifi- cant time between observations, and we cannot necessarily use an- cillary datasets with measured proper motions, how can we recover robust catalogue counterpart assignments through cross-matching?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Naively, we might think that we can ‘re-center’ the distribution of offsets, by subtracting some mean separation between all of our counterpart pairings to account for the drift of our objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' How- ever, different average proper motions across the dynamic range of the two catalogues would cause further systematics, affecting the distribution of counterpart separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' One may also think to use a Galactic model that calculates stellar velocities, and hence provides proper motions as viewed from Earth, for example the Besançon model (Robin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, as discussed in more detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8, there are various reasons that these proper motions do not provide robust enough statistics for the determination of a statistical separation drift between two potential counterpart stars during a probabilistic cross-match process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, in this paper we put forward a model to build a statistical distribution of proper motions of sources, based on their Galactic motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Modelling all sources of motion a star orbiting the Galactic center is subject to – its bulk circular orbit, and any ‘random’ scatter of sources from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' stellar cluster interactions – combined with the Sun’s motion, we are able to build a picture of the apparent motion of the given object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To do so, we begin with the velocity of the star as it orbits around the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The conversion from velocity – in units like km s−1 – to proper motion – in units like arcsec yr−1 – is, roughly speaking, an inverse relation with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As we detail later, instead of using distance directly, we intend to use the brightness of an object as its more readily available surrogate, accepting that this is only an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, combining the apparent motions of a wide range of objects at the same sky position and brightness we obtain all proper motions such a source might have – faint M dwarfs close by would have larger apparent motions than very intrinsically bright super- RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 3 giants, but all ‘types’ of object contribute to the spread of motion drifts a source of this particular brightness could have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We are not particularly interested in a precise reconstruction of Galactic orbital dynamics – not being overly concerned with the details of spiral arm dynamics, or the specifics of the Milky Way Bar shape, for ex- ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The use of the proper motions here is to ‘spread’ the motion drift, the separation between potential counterparts, allowing for the recovery, and increasing the reliability of match probability, of these objects within a Bayesian cross-matching framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' There- fore, the exact shape is less important than its central location and width;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' so long as these match reality to a fraction of the precision of the objects’ positions and the underlying distribution width to a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 − 2, the model has served its purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The probability of two stars being counterpart ranges over many orders of magnitude, and hence the resulting probability density functions (PDFs) we derived to model the statistical proper motions having widths, and overall PDF heights, correct to a factor two is sufficient to improve counterpart recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also desire computational simplicity, and thus speed, over an overly prescriptive or detailed exact model of the Galaxy, as this model must fit within a wider counterpart assign- ment framework and be able to be run on-the-fly for some arbitrary sets of sky positions and brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once we have constructed our distribution of theoretical proper motions, we can then consider their effect on our potential cross- match pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we can, in a similar way to how we would han- dle known Gaia proper motions, translate one object’s position into the epoch of the second catalogue observations, and consider the additional separation caused by the motion of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We must then test all modelled proper motions, with appropriate weighting, ultimately accounting for these potential additional time-based sep- arations in answering the question of whether these two detections are two physical sky objects or one source viewed twice in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Paper Layout This paper is split into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, we detail the construction of a simple analytic model for the statistical distribution of potential proper motions of a source of a given magnitude and sky coordi- nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In Section 2 we detail the constituent parts necessary to build the model of proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We describe the process of building the distributions in Section 3, while Section 4 discusses the preci- sion and accuracy of the model at various Galactic sightlines and brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we will return to Gaia, using its high-precision stellar proper motions across many Galactic sightlines to evaluate and corroborate our model distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The second part of this work describes how to include this unknown proper motion distribution – or any distribution of proper motions, theoretical or poorly constrained yet detected – in the cross-matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We discuss the mathematical framework necessary for including proper motion drift in the evaluation of the separation between potential counterpart detections in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also touch upon how to use these statistical distributions of proper motion drift as a discriminator between stars and galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Concluding remarks are given in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In Appendix A we outline the various coordinate systems and derive the transformation matrices used throughout this work, while in Appendix B we detail the inclusion of the proper motion PDFs within a probabilistic cross-match astrometric separation likelihood framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2 CONSTRUCTING THE PROPER MOTIONS We need to build a model to describe the observed motions of sources across the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' There are, essentially, three components that matter: first, the ‘peculiar’ motion of the Sun itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' second, the expected velocity of a source moving with the Galactic rotation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' and third, the random velocities of the Galactic sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' these will be discussed individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, we must consider how we will build this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As the model involves consideration of the Galactic rotation (and we will see later that source random motion is location depen- dent), we will require a description of the position of the source in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The first two components are easy: sky coordinate in Galactic coordinates (converting Equatorial 𝛼 and 𝛿 by rotation to longitude 𝑙 and latitude 𝑏, if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The only other component we would need is a distance, or parallax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' however, if we have paral- lax we likely have a unique proper motion, as these are generally fit for simultaneously, and hence we use the next best proxy: magni- tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This will blend several ‘types’ of source together (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' dwarfs and giants of the same brightness are at different distances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We will see that while this might introduce extra scatter in the proper motion drifts, our models match Gaia proper motions at magnitude cuts well – and indeed account for the fact that we do not know the type of any individual source from its photometry a priori!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we do need some distance metric, and hence we turn to the TRILEGAL simulations (Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2005) to provide a theoretical magnitude-distance relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, while our catalogue sources have their proper motions built as a function of sky coordi- nates and photometric brightness, our model is coordinate/distance based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In the following sections we describe how we formulate a description of the observed proper motion of a set of sources based on their given parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Solar Peculiar Motion The first component in the Galactic motion is the unique velocity of the Sun, relative to the local standard of rest (LSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Sun’s motion through the Galaxy will induce a ‘secular’ parallax effect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' distance-dependent, albeit non-periodic, as a trigonometric par- allax would be) in the apparent movement of all other sources in the sky, with opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence we need to know the Sun’s mo- tion, relative to this ‘zero point’ motion, the LSR, defined in the Heliocentric Cartesian coordinate frame, (𝑈, 𝑉, 𝑊) – velocities corresponding to the (𝑥, 𝑦, 𝑧) coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we use the values of Schönrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2010): 𝑈⊙ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 km s−1 𝑉⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 km s−1 𝑊⊙ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Galactic Rotation The main component of our model that will dictate the Galaxy-wide observed motions of sources is that of the Galactic rotation, and the stellar streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Descriptions of this motion go back to Oort (1927), with the Oort constants describing the motion of stars on closed orbits around the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, through modern kine- matic surveys, obtaining the three-dimensional velocities and in- dependent distances to a host of well-characterized objects, it is possible to directly measure the rotation curve of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) 4 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson Thus, we can derive the average tangential velocity of sources or- biting the Galactic center at a given radius, here following Model 3 of Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Obtaining the rotational velocity Θ at a given Galactocentric radius 𝑅𝑐, we can transform this Galactocentric Cylindrical az- imuthal velocity into a Galactocentric Cartesian coordinate frame and subtract the Solar peculiar and LSR motion, obtaining 𝑈1, 𝑉1, and 𝑊1 (Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019, equations 5-7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To do so, we use the transformation T𝑡 = ����� � 𝑅2 𝑐+𝑅2 ⊙−𝑑2 ip 2 𝑅𝑐 𝑅⊙ 𝑑ip 𝑅𝑐 sin(𝑙) 0 − 𝑑ip 𝑅𝑐 sin(𝑙) 𝑅2 𝑐+𝑅2 ⊙−𝑑2 ip 2 𝑅𝑐 𝑅⊙ 0 0 0 1 ����� � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2) here 𝑅⊙ is the Solar Galactocentric Cylindrical radius, 𝑑ip is the in-plane distance from the Sun to the particular location, and 𝑙 is Galactic longitude – see Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Deviating from Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019), we set (𝑈𝑠, 𝑉𝑠, 𝑊𝑠), the non-circular motion of the source, all to zero, and hence have 𝑈1 = Θ(𝑅𝑐) × 𝑑ip 𝑅𝑐 sin(𝑙) − 𝑈⊙ 𝑉1 = Θ(𝑅𝑐) × 𝑅2𝑐 + 𝑅2 ⊙ − 𝑑2 ip 2 𝑅𝑐 𝑅⊙ − 𝑉⊙ − Θ⊙ 𝑊1 = −𝑊⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (3) Once we have the Galactocentric Cartesian components of the rota- tional velocity, relative to the Sun, we can transpose into the in-plane Heliocentric radial and tangential velocities, 𝑣𝑑 = 𝑈1 cos(𝑙) + 𝑉1 sin(𝑙) 𝑣𝑙 = −𝑈1 sin(𝑙) + 𝑉1 cos(𝑙), (4) along with 𝑣𝑧 = 𝑊1, since the two axes are still in alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, once we have appropriate Heliocentric Cylindrical velocities, we can construct our latitudinal velocity as 𝑣𝑏 = 𝑣𝑧 cos(𝑏) − 𝑣𝑑 sin(𝑏) (5) and relate the Heliocentric longitudinal and latitudinal velocities to their proper motions through 𝜇𝑙∗ ≡ 𝜇𝑙 cos(𝑏) = 𝑘 × 𝜋𝑣𝑙, (6) 𝜇𝑏 = 𝑘 × 𝜋𝑣𝑏, (7) where 𝜋 is the parallax of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As our distances come from Galactic models, we assume they are not subject to any observational bias or uncertainty, and simply treat 𝜋−1 = 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The factor 𝑘 describes the translation from units of km s−1 kpc−1 to mas yr−1, and is given by 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2108 mas yr−1 km−1 s kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In practice, the TRILEGAL simulations do not provide either distance or parallax, but provide its distance modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, to obtain a (three-dimensional) distance 𝑑 in kpc, we invert the absolute magnitude equation: 𝑑 = 10−3 × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2(𝑚−𝑀)+1 (8) where 𝑚 − 𝑀 is the distance modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 Asymmetric Drift Velocity The above equations describe the average Galactic rotation velocity around the center of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, objects have other sources of velocity that impact their observed proper motions, and this leads to a deviation away from the expected velocity, and thus proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is termed the asymmetric drift velocity, and essentially controls how much of the theoretical velocity a source should have is taken by other, random motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, we must include this component in the velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We model three Galactic components (see Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 for more details of their construction) in our simulations: the Galac- tic thin and thick discs, and a single Galactic (outer) halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Each of these is given their own asymmetric drift, as a measure of the levels to which their motions are different from ‘pure’ streaming motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We assume the thin disc of the Galaxy has an azimuthal asymmetric drift velocity of 10 km s−1 (Robin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019) used Classical Cepheids in the derivation of their ro- tation curve, we also assume the rotation curve is valid for the thin disc and already folds in any drift velocity, and therefore just need to consider the relative drift velocities of the thick disc and the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We use a thick disc drift velocity of 49 km s−1 (Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2012a), and give the halo a drift velocity of 240 km s−1 (Golubov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2013), to essentially counteract the motion of the LSR, mod- elling the halo as stationary relative to the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore use 𝑣𝑎,𝜙 = {0 , 39 , 230} km s−1 for the relative thin disc, thick disc, and halo drift velocities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For a given location in the Galaxy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' our decomposition of the drift velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' from Galactocentric Cylindrical coordinates into He- liocentric Cylindrical coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' is given by the transformation 𝒗′drift = T𝑐 𝒗drift,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (9) with T𝑐 = ����� � 𝑅2 𝑐+𝑑2 ip−𝑅2 ⊙ 2𝑅𝑐𝑑ip 𝑅⊙ 𝑅𝑐 sin(𝑙) 0 𝑅⊙ 𝑅𝑐 sin(𝑙) − 𝑅2 𝑐+𝑑2 ip−𝑅2 ⊙ 2𝑅𝑐𝑑ip 0 0 0 1 ����� � (10) and 𝒗drift = �� � 0 𝑣𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='𝜙 0 �� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (11) with 𝑣𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='𝜙 taking on any one of the three given drift velocities above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' depending on which component of the Galaxy is being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For details on the derivation of this transformation (rotation and mirror) matrix, see Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 Velocity Dispersion The asymmetric drift velocity suggests that some of the motion that ought to be used by a given source in its rotation around the Galaxy is being used otherwise, in a random component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, a collection of sources in a particular part of the Galaxy will have some spread of their velocities around some mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, to be able to model our collection of sources in the Galaxy we require a description of the dispersion of the velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Each component of the Galaxy modelled – thin and thick discs, and halo – have their own prescription of velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In addition, when simulating sources, we do not initially know to which component to assign a given simulated object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Therefore we simply generate a set of proper motion realisations for each of the three components, weighted according to their a priori density at that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, in the following sections we also describe the formulation of each component’s density profile, which are simply re-normalised by the sum of their densities to provide a prior probability, used as the weight for the proper motion distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 5 We currently do not consider the Bulge, a common component of Galactic simulation models such as TRILEGAL, in our proper motion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We chose to ignore this additional component for this initial, exploratory model, focussing on relatively bright sources, with detected Gaia proper motions to compare and verify our model against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This should mean they are sufficiently far from the Galactic center to not be influenced by the Bulge, avoiding the complexities that the inner region of the Galaxy impose on velocities of orbiting stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, with the upcoming LSST survey and the need to model much fainter, more distant objects in the next few years, we will investigate a robust Galactic Bulge/Bar model for inclusion within this proper motion framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We simply conclude, for now, that it would be easy to add additional components, and we could model a simple Bulge component after e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once a given Galactic component has its dispersion vector – its covariance matrix – in the Heliocentric Cylindrical coordinate system, then a realisation is drawn from a multivariate normal, given by 𝒗noisy,𝑖 ∼ N (𝒗 − 𝒗′drift,𝑖, 𝚺′𝑖) (12) where 𝑖 ∈ {thin, thick, halo} and 𝚺′𝑖 is the Cylindrical frame rotated covariance matrix of the 𝑖th component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Thin Disc The thin disc is modelled as an exponentially decaying density profile with given radial and vertical scale heights, as per Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) and Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008): 𝜌(𝑅𝑐, 𝑧) = Γ exp � − 𝑅𝑐 − 𝑅⊙ 𝑙thin − 𝑧 + 𝑧⊙ ℎthin � , (13) with 𝑅⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='09 kpc (Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019), 𝑧⊙ = 25 pc (Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Γ is an irrelevant normalisation constant, used simply to explicitly cancel in the re-normalisation from density to weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The radial and vertical scale lengths we use are the bias-corrected values calculated by Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' : 𝑙thin = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 kpc, and ℎthin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The dispersion vector for the thin disc is based on observations of RAVE stars from Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2012b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, the nature of the observations limit their calculation of covariances to approximately 1 kpc from the Sun, and we need to extrapolate these dispersions out to perhaps five times that distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus we turn to Amendt & Cuddeford (1991) for relations between the various (co-)variances in the dispersion vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, we assume that the variance in the vertical direction, 𝜎2𝑧𝑧, scales with radial distance in the mid-plane of the Galaxy: 𝜎2 𝑧𝑧 (𝑅𝑐, 𝑧 = 0) = 𝜎2 𝑧𝑧(0, 0) exp � − 𝑅𝑐 𝑙thin � = 𝜎2 𝑧𝑧(𝑅′⊙, 0) exp � − 𝑅𝑐 − 𝑅′⊙ 𝑙thin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (14) As discussed by Amendt & Cuddeford, this scaling relation is also sometimes assumed for 𝜎2 𝑅𝑐𝑅𝑐, but a second, valid formalism can be used where the rotation curve of the Galaxy is flat – which it should be safe to assume given the very small gradient from Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019) for most of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This formalism is based on a constant Toomre (1964) local stability parameter, and gives 𝜎2 𝑟𝑟 ≡ 𝜎2 𝑅𝑐𝑅𝑐 ∝ 𝑅2 𝑐 exp � −2 𝑅𝑐 ℎthin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (15) Hence we use1 𝜎2 𝑟𝑟 (𝑅𝑐, 0) = 𝜎2 𝑟𝑟 (𝑅′⊙, 0) � 𝑅𝑐 𝑅′⊙ �2 exp � −2 𝑅𝑐 − 𝑅′⊙ ℎthin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (16) For the vertical extrapolation, we assume a local Taylor expan- sion to first order (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' we extrapolate linearly to above and below the plane, from 𝑧 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We limit this extrapolation to the inner one kiloparsec of the plane, and assume a constant dispersion beyond that, and hence: 𝜎2 𝑧𝑧(𝑅𝑐, 𝑧) ≃ 𝜎2 𝑧𝑧(𝑅𝑐, 0) + min (1 kpc, |𝑧|) 𝜕𝜎2𝑧𝑧 (𝑅𝑐, 0) 𝜕|𝑧| (17) 𝜎2 𝑟𝑟 (𝑅𝑐, 𝑧) ≃ 𝜎2 𝑟𝑟 (𝑅𝑐, 0) + min (1 kpc, |𝑧|) 𝜕𝜎2𝑟𝑟 (𝑅𝑐, 0) 𝜕|𝑧| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (18) Using the Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' data in the range 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 kpc ≤ 𝑅𝑐 ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8 kpc, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 kpc ≤ 𝑧 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 kpc, we find 𝜎2 𝑧𝑧(𝑅′⊙, 0) = 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='71 km2 s−2, 𝜕𝜎2𝑧𝑧(𝑅𝑐, 0) 𝜕|𝑧| = 306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='84 km2 s−2 kpc−1, 𝜎2 𝑟𝑟 (𝑅′⊙, 0) = 715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='93 km2 s−2, 𝜕𝜎2𝑟𝑟 (𝑅𝑐, 0) 𝜕|𝑧| = 1236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='97 km2 s−2 kpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (19) Additionally, to be consistent with the data as derived by Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', we use 𝑅′⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 kpc – note that 𝑅′⊙ ≠ 𝑅⊙ – for extrapolating the dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we have assumed their quoted location of the Sun ‘in the range 𝑅 ∈ ]8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6] kpc’ implies2 an assumed default location in the middle of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We assume, following Amendt & Cuddeford (1991) and Val- lenari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2006), that 𝜎2 𝑟𝑧 (𝑅𝑐, 𝑧) ≃ 𝜎2 𝑟𝑧(𝑅𝑐, 0) + 𝑧 𝜕𝜎2𝑟𝑧 (𝑅𝑐, 0) 𝜕𝑧 (20) where the first term on the right hand side vanishes by symmetry at 𝑧 = 0, and the derivative is given by 𝜕𝜎2𝑟𝑧 (𝑅𝑐, 0) 𝜕𝑧 = 𝜆(𝑅) 𝜎2𝑟𝑟 (𝑅𝑐, 0) − 𝜎2𝑧𝑧(𝑅𝑐, 0) 𝑅𝑐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (21) Given no information on the radial dependence of 𝜆, we fix it to the local value of 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 (Amendt & Cuddeford 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In cases where the linear extrapolation would result in a correlation � 𝜌𝑟𝑧 ≡ 𝜎2 𝑟𝑧 𝜎𝑟𝑟 𝜎𝑧𝑧 � larger in absolute value than one, we force the correlation back to either +1 or -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Following Vallenari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2006), this is the only off-diagonal term we consider for the thin disc covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, again following the prescription of Amendt & Cuddeford, we assume that the azimuthal and radial dispersions are related by a constant, and hence use 𝜎2 𝜙𝜙 = −𝐵 𝐴 − 𝐵 𝜎2 𝑟𝑟, (22) with 𝐴 and 𝐵 the Oort (1927) constants, for all 𝑅𝑐 and 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We use the Olling & Dehnen (2003) Oort constant values (their table 5, figure 1 Using 𝜎2𝑟𝑟 for the Galactocentric Cylindrical frame radial dispersion component, to avoid the slightly clunky notation 𝜎2 𝑅𝑐 𝑅𝑐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2 Inclusive of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 kpc but exclusive of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 kpc, equivalent to (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) 6 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson 6), as a function of intrinsic colour 𝐵 − 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we interpolate 𝐴 and 𝐵 as a linear function of (𝐵 − 𝑉)0, fitting: 𝐴 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='94553 × (𝐵 − 𝑉)0 + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='33138 𝐵 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='63360 × (𝐵 − 𝑉)0 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='60611 (23) as shown in Figure 1 (left-hand panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As we are using TRILEGAL simulations, we require a conversion from available TRILEGAL parameters to (𝐵 − 𝑉)0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' for this we use the dwarf colour sequence of Pecaut & Mamajek (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We fit a two-step function to the intrinsic B-V colour as a function of effective temperature (with 𝑇 in units of Kelvin), (𝐵 − 𝑉)0 = ����� ����� − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='40739 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='07836 × exp(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='27083 × 𝑇/1000K) 𝑇 < 10000 K − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='35093 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='69012 × exp(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='08179 × 𝑇/1000K) 𝑇 ≥ 10000 K (24) as shown in Figure 1, right-hand panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once we have all of the terms, we rotate the covariance ma- trix from its Galactocentric cylindrical coordinate frame into the Heliocentric coordinate system by 𝚺′ = T𝑐 𝚺 T𝑇 𝑐 = T𝑐 ��� � 𝜎2𝑟𝑟 0 𝜎2𝑟𝑧 0 𝜎2 𝜙𝜙 0 𝜎2𝑟𝑧 0 𝜎2𝑧𝑧 ��� � T𝑇 𝑐 (25) using the rotation matrix as defined in equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Thick Disc The formalism for the thick disc is very similar to that of the thin disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also use the exponential decay model for the density profile, albeit with different scale lengths: 𝜌(𝑅𝑐, 𝑧) = Γ 𝑓thick exp � − 𝑅𝑐 − 𝑅⊙ 𝑙thick − 𝑧 + 𝑧⊙ ℎthick � , (26) again following the Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) and Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) formalism, with 𝑅⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='09 kpc and 𝑧⊙ = 25 pc again, and 𝑙thick = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 kpc, and ℎthick = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='9 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In addition, the parameter 𝑓thick = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='13 sets the relative densities of the thin and thick discs, and Γ again is an arbitrary normalisation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The thick disc dispersion vector uses the data from Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2012a), again following the same radial scaling relations for the thin disc: 𝜎2 𝑟𝑟 (𝑅𝑐, 0) = 𝜎2 𝑟𝑟 (𝑅′⊙, 0) � 𝑅𝑐 𝑅′⊙ �2 exp � −2 𝑅𝑐 − 𝑅′⊙ ℎthick � (27) 𝜎2 𝜙𝜙(𝑅𝑐, 0) = 𝜎2 𝜙𝜙(𝑅′⊙, 0) � 𝑅𝑐 𝑅′⊙ �2 exp � −2 𝑅𝑐 − 𝑅′⊙ ℎthick � (28) 𝜎2 𝑧𝑧(𝑅𝑐, 0) = 𝜎2 𝑧𝑧(𝑅′⊙, 0) exp � − 𝑅𝑐 − 𝑅′⊙ ℎthick � (29) where, once again, we use 𝑅′⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 kpc from Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2012b), assuming the two papers were jointly analysed and hence have the same 𝑅′⊙, although neither paper in the series quote a specific value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This time, we do not describe any vertical dependency of the dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, we take the diagonal terms as presented by Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2012a) within or without the Solar circle as the values approximately at (𝑅′⊙, 0), as given by their tables 3 and 4 respectively, but ignore the off-diagonal terms, which are all within ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5𝜎 of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Exactly the same as with the thin disc, we rotate the Galac- tocentric Cylindrical reference frame into Heliocentric Cylindrical coordinates by 𝚺′ = T𝑐 𝚺 T𝑇 𝑐 = T𝑐 ��� � 𝜎2𝑟𝑟 0 0 0 𝜎2 𝜙𝜙 0 0 0 𝜎2𝑧𝑧 ��� � T𝑇 𝑐 (30) again using the cylindrical rotation matrix of equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 Halo The halo Galactic component follows the density profile 𝜌(𝑅𝑐, 𝑧) = Γ 𝑓ℎ ����� � 𝑅⊙ √︂ 𝑅2𝑐 + � 𝑧 𝑞 �2 ����� � 𝑛 , (31) again using the Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) and Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) formalism, with 𝑓ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0051, 𝑞 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='64, 𝑛 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Γ is a normalising constant once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This parameterization of an inverse power law leads, at 𝑅𝑐 = 0, 𝑧 = 0, to an infinite halo density, and hence unphysical normalising weighting in the Galactic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore truncate the halo density within the solar circle, 𝑅⊙, fixing it at its value at 𝑅𝑐 = 𝑅⊙ at smaller radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This ought to be possible because the old Galactic halo should be negligible in relative density by the solar circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The dispersion vector for the halo is derived from King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2015), given in spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We take the full covariance matrix from the closest radial bin from the ‘Equally Populated Bins’ in their table 3 for a given set of (𝑅𝑠, 𝜙, 𝜃) parameters for a given source, with the exception of their 𝑅𝑠 = 12 kpc bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This bin gives a covariance matrix that is not positive semi-definite, and hence we ignore the off-diagonal terms for that individual bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These have no scaling applied to them and are taken exactly as quoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To rotate into the Heliocentric Cylindrical reference frame, we use 𝚺′ = R𝑠𝑐 𝚺 R𝑇 𝑠𝑐 (32) where3 𝚺 = ��� � 𝜎2𝑟𝑟 Σ𝑟 𝜙 Σ𝑟 𝜃 Σ𝑟 𝜙 𝜎2 𝜙𝜙 Σ𝜙𝜃 Σ𝑟 𝜃 Σ𝜙𝜃 𝜎2 𝜃 𝜃 ��� � , (33) R𝑠𝑐 = T𝑐R𝑠, (34) R𝑠 = �� � cos(𝛽) 0 −𝑑/𝑅𝑠 sin(𝑏) 0 1 0 𝑑/𝑅𝑠 sin(𝑏) 0 cos(𝛽) �� � , (35) and where Σ𝑟 𝜃 ≡ 𝜎2 𝑟 𝜃, following the King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' R𝑠 describes the rotation from Galactocentric Spherical coordinates to Galactocentric Cylindrical coordinates, with T𝑐, as before, defining the rotation from Galactocentric Cylindrical to Heliocentric Cylin- drical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝛽 is defined as the angle between the spherical radial vector and the Galactic plane (𝑏 = 0◦), with 𝑑 the three- dimensional distance to the source in question, and 𝑅𝑠 the three- dimensional Galactocentric distance to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For more details on the derivation of this transformation matrix, see Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 3 Once again, 𝑟 has been used instead of 𝑅𝑠 for notation’s sake, analogous to the thin and thick disc notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 7 0 10 20 30 40 Teff / 103 K 0 1 2 (B - V)0 / mag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 (B - V)0 / mag −10 0 10 20 Oort Constant / km s−1 kpc−1 A B Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Relationships used to derive the dependencies of 𝐴 and 𝐵 on intrinsic colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Left: linear relationships between intrinsic B-V and Oort constants, using the Oort constants as derived by Olling & Dehnen (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Right: a two-piece fit between effective temperature and intrinsic B-V, using the empirical colour sequence of Pecaut & Mamajek (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 3 CREATING A PROPER MOTION DISTRIBUTION Now that we have described the model for simulating the velocity of a source at a given position in the Galaxy, we can create a theoretical distribution of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In a small sky coordinate window (in our tests limiting ourselves to a few square degrees in the Galactic plane, and relatively small polar cap latitude windows), we run a TRILEGAL simulation in the center of the defined region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We simulate either 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 million sources down to Gaia 𝐺 = 25, or as many as we are allowed within 10 square degrees, the maximum limit of the public simulation API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distances for these simulated sources are derived from their absolute distance modulus, and – with no positional information in the simulated dataset – we randomly place the sources within the rectangle defining the coordinate window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For a given small magnitude range of sources, each source then has its proper motions calculated as though it were from each of the three Galactic components in turn, with some number of realisations (𝑁 ≈ 1000) of the multivariate dispersion drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We then calculate a weighted histogram of proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For each source, 𝑗 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 𝑀, where 𝑀 is the number of simulated stars (and thus distances), the three Galactic components at the given Galactic longitude, latitude, and distance have their respective weights 𝑤𝑖 𝑗 (𝑖 ∈ {1, 2, 3}, or 𝑖 ∈ {thin, thick, halo}) calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The weighted histogram is therefore built with each derived proper motion being given weight 𝑤𝑖 𝑗/𝑁 (𝑁 the number of derived Galactic velocities for the 𝑗th source, in each of the three components), for each of the 3 × 𝑁 × 𝑀 derived proper motions, across all 𝑀 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The histogram (which will contain 𝑀 weighted counts across all bins) is then converted to a PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For the purposes of visualisation and testing, we extract all of the proper motions of Gaia eDR3 sources with flux SNRs greater than five in the same coordinate window and magnitude range de- fined for the simulated proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, one additional step is taken, solely for the purposes of distribution comparison: we convolve the model with the median uncertainty of the Gaia proper motions in the dataset for this magnitude cut and sightline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This allows for the inclusion of non-negligible Gaussian uncertainties in our comparison of our generated model to the Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This step was purely for visualisation purposes, and is not part of the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4 ASSESSING THE ACCURACY AND PRECISION OF THE PROPER MOTION MODEL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Overall Model Shape The simulated proper motions are good across all sightlines and brightnesses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' some examples are shown in Figures 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We get good agreement in the mean proper motion, and shape of the dis- tributions, of bulk source motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For most sightline-brightness combinations the agreement is quantitative, while sometimes the shapes are merely broadly in agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Disagreement in modal proper motion drift is likely largely caused by our Galactic rotation model not capturing the fine detail of Galactic potentials or inac- curacies in our asymmetric drift velocity, while distribution width issues can mostly be explained by the extrapolation of the velocity dispersion vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we stress that the model’s simplicity is one of its strengths in the context of inclusion within a larger cross-match process, and that these minor differences are more than acceptable for the purpose of improving Bayesian match likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These distributions are intended to reflect a wide range of potential positional shifts through time, rather than model any one specific proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence so long as the rough widths – to within some- thing like a factor two, which we achieve – and mean offsets – good to high accuracy using the Galactic rotation curve – are modelled to reasonable accuracy, our distributions are good enough for our work, and as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 5 shows some reduced statistics for the entire set of sightline-brightness combinations we tested in the Galactic plane – 𝐺 = {12, 15, 18, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5}, 𝑙 in the range from 0◦ to 345◦ in 15 degree intervals, and 𝑏 = {−50◦, −30◦, 0◦, 25◦, 40◦}, as well as the Galactic north and south poles at −90◦ ≤ 𝑏 ≤ −80◦, −80◦ ≤ 𝑏 ≤ −70◦, 70◦ ≤ 𝑏 ≤ 80◦, and 80◦ ≤ 𝑏 ≤ 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall, we find that the widths of the Gaia data are approximately 80% that of our model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' our model is too wide by 25%) across all positions and brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' There is a roughly 10% spread in relative widths – middle column, Figure 5, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 2, bottom right panel, where our red model has a slightly wider wing than the histogram of the black Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also see evidence for overly narrow simulated proper motion distributions (Gaia-to-model ratios larger than one) along various sightlines, in approximately 8% of cases – but, again, get extremely good agreement along others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These slightly-too-wide RASTI 000, 1–20 (2022) 8 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 ∆l / arcsecond [10yr baseline] 0 5 10 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 ∆l / arcsecond [10yr baseline] 0 5 10 15 PDF / arcsecond−1 G = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆b / arcsecond [10yr baseline] 0 10 20 30 40 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 ∆b / arcsecond [10yr baseline] 0 20 40 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0 10 20 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 ∆b / arcsecond [10yr baseline] 0 20 40 PDF / arcsecond−1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions (Galactic longitude, left-hand columns, and latitude, right-hand columns) for sources at 𝑙 = 90◦, 𝑏 = 0◦, for 𝐺 = 12, 𝐺 = 15, 𝐺 = 18, and 𝐺 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 (each respective row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Proper motions have been converted from a per-year drift to decadal positional change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Gaia proper motions are shown in the black histogram, with sim- ulated distributions of proper motions in the red solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Errorbars in the corner of each subplot show the typical uncertainty of each individual Gaia proper motion, while the plot labels show the Galactic longitude and latitude, and 𝐺 magnitude, of the subset of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' distributions are likely related to our modelled radial and vertical dependencies of the thin disc dispersion vector, as the thin disc is the dominant term at the distances our Gaia data probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Additionally, as can be seen in the top row of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 2, low-number statistics of brighter Gaia stars could be interpreted as lower standard deviations, as the ‘real’ distributions fail to probe the wings of the simulated distributions to high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Relative to the standard deviation of the distributions, our bi- ases – the mean motion offsets – are within ∼ 10% two-thirds of the time, and almost always within 15 − 20%, as shown by right-hand column, Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Absolutely, on a decade baseline, we find most of our mean motion offsets are within approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 arcseconds (or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 arcseconds on a 25-year baseline, as maybe be important for LSST), well within the ‘centroid’ precision of most photometric catalogues – left-hand column, Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These results – even where qualitative (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 4, bottom-left panel) as opposed to quantita- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 ∆l / arcsecond [10yr baseline] 0 5 10 15 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 ∆b / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 ∆b / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 ∆l / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆b / arcsecond [10yr baseline] 0 10 20 30 40 PDF / arcsecond−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 ∆l / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆b / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 180◦, 𝑏 = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' tively (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 4, top-left panel) good fits – are satisfactory, and we therefore have chosen not to over-explore the residuals, as the subtleties of the spiral arm structure of the Milky Way are outside of the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Galactic Poles All previous examples shown (Figures 2-4) were limited in Galac- tic latitude to |𝑏|≤ 50◦, exploring primarily the proper motions of sources roughly in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we must also verify that our model is good at high absolute Galactic latitudes, where we are viewing sources orbiting around the Galactic cen- ter ‘above’ us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As shown in Figure 6, we get good agreement for the Galactic longitudinal and latitudinal proper motions, af- ter removing objects with parallax 𝜋 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 mas (𝑑 ≳ 20 kpc) or 𝜋/𝜎𝜋 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here, close to 𝑏 = 90◦, our equations for the average rotational velocities (equations 3-7) simplify somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, look- ing straight up out of the Galactic plane, we have 𝑑ip ≈ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' we also have 𝑅𝑐 ≈ 𝑅⊙, and hence (𝑅2𝑐 + 𝑅2 ⊙ − 𝑑2 ip)/(2 𝑅𝑐 𝑅⊙) ≈ 1, and can assume Θ(𝑅𝑐) = Θ⊙, giving 𝑈1 = −𝑈⊙, 𝑉1 = −𝑉⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This further gives 𝑣𝑑 = −𝑈⊙ cos(𝑙) −𝑉⊙ sin(𝑙), 𝑣𝑙 = 𝑈⊙ sin(𝑙) −𝑉⊙ cos(𝑙), and RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0 2 4 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 2 4 6 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 ∆l / arcsecond [10yr baseline] 0 2 4 6 PDF / arcsecond−1 G = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 5 10 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 5 10 15 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0 2 4 6 8 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 5 10 15 PDF / arcsecond−1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 255◦, 𝑏 = 25◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' hence, along with a simplified 𝑣𝑏 = −𝑣𝑑, 𝜇𝑙∗ = 𝑘 𝑑 [𝑈⊙ sin(𝑙) − 𝑉⊙ cos(𝑙)] , (36) 𝜇𝑏 = 𝑘 𝑑 [𝑈⊙ cos(𝑙) + 𝑉⊙ sin(𝑙)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (37) This renders the orbital motions effectively just those of the Sun, with our dispersion vector giving good shape agreement to the Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall, we see good agreement in the shape of our model and the Gaia proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 Widths of Distributions vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Position and Proper Motion Precisions At this point it is worth briefly considering if, or at what bright- nesses, this additional information is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figures 2-6 show a representative sample of Galactic sightlines and the various widths of the distributions of potential proper motions in each sightline- magnitude combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall, the widths of these distributions are approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 arcsecond drifts over a 10 yr baseline (20 − 50 mas yr−1) at the bright end of our tests (𝐺 = 12), reduc- ing to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 arcsecond drifts in 10 years (10 − 30 mas yr−1) at 𝐺 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The first parameter we should compare the proper motions to is the precision on an individual position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If one or both of the positions in a given cross-match were highly uncertain, factors 10 or higher than the proper motion drift, this would dominate over the extra positional spread caused by the potential proper motion of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Its inclusion would then not contribute to the determination of potential counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, for a decade-long baseline, the spread of separations induced by unknown proper motion is at least a factor two or three higher than typical astrometric precisions, with even small motions over long enough baselines moving objects several astrometric precisions apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For Gaia astrometric precisions are vastly higher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' while 80% of its sources will also have incredibly high precision proper motions even the remaining sources will have coordinate positions significantly higher than the unknown proper motion distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' A more typical ground-based survey, LSST should have at worse 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='07 arcsecond precision on each individual visit at 𝑟 = 24 (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' While this is a factor ≈ 3 times smaller than the widths of our proper motion distributions on decade baselines, the real power of LSST lies in its repeated observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Depending on whether the object is in the full ‘Wide-Fast-Deep’ (WFD) survey or in the Galactic Plane footprint, it will either be observed approximately 800 or 150 times across LSST’s full survey lifetime (Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence the statistical precision on a co- added detection at 𝑟 = 24 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='003−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='006 arcseconds depending on the exact number of visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Even including ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 arcsec systematic precision (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019) this is far below the widths of our models for proper motion drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' While those objects will likely also have proper motions after LSST DR3-4, 𝑟 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 coadded detections will have statistical astrometric precisions a factor √ 10 higher, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='008 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 arcseconds, still a factor 10 or more below our 10-year baseline drift spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It will be therefore important to take these long-baseline drifts into account for faint LSST objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For current-generation surveys such as SDSS, its very faintest sources have statistical positional uncertainties comparable to the tightest of our proper motion distribution widths (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 arcsec) so 𝑟 = 24 objects in SDSS may only see limited gains matching across a 10-year timespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Of course, the drifts increase linearly with time, and so a 15-year baseline (2015-2030, for example, in the case of SDSS-LSST) increases the potential proper motion drifts to a larger impact than astrometric precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Additionally, in the context of crowded field Bayesian cross-matching, even a ‘one- sigma’ positional movement will be enough to significantly disrupt match likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is also useful to ask if proper motion precisions are ever com- parable to the width of potential unknown proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Again, for Gaia this is not the case due to its extremely high precision and repeated observations of all objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At 𝐺 = 20 the median precision on its proper motions are of order 1 mas yr−1 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For LSST, quoted proper motion uncertainties are also of order 1 mas yr−1 (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019) – but these assume ≈ 800 visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence the stellar proper motion precisions for Galactic Plane objects will be a factor ≈ 3 higher due to the reduced number of observations within the same timeframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, even 5 mas yr−1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 arcsec over a 10-year timeframe and therefore a smaller, but sizeable, fraction than the unknown proper motion distribution widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Right at the detection limit of proper motions with LSST it may be the case that it is preferable to not use the detected-but- unconstrained proper motions, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' SDSS has typical limiting proper motion precisions of 5 mas yr−1 by 𝑟 ≈ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, like LSST, where constrained individual SDSS proper motions brighter than about 20th magnitude are likely preferable to unknown proper motions, but below this limit and the single-visit detection limit of RASTI 000, 1–20 (2022) 10 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='03 Gaia − Model Average / arcsecond [10yr baseline] 0 25 50 75 100 125 150 N Mean Median Mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Ratio of Gaia-to-Model Distribution Widths 0 20 40 60 80 100 N StDev 90th−10th 84th−16th 75th−25th −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 Gaia − Model Median / Gaia StDev 0 10 20 30 40 50 60 70 N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='03 Gaia − Model Average / arcsecond [10yr baseline] 0 25 50 75 100 125 150 N Mean Median Mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Ratio of Gaia-to-Model Distribution Widths 0 20 40 60 80 N StDev 90th−10th 84th−16th 75th−25th −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 Gaia − Model Median / Gaia StDev 0 10 20 30 40 50 60 N Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Comparison between Gaia proper motions and those of our model across all sightlines and brightnesses in the Galactic plane, in Galactic longitude (top row) and latitude (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Left: Data-to-model average proper motion drift offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Middle: Ratio of data and model proper motion drift distribution widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Right: Ratio of the median proper motion drift offset between Gaia data and the model distribution, normalised by the standard deviation of the Gaia proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑟 = 22 unknown proper motions may be the more precise constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In the IR, CatWISE (Eisenhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2020) has proper motion un- certainties of 20 mas yr−1 at 𝑊1 ≈ 15 (Marocco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2021) and the VVV survey (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2018) cites uncertainties of 10 mas yr−1 around 𝐾𝑠 ≈ 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' below these brightnesses the precisions on in- dividual proper motions become comparable to or larger than the widths of typical unknown proper motion distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 Missing Galactic Components As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4, we do not currently include a full prescription for the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In particular, we do not model the Galactic Bulge (or Bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This may have an effect at very low Galactic longitudes and latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Gaia data show a broader, almost flat distribution of longitudinal proper motions, where our simpler Galactic model, using the thin disc as the dominant term, has a bi- modal distribution of two narrower peaks, as shown in Figure 7, left hand panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It can also be seen in the data (Figure 7, right hand panel) that there is a slightly too narrow distribution of latitudinal proper motions, as compared to the Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This likely comes back to the minor effects of radial dependencies of the 𝜎2𝑟𝑟 term, either following a Gaussian- or Rayleigh-like distribution (as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, it could also be the case that our radial and vertical dispersion scalings are failing at these smaller Galactic radii, as a significant fraction of Gaia sources ought to be sufficiently far removed from the Galactic center to be Bar or Bulge objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once again, we deem these minor issues beyond the scope of this preliminary work – the combined bi-modal longitudinal distribution almost entirely covers the distribution of Gaia motion drifts, to within better than a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 or so, which is our goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we highlight the issue that the very inner few degrees of the Galactic center may suffer systematic proper motion effects due to the nature of the Galactic Bulge and Bar complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also have not modelled the Magellanic Clouds, and in- deed during testing found that several of our test fields are heavily ‘contaminated’ by sitting on the Small Magellanic Cloud (SMC) and Large Magellanic Cloud (LMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These extra terms, as with the Bulge, would be easy to implement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' provided a relative num- ber density of sources, with some positional distribution, and bulk and dispersal proper motion – assuming the Magellanic Clouds are orbiting internally, and around the Milky way – the proper motions can be modelled in much the same way with the Galactic discs and halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For now, we also simply urge the reader to take care when sim- ulating sources centered on the Magellanic Clouds (SMC 𝑙 ∼ 300◦, 𝑏 ∼ −45◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' LMC 𝑙 ∼ 280◦, 𝑏 ∼ −35◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 Missing Binarity Perturbation Our model for motions of objects in the plane of the sky assumes all sources are single stars, subject solely to the Galactic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, half of objects are in some form of higher-order system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Raghavan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2010) and should therefore be subject to ad- ditional on-sky motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is therefore reasonable to ask whether the non-inclusion of this effect, of unresolved binary objects, would have any impact on our derived proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If the binary were equal mass, any orbital motion of the two sources around their common barycentre would completely cancel by symmetry, and show no impact on the photocentre and proper motion of the blended sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' On the other hand, if the objects were very unequal in mass then both the barycentre and photo- centre of the pair will be dominated by the larger, brighter main source, and effectively reduce to a singular object for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 ∆l / arcsecond [10yr baseline] 0 1 2 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = -75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 ∆b / arcsecond [10yr baseline] 0 1 2 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 ∆l / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 G = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 ∆b / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 ∆l / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 ∆b / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 ∆l / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 ∆b / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 180◦, 𝑏 = −75◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Additionally, Gaia objects were filtered for parallaxes consistent with zero to remove extragalactic contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 ∆b / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 ∆b / arcsecond [10yr baseline] 0 10 20 PDF / arcsecond−1 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 0◦, 𝑏 = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' An object of approximately half the mass of the primary, however, contributes very little in luminosity but significantly in astrometric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Placing such an object on a worse-case orbit of approxi- mately 10 AU would give an orbital period around 25 years, and a half-phase orbit on our key decade-long time interval between pho- tometric catalogue ‘generations’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In that time, the primary object would travel halfway around the orbit, appearing to move a total of ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5AU, twice its orbital distance from the barycentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At a typical distance of roughly 1 kpc this is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 mas or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='54 mas yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Such a perturbation is well below the of order 10 mas yr−1 widths to the proper motion distributions observed for faint objects in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If the object were significantly closer – say 100 pc instead – the motion effects would be 10 times higher, and comparable to the model widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In those cases the object would be much brighter, and likely have an individually measured proper motion or be known to be a multiple system through other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore believe the non-inclusion of higher-order systems is justifiable at the resolution we are aiming to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 Random Positions of Sources As noted in Section 3, the TRILEGAL simulations we use to con- struct our models of proper motions do not provide individual posi- tions for simulated sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To overcome this, we simply uniformly distributed sources within the rectangular area we sampled our Gaia proper motions in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This effect may explain some small disagree- ments between our simulated and Gaia proper motion distributions, as we are therefore not properly modelling any clustering, extinction effects, or other non-uniformity and correlations in the distances and positions of Galactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, the effect on each individual proper motion should be relatively small, as the regions in question were mostly limited to several degrees in extent, and cos(𝑥 + 5◦) − cos(𝑥) ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='08, sin(𝑥 + 5◦) − sin(𝑥) ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='08 over the entire Galactic longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus our values for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the decomposition of 𝑈⊙, or Θ, within our proper motion equations, are of order 8% wrong at their most extreme, in the case of a simulated patch of sky five degrees wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As an ensemble, however, this assumption should be a reasonable one, and the ‘incorrectness’ should average out, with uniformity of source distribution acceptable for small enough patches of sky, providing a statistical distribution of variations of velocity decomposition across the whole region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='7 Gaia Proper Motion Uncertainty To compare our ensemble proper motion distribution with the distri- bution of Gaia proper motions, we included the Gaia measurement uncertainty in our theoretical distribution of drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Gaia data have individual uncertainties but to smooth the model with the uncertainty we had to select a single average value (the error bar included in the corners of sub-plots in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' A small part of the discrepancies between model and data in our analysis could therefore stem from this simplifying assumption, with no bearing on the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If the Gaia data have a particularly broad dis- tribution of measurement uncertainties, as they tend to at fainter magnitudes, our single uncertainty value would not produce an uncertainty-convolved motion drift distribution that reflected that of the Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Testing more complex treatments of Gaia uncer- tainty distributions in the comparison between model and data, we found that more fully describing the non-singular value of mea- surement precision did produce theoretical drift distributions that RASTI 000, 1–20 (2022) 12 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0 5 10 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆b / arcsecond [10yr baseline] 0 10 20 30 40 PDF / arcsecond−1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 270◦, 𝑏 = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In addition, the dashed blue line shows simulated Besançon proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' better matched the expected data, but not completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore still find a few sightlines with slight differences in central proper motion or distribution width, but perhaps 30% of these tensions are explainable by the different measurement precisions of faint Gaia proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Ultimately, however, as mentioned in Section 3, we do not include this measurement uncertainty in the final model, just performing the convolution to compare to the Gaia distributions more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8 Comparison with the Besançon Model Throughout this work we have used the TRILEGAL simulations to provide a set of theoretical distances for sources of a particular Galactic sightline and magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We could, of course, use any model of the Milky Way to achieve this, such as the Besançon model (Robin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2003, 2012, 2014, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Czekaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Bienaymé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' With the Besançon models, however, we receive simulated proper motions for the objects returned in our query, unlike with TRILEGAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We can use these simulated proper motions to further verify the robustness of our proper motion model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' but this then raises the question of why we simply do not use these simulated proper motions for use in our cross-matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We will address both of these issues in the next two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Verifying the Accuracy of Our Model with Besançon With simulated Besançon proper motions, we can compare our model’s statistical distribution of proper motions with those of the Galactic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Shown in Figure 8 are distributions of proper mo- tions at 𝑙 = 270◦, 𝑏 = 0◦ for Gaia, our simple model for stellar velocities, and Besançon proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall, at fainter mag- nitudes (bottom row), both models are in agreement with the Gaia data, with our distribution a slightly better match in Galactic latitude than the Besançon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, there are some sightlines within the Galaxy where our model has some mismatches to the Gaia data – an example sightline demonstrating this effect is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here, at faint magnitudes (𝐺 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5), the Besançon model better reproduces the Galactic longitude proper motion distribution seen with Gaia, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0 10 20 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 ∆b / arcsecond [10yr baseline] 0 20 40 PDF / arcsecond−1 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 60◦, 𝑏 = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' where our model shows a slight bias, and a distribution slightly too broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Neither model can reproduce the Galactic latitude Gaia proper motions, and both look very similar in their over-broad dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At bright magnitudes (𝐺 = 12), however, we can see that our distribution (red solid lines) much better matches the Gaia data points than the Besançon simulation (blue dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In almost all cases, 𝑙 = 60◦ and 𝑙 = 270◦ in Figures 8 and 9, but more generally across multiple sightlines, the Besançon models are too sharp in distribution, and fail to match the Gaia data as well as our model for proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We discuss this magnitude-dependence of the Besançon model fits further in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we conclude that our model matches the Besançon models very well, as it does the Gaia data, and see cases where both our model and the Besançon model fail to match the Gaia data perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Why Not Just Use the Besançon Proper Motions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We have used TRILEGAL simulations to construct our Galactic model throughout this work, but we could have used any Galactic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If we had used the Besançon model, we would also have been provided with simulated proper motions for the objects we use for their distances in constructing our proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is therefore reasonable to ask why we would go to the effort of using another model, if we already had a set of proper motions from which to construct a PDF of unknown proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, as our model is broken up into separate smaller sub- models, as opposed to being wrapped in a full Galaxy model, our magnitude-to-distance relation is flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As mentioned, we have been using TRILEGAL simulations to get our potential distribution of distances of sources of a given magnitude, but we could use any Galactic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Indeed, we do not need to use a model at all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' if we instead had a known distribution of tip of the red-giant branch stars, or some other class of standard candle, we would immediately know the distance of our sources from their brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore do not necessarily need to rely on fully resolved Galactic models to provide proper motions or distances with our simple model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' On the other hand, our options become slightly more limited if we wish to use a full Galactic model to obtain simulated proper motions in one pass, as opposed to generating more ‘static’ distributions of RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 13 object brightnesses (or distances), and using other functionality to continue on to create our final proper motion distributions, as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The second consideration is that of dimensionality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' each Be- sançon source is provided with a simulated proper motion – but only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Our model uses the simulated distance for each source once, but draws 𝑁 simulated velocities – and hence 𝑁 simulated proper motions – for each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore much more com- pletely sample the 3-D velocity space than any one simulation from the Besançon Galactic model will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This effect can be seen in the 𝐺 = 12 panels of Figures 8 and 9, where our model (red solid lines) much better agrees with the Gaia proper motions, where limited sample size means the Besançon model is not fully populating the velocity dispersion dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At 𝐺 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5, number counts have increased by a factor 100, and the velocity dispersion, having an inverse-distance component (and fainter stars being further away, on average), has reduced in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This reduces the effect of the lack of realisations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the Besançon models therefore agree much better at these fainter magnitudes than they do at bright ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' By using each source only for its distance, as opposed to using it to sample the 4-D distance-velocity dimensionality, we much more accurately sample from the full potential proper motion distribution at bright magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is crucial for bright sources, being closer to the Sun on average, which have larger proper motions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the 𝑥 axis ranges on the top and bottom rows of Figures 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is here where our constructed model has the edge on the proper motions constructed from large-scale Galactic simulations, although brighter objects are, of course, more likely to have a robustly detected proper motion from other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 5 INCLUDING PROPER MOTIONS IN PROBABILISTIC CATALOGUE CROSS-MATCHING No matter how you construct your proper motion distributions, it is still important to consider them in a match between two photo- metric catalogues of differing epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Astrometric Uncertainty Function (AUF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson & Naylor 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson & Naylor 2018b) is the description of the belief as to a true position of the source, given its measured position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is typically assumed to be a Gaussian, which describes the most obvious term affecting the measured posi- tions of sources in photometric catalogues, and hence the separation between two potential counterparts: the noise-based centroiding of the individual objects during the catalogue creation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The function representing the likelihood of two sources having a given separation under the assumption that they are counterparts to one another – two detections of the same physical object – is given by 𝐺(Δ𝑥, Δ𝑦) = (ℎ𝛾 ∗ ℎ𝜙)(Δ𝑥, Δ𝑦) (38) (Wilson & Naylor 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here Δ𝑥, Δ𝑦 are the two-dimensional sky offsets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' right ascension and declination, or Galactic longitude and latitude), ℎ𝛾 and ℎ𝜙 the AUFs of the sources from the two catalogues respectively, and ( 𝑓 ∗ 𝑔)(𝑥, 𝑦) denotes the convolution of two arbitrary functions 𝑓 and 𝑔 evaluated at 𝑥 and 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As discussed by Wilson & Naylor (2018b), the AUF ℎ can be extended with any additional terms, ℎ𝛾 = ℎ𝛾,1∗ℎ𝛾,2∗ℎ𝛾,3 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here the additional ℎ𝛾,𝑖 components, after the first noise-based centroid term, describe extra potential movement away from the ‘true’ sky position of the source in the limit of infinite precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These could include, for example, stochastic processes such as the perturbation of objects due to hidden contaminants affecting the center-of-light of sources, or systematic effects like offsets of the coordinate frame of the catalogue from a common reference frame, such as the ICRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Whatever the effects, the point is that each source is considered individually, and has all of its ℎ𝛾,𝑖 components applied to it on an isolated, per-source basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Proper motion, however, does not work like this;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the effect of proper motion drift works on offsets between two positions, as opposed to affecting the absolute position of one source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus the proper motion drift must be applied to 𝐺, giving, effectively 𝐺′ = 𝐺 ∗ ℎ′pm (see Appendix B1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' ℎ′pm should be calculated in the sense of mapping from oldest to youngest epoch, in units of distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' thus for Δ𝑡 > 0 we have, crudely, Δ𝛿 = 𝜇𝛿 × Δ𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Mapping from most recent to older data would have a negative Δ𝑡, but the proper motion would have to be of the opposite sign as well (being a ‘rewind’ of the motion), and thus the sign of Δ𝛿 would be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If a source has a purely positive proper motion distribution, such that all 𝜇𝑙∗ > 0 for this simulated source, then we would expect a source observed in the year J2000 to have a smaller Galactic longitude than a source observed at J2015, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This convolution can be performed as any other convolution done to calculate 𝐺 by the convolution of all ℎ components – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' either numerically, or through expression as a mixture of analytically convolvable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also highlight here that while Sections 2-4 detail a method for the construction of a distribution of unknown proper motions, ℎ′pm can be constructed through any available means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For exam- ple, Kerekes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2010) construct sets of data-driven proper mo- tion distributions for the purpose of improving cross-matches, using available proper motions to construct priors for weighting the search for unknown proper motions between potential source counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' On the other hand, the faint end of a photometric catalogue will sys- tematically have worse precision on its measurements (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3), and at some point will have detected the proper motion of an object but be unable to constrain it with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases, ℎ′pm could very well be constructed as a Gaussian PDF with mean and covariance matrix that of the best-fit and uncertainty of the proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Star-Galaxy Separation Our model for proper motions assumes the source in question is a star – objects orbiting the Galactic center in some fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, for an all-sky catalogue cross-match we will also, at high Galactic latitudes, be matching a considerable number of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We there- fore need to model the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, that the sources being matched are stars, and hence have the statistically modelled un- known proper motion distribution, with which we wish to ‘blur’ out our potential match separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Second, they are galaxies, which have zero proper motion, being altogether too distant to have visibly moved anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore have a slightly different probability of match (sources being ‘counterparts’, under hypothesis 𝑐) given separation 𝑑, now also conditioned on the ‘type of source’ hypoth- esis, which we will denote as 𝑝(𝑐|𝑑, S) and 𝑝(𝑐|𝑑, G) for a ‘star’ and ‘galaxy’ pairing respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' When matching, we are generally only concerned with the overall probability of the two sources having a given sky separation under the hypothesis of their being matched, 𝑝(𝑑|𝑐) – this term is denoted 𝐺 by Wilson & Naylor (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Note that this differs from 𝑝(𝑐|𝑑), the probability of the two sources being counterparts given their sky separation, Wilson & Naylor (2018a)’s 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑝(𝑑|𝑐) we can RASTI 000, 1–20 (2022) 14 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson obtain by the marginalisation over the two hypotheses: 𝑝(𝑑|𝑐) = 𝑝(𝑑, S|𝑐) + 𝑝(𝑑, G|𝑐) = 𝑝(𝑑|𝑐, S)𝑃(S|𝑐) + 𝑝(𝑑|𝑐, G)𝑃(G|𝑐) (39) where 𝑃(S|𝑐) is the prior probability that these counterparts (with given sky positions, brightnesses, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=') are stars (or galaxies, in the opposite case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We work under the assumption there is no third type of object – crudely labelling objects as ‘in the Milky Way’ or ‘outside the Milky Way’ – and thus 𝑃(S|𝑐) + 𝑃(G|𝑐) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we can also ask a related but separate question: ‘what is the probability that these two detections are of a star, given that they are counterparts with a given separation?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', which looks like 𝑃(S|𝑑, 𝑐) = 𝑝(𝑑|𝑐, S)𝑃(S|𝑐) 𝑝(𝑑|𝑐) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (40) Here we have the likelihood of the separation given the hypoth- esis that the sources are counterparts and stars, multiplied by the prior chance of the sources being stars given they are counterparts, normalised by the overall chance of either a galaxy or star pair hav- ing this particular detection offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To calculate both 𝑃(S|𝑑, 𝑐) and 𝑝(𝑑|𝑐) we therefore need both prior and likelihood terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Calculating the likelihood terms 𝑝(𝑑|𝑐, S) and 𝑝(𝑑|𝑐, G) is relatively straightforward, simply being the convolution of the re- spective AUFs (containing all relevant AUF components for the two catalogues) of the sources in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For 𝑝(𝑑|𝑐, G) this does not include any proper motion terms, as the ‘proper motion model’ for galaxies is a static one – mathematically, this is equivalent to the convolution of 𝐺 and a delta function at zero proper motion, with 𝑓 ∗ 𝛿 = 𝑓 – and hence 𝑝(𝑑|𝑐, G) = 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For 𝑝(𝑑|𝑐, S), how- ever, we wish to include the motion of Galactic sources, and hence subsequently convolve by the ℎ′pm PDF, describing the potential additional on-sky movement due to the epoch difference between the two sets of observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑝(𝑑|𝑐, S) = 𝐺′ ≡ 𝐺 ∗ ℎ′pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, the likelihood for our new question is easy to calculate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' we are therefore left with the derivation of the prior, 𝑃(S|𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The ‘conditioned on the fact that the sources are counterparts’ aspect of the prior is tricky to implement in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We wish to know, analogous to Wilson & Naylor (2018a)’s derivation of photomet- ric likelihoods, the distribution of stars and galaxies as a function of the two bandpasses in question – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑟 and 𝐽, for a match be- tween optical and infrared data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, 𝑃(S|𝑐) is really ‘what is the probability that these two sources are stars given that they are counterparts with magnitude limits (or dynamic ranges) in their re- spective bandpasses?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 𝑃(S|𝑐, 𝑚lim,r, 𝑚lim,J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Due to the nature of the simulated objects – being derived from one-sided distributions, a function of just a single magnitude in one bandpass in one of the two catalogues – we are unable to create two-dimensional relation- ships between stars and galaxies in the construction of these priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In fact, our likelihoods, 𝑝(𝑑|𝑐, S), should implicitly assume coun- terparts for sources, but are built from the full distribution of sources of just a single magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we could have, for example, a case where sources of 𝐽 = 17 either have optical brightnesses 𝑟 = 18 or 𝑟 = 25 (being two classes of objects at differing distances, say);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' this distance distribution is blurred into a bimodal proper motion distri- bution in the IR, but one class of object is rejected if we consider the dynamic range of the optical data for an example 𝑚lim,𝑟 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At present, the explicit dependency on the two-sided, magnitude-magnitude relationship between sources in our two cat- alogues is beyond the scope of this work, due to the nature of the outputs available from most Galactic simulations being limited to a particular set of bandpasses for a specific catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore simply note here that for now, the construction of these models is one-sided – in contrast to the cross-matching algorithms of Wilson & Naylor (2018a), taking into account both catalogues symmet- rically, in both AUF-based astrometry and photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We thus sidestep this dependency by constructing our priors on star and galaxy counts on single magnitude source counts, effectively cre- ating 𝑃(S) and 𝑃(G), removing the dependency on 𝑐 within the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We can still, however, account for the dynamic range of each bandpass within its given catalogue on a per-filter basis, and hence implicitly use 𝑃(S|𝑚lim) in a practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' With this minor practical dependency removed, we conclude that with the inclusion of a distribution of unknown proper motions for Galaxy-based stars, it is possible to discriminate between stars and galaxies in photometric catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The equations 𝑃(S|𝑑, 𝑐) = 𝑝(𝑑|𝑐, S)𝑃(S) 𝑝(𝑑|𝑐) (41) and 𝑃(G|𝑑, 𝑐) = 𝑝(𝑑|𝑐, G)𝑃(G) 𝑝(𝑑|𝑐) (42) allow for the drift of Galactic sources with time, recovering them as non-static sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is the most certain question that can be answered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' stars, as shown in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 3, can have a very high probability of small proper motions in certain sightlines in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, zero proper motion does not necessarily mean galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' but a combination of delta-function likelihood for Galactic proper motion and imbalanced priors at high Galactic latitudes mean that zero proper motion objects will bias towards being extragalactic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' On the other hand, if a source has a proper motion distribution which is significantly non-zero, as is the case for Galactic longitude proper motions at 𝑙 = 270◦, 𝑏 = 0◦ (Figure 8), then we should see a breaking of this degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The offset between the two sources being considered as potential counterparts should now be able to tell whether the sources are further apart than their respective AUFs would suggest – at which point they are very likely detections of a star – or if they have an offset compatible with their AUFs – at which point they are very likely a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Inclusion of Proper Motions in the Non-Match Hypothesis We also note that we should also consider the proper motions within the context of non-matches, but it is easy to see that this results in a trivial case, effectively ignoring the proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For the counter hypothesis of ‘these sources are unrelated to one another, and separate detections of two physical sky objects’, each source can have its own proper motion, based on its own statistical distribution of potential motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases, we need to compare to the hypothesis that these sources are not related to one another given the separation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This involves the double, but separate, marginalisation over all possible unknown locations and proper motions, for both objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Ultimately, as the integrals are separate the proper motions do not affect the end result – see Appendix B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is obvious intuitively: the distribution of separations of unrelated, randomly placed objects is independent of the unknown motion history of those objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 6 WHEN ARE UNKNOWN PROPER MOTION DISTRIBUTIONS NEEDED?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The theoretical framework for accounting for unknown proper mo- tions presented here is relatively indifferent to the type of surveys RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 15 being matched and brightnesses at which it is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, prac- tically it is more useful in some situations than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, we summarise here some key surveys, magnitude ranges, and science cases for which the inclusion of statistical proper motions may be most crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The main criterion for considering whether the inclusion of unknown proper motion distributions is important or not is the surveys being matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The model is most useful outside of Gaia dynamic ranges, as such high-precision individual proper motions may be too impor- tant to ignore at brighter magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The main downside here is that Gaia is quite a bright survey relative to the next generation of photometric catalogues, and therefore large numbers of objects won’t be detected in Gaia at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In the case of LSST, it will also offer proper motions down to perhaps 𝑟 = 24 (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019) but cannot offer proper motions for objects not detectable in its single-visit images, and thus those objects will have to rely solely on statistical proper motions to avoid risking underestimating match probability or unnecessary false match rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' More generally, any science done in the Northern Hemisphere, where LSST has no coverage, will be unable to take advantage of the dataset – for its increased proper motion coverage or otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These cases will more likely require the falling back on unknown proper motions when outside of Gaia or SDSS proper motion dy- namic ranges (𝐺 ≈ 20 and 𝑟 ≈ 21 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Where proper motions are not important to the science case, and any motion drift is a nuisance parameter, it may also be prefer- able to avoid relying on matching to an intermediate catalogue that contains proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' When trying to match catalogue 𝐴 to cata- logue 𝐵, we may not want to perform separate LSST-𝐴 and LSST-𝐵 (or Gaia-𝐴 and Gaia-𝐵, depending on your source of individual proper motions) matches, then join across common LSST (Gaia) objects to obtain a final cross-match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases, where the abil- ity to select high-quality matches using the added-value information from a probabilistic cross-match algorithm is important, reliance on proper motion distributions may suffice to gain in other areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' With the key exceptions of CatWISE, albeit with order-of- magnitude larger uncertainties than LSST or Gaia, and, but with much less sky coverage, VVV, most IR surveys are single-epoch, and matching longer wavelength surveys to one another therefore relies far more than optical catalogues on unknown proper motion distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In terms of science cases, the main areas that benefit from in- cluding unknown proper motions are those in which proper motions are crucial but lacking by other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Nearby faint objects, which LSST especially will find signifi- cant numbers of, will have appreciable on-sky motions that may not be derived as part of the survey’s dataset construction due to their faint fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Red objects will suffer a bias in current- and future-generation surveys such as LSST, Euclid, and Roman where they will system- atically be less likely to have measured proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Very faint transient progenitors will also suffer from a lack of known proper motions, and potentially may require matching back to a number of long-time-baseline surveys to probe progenitor characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For LSST, Galactic Plane science will systematically be af- fected due to the much lower number of visits currently planned than in the main WFD survey (Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Current simu- lated LSST precisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019, table 3) assume WFD cadences and hence numbers of observations, but reduced visit count will lead to worse proper motion accuracies and precisions by factors of a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, care should be taken when attempting to extrapolate reasonably uncertain, but ‘detected’ proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases it may be more advantageous to not use the best-fit value, but marginalise over all potential proper motions based on the likely more robustly determined position and brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Alternatively, the best-fit proper motion can be used, but ‘blurred’ out with the detection’s precision, representing the proper motion offset PDF ℎ′pm as a Gaussian with given mean proper motion and one-sigma uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 7 CONCLUSION We described a model of the bulk motion of a random set of sources through the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The model uses the rotation curve of the Galaxy, the Solar motion, and a prescription for the random motion of sources due to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' their interaction history to create a statistical distribution of potential proper motions of a source at a particular set of sky coordinates and brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We compared this model to Gaia sources in various sightlines across the Galactic plane – in the mid-plane and out of plane – in different magnitude regimes, and to the proper motions provided by the Besançon Galactic model, to verify its robustness and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall we find that our model matches the observed proper motions with a high degree of both accuracy and precision, and hence believe that our model is an acceptable description of the statistical proper motions of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This will be invaluable when matching the next generation of deep photometric surveys to other datasets, in the regime where Gaia cannot provide individual proper motions for sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Without the inclusion of unknown proper mo- tions we could be subject to a source separation bias that will impact the number of cross-matches reported between two such catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This will be particularly crucial in the coming years in light of the revolution in Galactic studies that the Rubin Observatory’s LSST will bring, where – with its long time baseline back to previous brighter infrared surveys – this effect has the potential to dominate a systematic search for classes of sources such as faint, red objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We have made a Python version of the model described in this paper available through the macauff GitHub codebase4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' ACKNOWLEDGEMENTS TJW thanks the reviewers for their useful comments and sugges- tions, which much improved the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' TJW would also like to thank Sergey Koposov for useful conversations and suggestions that improved the accuracy of the model, and Tim Naylor for his helpful discussions and proofreading assistance throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This work has been supported by STFC funding for UK participation in LSST, through grant ST/S 006117/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This work has made use of Python (Van Rossum & Drake 2009), and the SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2020), NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2020), Astropy (Astropy Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2018), astroquery 4 At this URL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) 16 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson (Ginsburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019), Matplotlib (Hunter 2007), and F2PY (Pe- terson 2009) Python modules, as well as NASA’s Astrophysics Data System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='int/ gaia), processed by the Gaia Data Processing and Analy- sis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='int/web/ gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Funding for the DPAC has been pro- vided by national institutions, in particular the institutions partici- pating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' DATA AVAILABILITY The datasets used in this manuscript were derived from sources in the public domain, from the Gaia archive (https://gea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='int/archive/), TRILEGAL (http://stev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='oapd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' it/cgi-bin/trilegal_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='7), and Besançon (https://model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' obs-besancon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='fr/modele_home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='php).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' CreateS- pace, Scotts Valley, CA Virtanen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 2020, Nature Methods, 17, 261 Wilson T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', Naylor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 2017, MNRAS, 468, 2517 Wilson T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', Naylor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 2018a, MNRAS, 473, 5570 Wilson T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', Naylor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 2018b, MNRAS, 481, 2148 York D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 2000, AJ, 120, 1579 APPENDIX A: COORDINATE SYSTEMS Here we define the coordinate systems we use in this paper, and how to transform from one to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We use several coordinate systems: Heliocentric Cartesian space (𝑥, 𝑦, 𝑧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the observable, He- liocentric Spherical coordinate space (𝑑, 𝑙, 𝑏);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Heliocentric Cylin- drical coordinates (ˆ𝑣𝑑, ˆ𝑣𝑙, ˆ𝑣𝑧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the Galactocentric Cylindrical co- ordinate system (𝑅𝑐, 𝜙, 𝑧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the Galactocentric Spherical coordinate system (𝑅𝑠, 𝜙, 𝜃);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' and the Galactocentric Cartesian coordinate sys- tem (𝑋, 𝑌, 𝑍).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Heliocentric Cylindrical coordinate system is defined as the radial and tangential velocity components of the in-plane stellar motions, as measured from the Sun in (and orthogonal to) the direc- tion towards the source, as well as the orthogonal, vertical compo- nent of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Its transformation from Heliocentric Spherical coordinates is a simple rotation from (𝑑, 𝑏) through the angle 𝑏 to (ˆ𝑣𝑑, ˆ𝑣𝑧), albeit with the caveat that the direction of rotation varies with the sign of 𝑏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' its transformation from Galactocentric Cartesian coordinates is a rotation through longitudinal angle 𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Heliocentric Cartesian coordinate system can be obtained from the Heliocentric Spherical coordinates, the observables, dis- tance 𝑑, and Galactic coordinates 𝑙 and 𝑏, with 𝑥 = 𝑑 cos(𝑙) cos(𝑏) (A1) 𝑦 = 𝑑 sin(𝑙) cos(𝑏) (A2) 𝑧 = 𝑑 sin(𝑏), (A3) where we have used the ‘right-handed’ system that defines 𝑥 as pointing towards the Galactic center from the Sun, to 𝑙 = 0◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑦 towards 𝑙 = 90◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' and 𝑧 towards 𝑏 = +90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Galactocentric Cartesian coordinates are a simple shift of zero-point, relative to the Heliocentric coordinates: 𝑋 = 𝑥 − 𝑅⊙ (A4) 𝑌 = 𝑦 (A5) 𝑍 = 𝑧 + 𝑧⊙ (A6) with a shift of the origin up by ≃ 8kpc and down ≃ 25pc in the 𝑋 and 𝑍 directions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For the Galactocentric non-Cartesian coordinate systems, we have to define new angles, as well as two additional radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The radii are fairly straightforward, being based simply on the Galactocentric Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, the Galactocentric Cylindrical radius, being defined as the in-plane radius, is given by 𝑅𝑐 = √︁ 𝑋2 + 𝑌2 (A7) RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 17 and the Galactocentric Spherical radius by 𝑅𝑠 = √︁ 𝑋2 + 𝑌2 + 𝑍2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A8) The angle 𝜙, used in both Galactocentric Cylindrical and Spherical coordinate systems, is defined as the angle around the Galaxy – if viewed top-down, from the Galactic North Pole – from the line running from the Sun through the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This an- gle, however, is defined as clockwise for the Galactocentric Cylin- drical coordinates (and during the derivation of the Heliocentric Spherical proper motions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3), but counter-clockwise for the Galactocentric Spherical coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Equivalently, this counter-clockwise angle can be considered as being measured in the 𝑋 𝑌 plane, from the 𝑋 axis towards the 𝑌 axis (or −𝑌 axis, for a clockwise defined 𝜙), analagous to how 𝑙 is defined as the angle from the 𝑥 axis towards the 𝑦 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, when in Galactocentric Spherical coordinates, we could calculate 𝜃 by 𝜃 = arccos(𝑍/𝑅𝑠), (A9) where 𝜃 is the co-latitude, the angle as measured from the Cartesian 𝑍-axis, which differs from the system defining the Galactic latitude 𝑏, measured from the (𝑥, 𝑦) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We will find that we never need to consider 𝜙 or 𝜃 themselves, as they will entirely be used to de- fine rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The rotation matrices to convert from Galactocentric Spherical to Galactocentric Cylindrical coordinates, or Galactocen- tric Cylindrical to Heliocentric Cylindrical coordinates, along with the conversion from Galactocentric Cylindrical to Galactocentric Cartesian coordinates, are derived in Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' A1 Rotating Covariance Matrices In this section we briefly outline the transformation, reflection, and rotation matrices used to convert between four coordinate sys- tems: the Galactocentric and Heliocentric Cylindrical, and Galac- tocentric Cartesian and Spherical frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, we need to convert from Galactocentric Cylindrical to Galactocentric Cartesian coor- dinates, following the rotation curve-based methodology of Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Additionally, to work entirely in Sun-based radial, tangential, and vertical velocity space, we need to rotate the co- variance matrices calculated from Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', and King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', in Galactocentric Cylindrical and Spherical coordinates respectively, to ˆ𝑣𝑑 − ˆ𝑣𝑙 − ˆ𝑣𝑧 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Galactocentric Cylindrical to Galactocentric Cartesian Rotation First we need to calculate the rotation matrix describing the change from Galactocentric Cylindrical to Galactocentric Cartesian coordi- nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Starting with Figure A1, we first consider the case of 𝑙 ≤ 180◦ (left-hand schematics), a counter-clockwise rotation from 𝑅 through angle 𝜔 to 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As these are left-handed cartesian coordinate systems, this is a negative rotation, and hence the rotation matrix is TCCW = � cos(𝜔) sin(𝜔) − sin(𝜔) cos(𝜔) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A10) We therefore need to calculate sin(𝜔) and cos(𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' sin(𝜔) can be derived using the law of sines, and is given by sin(𝜔) = 𝑑 𝑅 sin(𝑙), (A11) while the cosine can be calculated from its corresponding law, cos(𝜔) = 𝑅2 + 𝑅2 0 − 𝑑2 2 𝑅 𝑅0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A12) Sun GC R R0 d l ω Sun GC R R0 d l ω 360∘ − l V U ϕ R ω V U ϕ R ω Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Schematic showing the transformation from 𝑅 − 𝜙 Galactocen- tric Cylindrical coordinates to Galactocentric Cartesian 𝑈 − 𝑉 coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In the 𝑙 ≥ 180◦ case, right-hand side of Figure A1, we now have a clockwise rotation, which in our left-handed coordinate system is a positive rotation, TCW = �cos(𝜔) − sin(𝜔) sin(𝜔) cos(𝜔) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A13) We can, as before, calculate the sine and cosine of 𝜔: sin(𝜔) = 𝑑 𝑅 sin(360◦ − 𝑙) = − 𝑑 𝑅 sin(𝑙), (A14) cos(𝜔) = 𝑅2 + 𝑅2 0 − 𝑑2 2 𝑅 𝑅0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A15) We can therefore now see that the changing from positive to negative rotation in T, which changes the sign of sin(𝜔) in the rotation matrix, is correlated with a change of sign of sin(𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus we can simplify our matrices, giving us T𝑡 = �� � 𝑅2+𝑅2 0−𝑑2 2 𝑅 𝑅0 𝑑 𝑅 sin(𝑙) − 𝑑 𝑅 sin(𝑙) 𝑅2+𝑅2 0−𝑑2 2 𝑅 𝑅0 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A16) Expanding to the full three-dimensions of our problem, we note that the third axis is unchanged by the rotation within the plane of the Galaxy, and therefore the final axis has a trivial transformation, giving T𝑡 = ���� � 𝑅2+𝑅2 0−𝑑2 2 𝑅 𝑅0 𝑑 𝑅 sin(𝑙) 0 − 𝑑 𝑅 sin(𝑙) 𝑅2+𝑅2 0−𝑑2 2 𝑅 𝑅0 0 0 0 1 ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A17) A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Galactocentric Cylindrical to Heliocentric Cylindrical Rotation Here we calculate the Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' rotation from the Galactic center- based cylindrical frame on to one centered on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Consider the RASTI 000, 1–20 (2022) 18 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson Sun GC R R0 d l θ ̂vl R ϕ ̂vd α Sun GC R R0 d l θ ̂vl R ϕ ̂vd α 360∘ − l Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Schematic showing the transformation from 𝑅 − 𝜙 Galactocen- tric coordinates to a Heliocentric ˆ𝑣𝑑 − ˆ𝑣𝑙 coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' left-hand panel of Figure A2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' to rotate from 𝑅 − 𝜙 coordinates to ˆ𝑣𝑑 − ˆ𝑣𝑙 is a negative (clockwise) rotation – working in the more traditional right-handed coordinate system – through 𝛼, as well as a mirroring around the 𝑅 axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' a flip of the 𝜙 axis on to the 𝑣𝑙 axis, after rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence a rotation-then-mirror transformation matrix would look like TCW = �1 0 0 −1 � � cos(𝛼) sin(𝛼) − sin(𝛼) cos(𝛼) � = �cos(𝛼) sin(𝛼) sin(𝛼) − cos(𝛼) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A18) As can be seen in Figure A2, 𝛼 = 𝜃, and hence cos(𝛼) = cos(𝜃) = 𝑅2 + 𝑑2 − 𝑅2 0 2𝑅𝑑 , (A19) sin(𝛼) = sin(𝜃) = 𝑅0 𝑅 sin(𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A20) In the right-hand case of Figure A2, where 𝑙 ≥ 180◦, we now have a counter-clockwise, positive rotation from 𝑅 through ˆ𝑣𝑑, but still have a mirror reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This simply changes the sign of sin(𝛼) in the rotation matrix, and hence TCCW = �1 0 0 −1 � �cos(𝛼) − sin(𝛼) sin(𝛼) cos(𝛼) � = � cos(𝛼) − sin(𝛼) − sin(𝛼) − cos(𝛼) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A21) Again, we can consider the inner triangle of 𝑅0−𝑑−𝑅 and calculate angles for 𝛼 using 𝜃: cos(𝛼) = cos(𝜃) = 𝑅2 + 𝑑2 − 𝑅2 0 2𝑅𝑑 , (A22) sin(𝛼) = sin(𝜃) = 𝑅0 𝑅 sin(360◦ − 𝑙) = − 𝑅0 𝑅 sin(𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A23) Similar to Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1, we can see that no matter the di- rection of the rotation – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' if 𝑙 ≤ 180◦ or 𝑙 ≥ 180◦ – the sign R z θ ρ β R z θ ρ β Sun GC Sun GC ρ R0 d R0 b β d ρ b β Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Schematic showing the rotation from 𝜌−𝜃 Galactocentric Spher- ical coordinates to a Galactocentric Cylindrical 𝑅 − 𝑧 coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' of sin(𝛼) cancels with the sign within the sin(𝛼) elements of the transformation matrix, and hence TCCW = TCW = �� � 𝑅2+𝑑2−𝑅2 0 2𝑅𝑑 𝑅0 𝑅 sin(𝑙) 𝑅0 𝑅 sin(𝑙) − 𝑅2+𝑑2−𝑅2 0 2𝑅𝑑 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A24) We can also now explicitly include the third axis, the vertical coordinate in our three-dimensional cylindrical reference frame, a trivial continued alignment of the 𝑧 axis with our 𝑣𝑧 axis, giving the final transformation matrix as T𝑐 = ���� � 𝑅2+𝑑2−𝑅2 0 2𝑅𝑑 𝑅0 𝑅 sin(𝑙) 0 𝑅0 𝑅 sin(𝑙) − 𝑅2+𝑑2−𝑅2 0 2𝑅𝑑 0 0 0 1 ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A25) A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 Galactocentric Spherical to Galactocentric Cylindrical Rotation Finally, we consider the King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' rotation from a spherical refer- ence frame into a cylindrical one, albeit still centered on the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To do this, we consider the frame goes from 𝜌 − 𝜙 − 𝜃 to 𝑟 −𝜙−𝑧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' here, (𝜌, 𝜃) and (𝑅, 𝑧) are both in a right-handed cartesian coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore define our rotation matrices in the opposite sense to Section A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1, TCW = � cos(𝛽) sin(𝛽) − sin(𝛽) cos(𝛽) � , TCCW = �cos(𝛽) − sin(𝛽) sin(𝛽) cos(𝛽) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A26) The upper case of Figure A3 shows the rotation necessary for 𝑏 ≥ 0◦, with the left hand side showing the clockwise rotation through 𝛽, and the right hand side showing a schematic of the various known distances and angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here, considering a negative – clockwise in a right-handed frame – rotation through 𝛽, we can calculate sin(𝛽) RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 19 and cos(𝛽) as sin(𝛽) = 𝑑 𝜌 sin(𝑏), (A27) where 𝜌2 = 𝑅2 0 + 𝑑2 − 2𝑅0𝑑 cos(𝑏), and cos(𝛽) = 𝑅2 0 + 𝜌2 − 𝑑2 2𝑅0𝜌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A28) For the lower case of Figure A3, 𝑏 < 0◦, with a positive rotation, cos(𝛽) = (𝑅2 0 + 𝜌2 − 𝑑2)/(2𝑅0𝜌), as previously, as the triangle is unchanged, just mirrored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' sin(𝛽) is a little more complicated to derive, however, as the triangle in Figure A3 uses 𝑏 as its modulus value, but it is negative in value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Using |𝑏| explicitly, the law of sines gives sin(𝛽) = 𝑑 𝜌 sin(|𝑏|), (A29) as previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, if we use, as we will in practice, 𝑏′ = −|𝑏|, we get sin(𝑏′) = − sin(|𝑏|), and hence sin(𝛽) = −𝑑/𝜌 sin(𝑏′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once again, we find – as with Appendices A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 and A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 – that the sign of sin(𝛽) cancels with the sign of the term within the rotation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, for either orientation – positive and negative Galactic latitude – the rotation matrix from Galactocentric spherical to Galactocentric cylindrical coordinates (from the (𝜌, 𝜃) to (𝑟, 𝑧) plane) is given by R𝑠 = � cos(𝛽) 𝑑/𝜌 sin(𝑏) −𝑑/𝜌 sin(𝑏) cos(𝛽) � , (A30) with cos(𝛽) still defined consistently as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' While we are using 𝜙 to represent the two azimuthal angles, they are defined in the opposite sense (see Section A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore need to reflect the 𝜙 axis through the (𝑟, 𝑧) plane, after the rotation has occurred, given by R𝜙,reflect = �� � 1 0 0 0 −1 0 0 0 1 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A31) Thus, our full three-dimensional transformation matrix is given by R𝑠 = �� � cos(𝛽) 0 𝑑/𝜌 sin(𝑏) 0 −1 0 −𝑑/𝜌 sin(𝑏) 0 cos(𝛽) �� � , (A32) APPENDIX B: CONVOLUTION MATHEMATICS FOR COUNTERPART AND NON-COUNTERPART HYPOTHESES B1 Counterpart Likelihood Including Proper Motion In this Appendix we detail the derivation of the inclusion of the proper motion PDF in the hypothesis that two objects are one astro- physical object given their separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Starting from a similar place to Wilson & Naylor (2018a)’s equation 14, we have 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) +∞ ∬ −∞ ℎ𝛾(𝑥0 − 𝑥𝛾, 𝑦0 − 𝑦𝛾)× ℎ𝜙(𝑥𝜙 − 𝑥0 − Δ𝑢, 𝑦𝜙 − 𝑦0 − Δ𝑣) d𝑥0 d𝑦0 dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B1) Here we have the simultaneous marginalisation over an unknown common position – dropping the prior, 𝑝(𝑥0, 𝑦0) for being uniform and independent of unknown position (and proper motion), as per Wilson & Naylor (2018a) – and a marginalisation over the PDF of all unknown proper motions drifts 𝑝 (here representing proper motions in the two orthogonal sky directions with 𝑢 and 𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Substituting Δ𝑥 = 𝑥𝜙 − 𝑥𝛾 and Δ𝑦 = 𝑦𝜙 − 𝑦𝛾 into ℎ𝛾 we get 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) +∞ ∬ −∞ ℎ𝛾(𝑥0 − 𝑥𝜙 + Δ𝑥, 𝑦0 − 𝑦𝜙 + Δ𝑦)× ℎ𝜙(𝑥𝜙 − 𝑥0 − Δ𝑢, 𝑦𝜙 − 𝑦0 − Δ𝑣) d𝑥0 d𝑦0 dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B2) Now we change variables from 𝑥0 and 𝑦0 to 𝑥 and 𝑦 via 𝑥 = 𝑥𝜙−𝑥0− Δ𝑢, 𝑦 = 𝑦𝜙 − 𝑦0 −Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This rearranges such that 𝑥0 −𝑥𝜙 = −Δ𝑢 −𝑥, 𝑦0 − 𝑦𝜙 = −Δ𝑣 − 𝑦, and thus 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) +∞ ∬ −∞ ℎ𝛾(Δ𝑥 − Δ𝑢 − 𝑥, Δ𝑦 − Δ𝑣 − 𝑦)× ℎ𝜙(𝑥, 𝑦) d𝑥 d𝑦 dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B3) As per Wilson & Naylor (2018a), we note that the inner integral is the definition of a convolution, and thus setting 𝐺(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) ≡ (ℎ𝛾 ∗ ℎ𝜙)(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) = +∞ ∬ −∞ ℎ𝛾(Δ𝑥 − Δ𝑢 − 𝑥, Δ𝑦 − Δ𝑣 − 𝑦)ℎ𝜙(𝑥, 𝑦) d𝑥 d𝑦, (B4) we have 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣)𝐺(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B5) Now it is clear that this is itself a convolution, of 𝑝 and 𝐺, and hence we can now write 𝐺′(Δ𝑥, Δ𝑦) ≡ (𝑝 ∗ 𝐺)(Δ𝑥, Δ𝑦) = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣)𝐺(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B6) We note that our equation B1 is of similar form to equation 5 of Kerekes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2010), with the interchange of integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we have chosen to construct a semi-analytic simulated model for the construction of the distribution of unknown proper motions, while Kerekes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' built theirs from survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, as discussed in Section 5, we can substitute such a data-driven distribution of proper motions within our matches, using any valid distribution as 𝑝(Δ𝑢, Δ𝑣) (or ℎ′pm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, consistent with Wilson & Naylor (2018a)’s original derivation, we explicitly remind the reader that the AUFs ℎ must be defined such that ℎ(𝑥, 𝑦) = ℎ(−𝑥, −𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' B2 Unrelated Object Likelihood Including Proper Motion For the case where the sources are unrelated to one another, we have a slightly different equation to that of equation B1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' something more RASTI 000, 1–20 (2022) 20 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson like equation 10 of Budavári & Szalay (2008), 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) � +∞ ∬ −∞ ℎ𝛾(𝑥0 − 𝑥𝛾 − Δ𝑢, 𝑦0 − 𝑦𝛾 − Δ𝑣) d𝑥0 d𝑦0 � dΔ𝑢 dΔ𝑣 × +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) � +∞ ∬ −∞ ℎ𝜙(𝑥0 − 𝑥𝜙 − Δ𝑢, 𝑦0 − 𝑦𝜙 − Δ𝑣) d𝑥0 d𝑦0 � dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B7) Here, as with equation B1, we have explicitly assumed that 𝑝(𝑥0, 𝑦0) is independent of both unknown position and proper motion, and thus can be removed as a factor from the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As ℎ𝛾 and ℎ𝜙 are normalised PDFs, the inner integral is trivially integrable to unity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' but with 𝑝, the PDF of unknown proper motions, also normalised, the outer integral then also evaluates to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus we have the trivial case, for unrelated objects, that 𝐺 = 1 and 𝐺′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases, as with the counterpart hypothesis having a prior 𝑝(𝑥0, 𝑦0) = 𝑁𝑐 as per Wilson & Naylor (2018a), we can say that the equivalent priors in the ‘unrelated’ hypothesis case are 𝑝(𝑥0, 𝑦0) = 𝑁 𝑓 , Wilson & Naylor (2018a)’s ‘field’ source density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, for the hypothesis of two sources being unrelated to one another and two detections of different sky objects, the PDF describing the likelihood of the objects having some separation is independent of proper motion, just as it is independent of the respective sources’ AUFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} diff --git a/eNFAT4oBgHgl3EQf7B5d/content/tmp_files/2301.08742v1.pdf.txt b/eNFAT4oBgHgl3EQf7B5d/content/tmp_files/2301.08742v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b9fb80fc682a5709f05d85c33e977c841eab1ef --- /dev/null +++ b/eNFAT4oBgHgl3EQf7B5d/content/tmp_files/2301.08742v1.pdf.txt @@ -0,0 +1,234 @@ +Keywords: Consciousness, Time, AI, Relativity, Quantum Mechanics, Reality, Responsible AI +Unifying Consciousness and Time to Enhance Artificial +Intelligence +Mahendra Samarawickramaa) +Centre for Consciousness Studies, Australia +a)samarawickrama@gmail.com +Abstract. Consciousness is a sequential process of awareness which can focus on one piece of information at a time. This process +of awareness experiences causation which underpins the notion of time while it interplays with matter and energy, forming reality. +The study of Consciousness, time and reality is complex and evolving fast in many fields, including metaphysics and fundamental +physics. Reality composes patterns in human Consciousness in response to the regularities in nature. These regularities could be +physical (e.g., astronomical, environmental), biological, chemical, mental, social, etc. The patterns that emerged in Consciousness +were correlated to the environment, life and social behaviours followed by constructed frameworks, systems and structures. The +complex constructs evolved as cultures, customs, norms and values, which created a diverse society. In the evolution of responsible +AI, it is important to be attuned to the evolved cultural, ethical and moral values through Consciousness. This requires the advocated +design of self-learning AI aware of time perception and human ethics. +INTRODUCTION +The notion of time is an integral part of consciousness [1]. The consciousness experiences the causation or changes in +reality/environment and perceives the time. Therefore, in our previous publication [2], we assumed that consciousness +is a sequential process which is aware of a single piece of information at a time. Even though the brain processes sen- +sory data of five sensors (i.e., Sight, Sound, Smell, Taste, and Touch) in parallel in the neural network, the awareness +of causation is a sequential process following cause and effect. See the illustration of this idea in Figure 1, +5 Senses to observe the +external world +(Peripherals) +Brain function +which manages the +5 senses and the memory +(Parallel processing +neural network: operating in +low frequency) + +Consciousness +(Sequential processing of +information: +Electromagnetic energy +operating in very high +frequency, which +can exhibit properties +of both waves and particles.) +Awareness and +Reality +FIGURE 1: The interplay of five sensors, brain and consciousness. The brain processes sensory information in +parallel. However, the awareness of causation (i.e., consciousness) is a sequential process focusing on a single piece +of information at a time. This sequential process of awareness in consciousness operates fast and consistently, which +underpins our perception of reality. +The assumption of sequential awareness in consciousness enables mapping the perception of time into conscious- +ness. Based on the theory of relativity [3], the perception of time is relative to the frame of reference. Einstein +assumed that the speed of light is constant in all frames of reference, and the time is derived based on that fundamen- +tal assumption. In our paper, we defined the shortest time to be aware of reality as a consciousness cycle. Then based +arXiv:2301.08742v1 [q-bio.NC] 10 Jan 2023 + +mon relativity, this consciousness cycle is also subjected to dilation, like relativistic time +Tv = +T0 +� +1− v2 +c2 +, +(1) +where, Tv is the dilated period of the consciousness cycle related to the rest period of the consciousness cycle T0. Note +that the +� +1− v2 +c2 is the Lorentz factor, where v is the relative velocity between inertial reference frames, and c is the +speed of light in a vacuum. Then, we mathematically modelled [2] how consciousness would interplay with matter +and energy, forming reality, which can be adapted to understand limitations and opportunities in AI consciousness. +This paper extends our discussion towards the time perception of artificial intelligence systems (AIS). +THE NOTION OF TIME IN PERCEPTION AND REALITY +Humans, like any other life forms, experience time through causation. Patterns are composed in the human conscious- +ness in response to the regularities in nature [4]. Since the beginning of human civilisation, humans have learnt and +evolved complex concepts and constructs by incorporating time emerged through patterns in the consciousness. The +earth’s rotation around itself determines the day, and orbiting around the sun determines the year. The Moon takes +about one month to orbit the earth. The tilt of the earth’s spin axis with respect to its orbital plane causes the weather +seasons. These environmental patterns cause many biological patterns and lifestyle patterns in human life. To pre- +dict and organise these patterns effectively, humans introduce standard time with clocks, calendars and various other +frameworks. These artificial frameworks enable us to model time and objectively measure subjective experiences. +Physics has been evolved by observation of nature with various frameworks of time. In this way, time became +an essential construct and dimension of our understanding of reality. For example, Newtonian physics [5] evolved +assuming that time is absolute and flows consistently from past to present and into the future. That enables the +development of mathematical models for explaining patterns in reality with time. However, later observations, such +as the perihelion motion of Mercury, allow humans to understand time as a relativistic measure rather than an absolute. +The modern understanding of the universe is based on the theory of relativity [6, 7], which is completely articulated +by space-time principles. Based on relativity, John Wheeler [8] stated, “Space tells matter how to move. Matter +tells space how to curve”. Relativity enables us to accurately understand and predict the behaviours of black holes, +stars, and planets. Further, relativity enables humans to develop technologies like the atomic clock [9] and Global +Positioning System (GPS) [10] that are useful in everyday life. +The behaviour of particles is completely different to larger objects like planets, stars, etc. This led to the evolution +of Quantum physics [11] as opposed to relativity. Quantum physics exhibits amazing accuracy in predicted results in +particle physics. However, it greatly disturbs the notion of time modelled in relativity. For example, in the collapse +of the wave function in quantum entanglement, Einstein described that as a spooky action at a distance [12]. As +per relativity, information cannot transfer faster than the speed of light. As per the recent discoveries in quantum +entanglement, information can be transferred instantly, faster than the speed of light, making our reality non-local +[13]. The non-local reality contradicts relativity, which is now applied in quantum teleportation at the subatomic +level. On the other hand, at the quantum level, the reality is uncertain, as described by Heisenberg’s uncertainty +principle [14]. As per the uncertainty principle, it is impossible to precisely measure or be aware of the position and +speed of a particle in a given time. This brings the limitation of human awareness and perception of time. Therefore, +many believe now that consciousness is fundamental and that time and causation are derived from consciousness [15]. +THE IMPLICATION OF PRINCIPLES OF TIME FOR AIS +The inability to consolidate quantum physics and the theory of relativity makes our understanding of reality incom- +plete. Moreover, the new discoveries proving the idea of non-local reality shake the status quo of fundamental physics +[16]. Therefore, it is still impossible to supervise AI to experience the notion of time to understand reality precisely. +On the other hand, human understanding of reality is also about 5%, whereas most of the universe consists of dark +matter and dark energy, which humans do not understand [17]. Under these conditions, AI might be used to explore +reality and time in a way we have never imagined. Perhaps incorporating AI to understand reality and causation might +help humans to become fully aware of reality by overcoming inherent biases from evolution, culture and nature. + +Typical Reinforcement Learning (RL) technique can be adapted to automate the learning of AI. The RL process +can be mathematically formulated using Markov Decision Process (MDP) [18]. That is a sequential learning process +by trial and error. In this process, the learning agent (i.e., AI) sequentially interacts with the environment with an +intelligent decision (i.e. action) followed by receiving a reward or a penalty based on the policy imposed. There will +be no influence on the AI agent’s action, but convey the value of its action through feedback with reward or penalty. +This way, the AI agent will self-learn about the environment over time. The RL process is illustrated in Figure 2: +Agent +Environment +Action +At +State +St +Reward +Rt +Rt+1 +St+1 +FIGURE 2: Components of the Markov Decision Process (MDP) and its function in the agent-environment +interaction. The sequential step of time is represented by t. +THE IMPLICATION OF HUMAN BELIEFS, VALUES AND CULTURES FOR THE +PERCEPTION OF TIME IN AIS +Human beliefs, customs, culture and values are tightly linked with various dynamics and interpretations of the time +and periodicities based on the movement of the earth, Moon and other terrestrial bodies. From the beginning, humans +identified that time affects life and nature differently. Therefore, in the Greece era, early Western culture, there were +at least three gods representing different time forms: Chronos, Aion, and Kairos [19]. Chronos represented the linear +time flowing from past to present into the future. This is the time that humans feel when life passes. In contrast, Aion +represented the cyclical nature of time experienced from natural events such as weather patterns, rebirths, etc. The +third god Kairos represented the opportunist time, which reflects the appropriate time to achieve a task. In this way, +time, environment and beliefs were tightly linked with life and governed society and values. +On the other hand, in Eastern culture, the horoscope is one good example of a planetary and constellation frame- +work underpinning Astrology as a foundation of certain belief systems [20]. These beliefs assume that Astrology is +associated with time and causality, which can predict the future and guide humans. +The human observation of the night sky led to perceiving time from various cyclical patterns going far back in time. +For example, the Aboriginal Australians [21] observed the night sky and mapped them to the environment and life +stages that evolved various customs, arts and even religions. Not only by interacting planets and stars but the tilt of +the earth’s spin axis also significantly led to diversifying human cultures based on seasons, particularly when moving +away from the equator. +The notion of time and associated beliefs, customs, and values are important to consider when training AIS [22]. +That will help promote human cultural values, ethics, and diversity, equity and inclusion (DEI). AI development may +need to pay attention to and integrate the time attributes that emerged from nature, values and cultures. Humans may +include them in the policies for rewarding self-learning AI algorithms (e.g., in MDP). + +THE IMPLICATION OF BIOLOGICAL TIME ON AIS +The biological cycles play a fundamental role in human behaviours and the perception of time—for example, mood +cycles, circadian rhythms, and the menstrual cycle. Without understanding these biological time-keeping processes, +AI cannot seamlessly integrate with human society when creating values in health, culture, art, etc. These insights +are essential to realising emotional intelligence, empathy and awareness in AI. Literature shows the effective use of +Cyclic Hidden Markov Models (CyH-MMs) for detecting and modelling cycles in a multidimensional heterogeneous +biological time series data collection [23]. It is important to attribute the relevant features of biological processes +when training AIS, which raises more awareness about humans. +Recent discoveries in quantum physics argue that our reality is non-local, where awareness can happen instantly, +faster than the speed of light. Physicists and neurologists think brain neurons might be aware of the quantum world +through the orchestrated collapse of microtubules in the neurons in the brain [24, 25]. If this hypothesis is true, then +there are possibilities that human awareness can be linked with non-local realities to expand our consciousness across +the universe instantly. From this perspective, future AI might need to be evolved with the capabilities of biological +neurons, which interplay with the quantum realities. The recent development of neurotech realising brain-computer +interface (BCI) along with emerging quantum computers might enable such capabilities in the near future [26]. +CONCLUSION +Consciousness and perception of time and causation are key to awareness and understanding reality. The notion of +time emerged from causation, a perception relative to the observer as per the relativity principles. In relativity, it’s +not time but the light-speed constant in all frames of reference. In contrast, in quantum entanglement, the reality is +non-local, and information can be transferred instantly faster than light. While the principles of time contradict the +foundation of physics, time also influenced the formation of diverse customs, values and cultures based on patterns +that emerged from nature, particularly around the regularities in the earth’s movement, environment, astronomy and +biology. Therefore, understanding time and related artefacts (i.e., cultures, beliefs, values, customs, physics, health, +etc.) are very important to realise deep awareness of reality. From the AIS perspective, it will enhance the understand- +ing of AI in human health, cultures, customs, values and various other diversities. Bringing this awareness to AI will +be a challenging and complex yet rewarding milestone in the evolution of ethical and responsible AI. +REFERENCES +1. L. Kent and M. Wittmann, “Erratum to: Time consciousness: the missing link in theories of consciousness,” Neuroscience of Consciousness, +vol. 2021, 05 2021. +2. M. 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(German) [On the ideological content of +quantum theoretical kinematics and mechanics],” Z. Physik, vol. 43, pp. 172–198, Mar. 1927. +15. D. +D. +Hoffman, +“The +Origin +of +Time +In +Conscious +Agents,” +Cosmology, +vol. +18, +pp. +494–520, +2014. +https://www.cogsci.uci.edu/ ddhoff/HoffmanTime.pdf. + +16. W. Sulis, “Locality Is Dead! Long Live Locality!,” Frontiers in Physics, vol. 8, 2020. +17. E. Oks, “Brief review of recent advances in understanding dark matter and dark energy,” New Astronomy Reviews, vol. 93, p. 101632, 2021. +18. M. van Otterlo and M. Wiering, “Reinforcement Learning and Markov Decision Processes,” Reinforcement Learning: State-of-the-Art, pp. 3– +42, 2012. +19. J. E. Smith, “Time, Times, and the ‘Right Time’; Chronos and Kairos,” The Monist, vol. 53, no. 1, pp. 1–13, 1969. +20. N. Campion, “Astrology as cultural astronomy,” Handbook of Archaeoastronomy and Ethnoastronomy, pp. 103–116, 2015. +21. D. W. Hamacher, “Comet and meteorite traditions of aboriginal australians,” Encyclopaedia of the History of Science, Technology, and +Medicine in Non-Western Cultures, pp. 1–4, 2008. +22. K. Lee and K. Joshi, “Understanding the Role of Cultural Context and User Interaction in Artificial Intelligence Based Systems,” Journal of +Global Information Technology Management, vol. 23, no. 3, pp. 171–175, 2020. +23. E. Pierson, T. Althoff, and J. Leskovec, “Modeling Individual Cyclic Variation in Human Behavior,” in Proceedings of the 2018 World Wide +Web Conference, p. 107–116, 2018. +24. S. Hameroff and R. Penrose, “Consciousness in the universe: A review of the ‘Orch OR’ theory,” Physics of Life Reviews, vol. 11, no. 1, +pp. 39–78, 2014. +25. S. Hameroff, “‘Orch OR’ is the most complete, and most easily falsifiable theory of consciousness,” Cognitive Neuroscience, vol. 12, no. 2, +pp. 74–76, 2021. +26. S. Saha, K. A. Mamun, K. Ahmed, R. Mostafa, G. R. Naik, S. Darvishi, A. H. Khandoker, and M. Baumert, “Progress in Brain Computer +Interface: Challenges and Opportunities,” Frontiers in Systems Neuroscience, vol. 15, Feb 2021. + diff --git a/eNFAT4oBgHgl3EQf7B5d/content/tmp_files/load_file.txt b/eNFAT4oBgHgl3EQf7B5d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..18e6da097cdafe25494e18584af45b19d274cbba --- /dev/null +++ b/eNFAT4oBgHgl3EQf7B5d/content/tmp_files/load_file.txt @@ -0,0 +1,298 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf,len=297 +page_content='Keywords: Consciousness, Time, AI, Relativity, Quantum Mechanics, Reality, Responsible AI Unifying Consciousness and Time to Enhance Artificial Intelligence Mahendra Samarawickramaa) Centre for Consciousness Studies, Australia a)samarawickrama@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='com Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Consciousness is a sequential process of awareness which can focus on one piece of information at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' This process of awareness experiences causation which underpins the notion of time while it interplays with matter and energy, forming reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The study of Consciousness, time and reality is complex and evolving fast in many fields, including metaphysics and fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Reality composes patterns in human Consciousness in response to the regularities in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' These regularities could be physical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=', astronomical, environmental), biological, chemical, mental, social, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The patterns that emerged in Consciousness were correlated to the environment, life and social behaviours followed by constructed frameworks, systems and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The complex constructs evolved as cultures, customs, norms and values, which created a diverse society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' In the evolution of responsible AI, it is important to be attuned to the evolved cultural, ethical and moral values through Consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' This requires the advocated design of self-learning AI aware of time perception and human ethics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' INTRODUCTION The notion of time is an integral part of consciousness [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The consciousness experiences the causation or changes in reality/environment and perceives the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Therefore, in our previous publication [2], we assumed that consciousness is a sequential process which is aware of a single piece of information at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Even though the brain processes sen- sory data of five sensors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=', Sight, Sound, Smell, Taste, and Touch) in parallel in the neural network, the awareness of causation is a sequential process following cause and effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' See the illustration of this idea in Figure 1, 5 Senses to observe the external world (Peripherals) Brain function which manages the 5 senses and the memory (Parallel processing neural network: operating in low frequency) Consciousness (Sequential processing of information: Electromagnetic energy operating in very high frequency, which can exhibit properties of both waves and particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=') Awareness and Reality FIGURE 1: The interplay of five sensors, brain and consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The brain processes sensory information in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' However, the awareness of causation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=', consciousness) is a sequential process focusing on a single piece of information at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' This sequential process of awareness in consciousness operates fast and consistently, which underpins our perception of reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The assumption of sequential awareness in consciousness enables mapping the perception of time into conscious- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Based on the theory of relativity [3], the perception of time is relative to the frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Einstein assumed that the speed of light is constant in all frames of reference, and the time is derived based on that fundamen- tal assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' In our paper, we defined the shortest time to be aware of reality as a consciousness cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Then based arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='08742v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='NC] 10 Jan 2023 mon relativity, this consciousness cycle is also subjected to dilation, like relativistic time Tv = T0 � 1− v2 c2 , (1) where, Tv is the dilated period of the consciousness cycle related to the rest period of the consciousness cycle T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Note that the � 1− v2 c2 is the Lorentz factor, where v is the relative velocity between inertial reference frames, and c is the speed of light in a vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Then, we mathematically modelled [2] how consciousness would interplay with matter and energy, forming reality, which can be adapted to understand limitations and opportunities in AI consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' This paper extends our discussion towards the time perception of artificial intelligence systems (AIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' THE NOTION OF TIME IN PERCEPTION AND REALITY Humans, like any other life forms, experience time through causation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Patterns are composed in the human conscious- ness in response to the regularities in nature [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Since the beginning of human civilisation, humans have learnt and evolved complex concepts and constructs by incorporating time emerged through patterns in the consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The earth’s rotation around itself determines the day, and orbiting around the sun determines the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The Moon takes about one month to orbit the earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The tilt of the earth’s spin axis with respect to its orbital plane causes the weather seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' These environmental patterns cause many biological patterns and lifestyle patterns in human life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' To pre- dict and organise these patterns effectively, humans introduce standard time with clocks, calendars and various other frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' These artificial frameworks enable us to model time and objectively measure subjective experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Physics has been evolved by observation of nature with various frameworks of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' In this way, time became an essential construct and dimension of our understanding of reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' For example, Newtonian physics [5] evolved assuming that time is absolute and flows consistently from past to present and into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' That enables the development of mathematical models for explaining patterns in reality with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' However, later observations, such as the perihelion motion of Mercury, allow humans to understand time as a relativistic measure rather than an absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The modern understanding of the universe is based on the theory of relativity [6, 7], which is completely articulated by space-time principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Based on relativity, John Wheeler [8] stated, “Space tells matter how to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Matter tells space how to curve”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Relativity enables us to accurately understand and predict the behaviours of black holes, stars, and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Further, relativity enables humans to develop technologies like the atomic clock [9] and Global Positioning System (GPS) [10] that are useful in everyday life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The behaviour of particles is completely different to larger objects like planets, stars, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' This led to the evolution of Quantum physics [11] as opposed to relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Quantum physics exhibits amazing accuracy in predicted results in particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' However, it greatly disturbs the notion of time modelled in relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' For example, in the collapse of the wave function in quantum entanglement, Einstein described that as a spooky action at a distance [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' As per relativity, information cannot transfer faster than the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' As per the recent discoveries in quantum entanglement, information can be transferred instantly, faster than the speed of light, making our reality non-local [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The non-local reality contradicts relativity, which is now applied in quantum teleportation at the subatomic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' On the other hand, at the quantum level, the reality is uncertain, as described by Heisenberg’s uncertainty principle [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' As per the uncertainty principle, it is impossible to precisely measure or be aware of the position and speed of a particle in a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' This brings the limitation of human awareness and perception of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Therefore, many believe now that consciousness is fundamental and that time and causation are derived from consciousness [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' THE IMPLICATION OF PRINCIPLES OF TIME FOR AIS The inability to consolidate quantum physics and the theory of relativity makes our understanding of reality incom- plete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Moreover, the new discoveries proving the idea of non-local reality shake the status quo of fundamental physics [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Therefore, it is still impossible to supervise AI to experience the notion of time to understand reality precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' On the other hand, human understanding of reality is also about 5%, whereas most of the universe consists of dark matter and dark energy, which humans do not understand [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Under these conditions, AI might be used to explore reality and time in a way we have never imagined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Perhaps incorporating AI to understand reality and causation might help humans to become fully aware of reality by overcoming inherent biases from evolution, culture and nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Typical Reinforcement Learning (RL) technique can be adapted to automate the learning of AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The RL process can be mathematically formulated using Markov Decision Process (MDP) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' That is a sequential learning process by trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' In this process, the learning agent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=', AI) sequentially interacts with the environment with an intelligent decision (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' action) followed by receiving a reward or a penalty based on the policy imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' There will be no influence on the AI agent’s action, but convey the value of its action through feedback with reward or penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' This way, the AI agent will self-learn about the environment over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The RL process is illustrated in Figure 2: Agent Environment Action At State St Reward Rt Rt+1 St+1 FIGURE 2: Components of the Markov Decision Process (MDP) and its function in the agent-environment interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The sequential step of time is represented by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' THE IMPLICATION OF HUMAN BELIEFS, VALUES AND CULTURES FOR THE PERCEPTION OF TIME IN AIS Human beliefs, customs, culture and values are tightly linked with various dynamics and interpretations of the time and periodicities based on the movement of the earth, Moon and other terrestrial bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' From the beginning, humans identified that time affects life and nature differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Therefore, in the Greece era, early Western culture, there were at least three gods representing different time forms: Chronos, Aion, and Kairos [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Chronos represented the linear time flowing from past to present into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' This is the time that humans feel when life passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' In contrast, Aion represented the cyclical nature of time experienced from natural events such as weather patterns, rebirths, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The third god Kairos represented the opportunist time, which reflects the appropriate time to achieve a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' In this way, time, environment and beliefs were tightly linked with life and governed society and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' On the other hand, in Eastern culture, the horoscope is one good example of a planetary and constellation frame- work underpinning Astrology as a foundation of certain belief systems [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' These beliefs assume that Astrology is associated with time and causality, which can predict the future and guide humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The human observation of the night sky led to perceiving time from various cyclical patterns going far back in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' For example, the Aboriginal Australians [21] observed the night sky and mapped them to the environment and life stages that evolved various customs, arts and even religions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Not only by interacting planets and stars but the tilt of the earth’s spin axis also significantly led to diversifying human cultures based on seasons, particularly when moving away from the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The notion of time and associated beliefs, customs, and values are important to consider when training AIS [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' That will help promote human cultural values, ethics, and diversity, equity and inclusion (DEI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' AI development may need to pay attention to and integrate the time attributes that emerged from nature, values and cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Humans may include them in the policies for rewarding self-learning AI algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=', in MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' THE IMPLICATION OF BIOLOGICAL TIME ON AIS The biological cycles play a fundamental role in human behaviours and the perception of time—for example, mood cycles, circadian rhythms, and the menstrual cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Without understanding these biological time-keeping processes, AI cannot seamlessly integrate with human society when creating values in health, culture, art, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' These insights are essential to realising emotional intelligence, empathy and awareness in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Literature shows the effective use of Cyclic Hidden Markov Models (CyH-MMs) for detecting and modelling cycles in a multidimensional heterogeneous biological time series data collection [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' It is important to attribute the relevant features of biological processes when training AIS, which raises more awareness about humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Recent discoveries in quantum physics argue that our reality is non-local, where awareness can happen instantly, faster than the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Physicists and neurologists think brain neurons might be aware of the quantum world through the orchestrated collapse of microtubules in the neurons in the brain [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' If this hypothesis is true, then there are possibilities that human awareness can be linked with non-local realities to expand our consciousness across the universe instantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' From this perspective, future AI might need to be evolved with the capabilities of biological neurons, which interplay with the quantum realities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The recent development of neurotech realising brain-computer interface (BCI) along with emerging quantum computers might enable such capabilities in the near future [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' CONCLUSION Consciousness and perception of time and causation are key to awareness and understanding reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' The notion of time emerged from causation, a perception relative to the observer as per the relativity principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' In relativity, it’s not time but the light-speed constant in all frames of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' In contrast, in quantum entanglement, the reality is non-local, and information can be transferred instantly faster than light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' While the principles of time contradict the foundation of physics, time also influenced the formation of diverse customs, values and cultures based on patterns that emerged from nature, particularly around the regularities in the earth’s movement, environment, astronomy and biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Therefore, understanding time and related artefacts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=', cultures, beliefs, values, customs, physics, health, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=') are very important to realise deep awareness of reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' From the AIS perspective, it will enhance the understand- ing of AI in human health, cultures, customs, values and various other diversities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' Bringing this awareness to AI will be a challenging and complex yet rewarding milestone in the evolution of ethical and responsible AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf'} diff --git a/g9E1T4oBgHgl3EQfzAXz/content/tmp_files/2301.03441v1.pdf.txt b/g9E1T4oBgHgl3EQfzAXz/content/tmp_files/2301.03441v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2bb381eac560d8aee44f0f3a7b93c9c6b7bdc73c --- /dev/null +++ b/g9E1T4oBgHgl3EQfzAXz/content/tmp_files/2301.03441v1.pdf.txt @@ -0,0 +1,2522 @@ +arXiv:2301.03441v1 [eess.SP] 9 Jan 2023 +1 +L-SeqSleepNet: Whole-cycle Long Sequence +Modelling for Automatic Sleep Staging +Huy Phan∗, Kristian P. Lorenzen, Elisabeth Heremans, Oliver Y. Ch´en, Minh C. Tran, +Philipp Koch, Alfred Mertins, Mathias Baumert, Kaare B. Mikkelsen, and Maarten De Vos +Abstract—Human sleep is cyclical with a period of approx- +imately 90 minutes, implying long temporal dependency in +the sleep data. Yet, exploring this long-term dependency when +developing sleep staging models has remained untouched. In this +work, we show that while encoding the logic of a whole sleep cycle +is crucial to improve sleep staging performance, the sequential +modelling approach in existing state-of-the-art deep learning +models are inefficient for that purpose. We thus introduce a +method for efficient long sequence modelling and propose a new +deep learning model, L-SeqSleepNet, incorporating this method +to take into account whole-cycle sleep information for sleep stag- +ing. Evaluating L-SeqSleepNet on a set of four distinct databases +of various sizes, we demonstrate state-of-the-art performance +obtained by the model over three different EEG setups, including +scalp EEG in conventional Polysomnography (PSG), in-ear EEG, +and around-the-ear EEG (cEEGrid), even with a single-EEG +channel input. Our analyses also show that L-SeqSleepNet is +able to remedy the effect of N2 sleep (the major class in terms +of classification) to bring down errors in other sleep stages and +that the network largely reduces exceptionally high errors seen +in many subjects. Finally, the computation time only grows at a +sub-linear rate when the sequence length increases. +Index Terms—Automatic sleep staging, deep neural network, +long sequence modelling, sequence-to-sequence. +I. INTRODUCTION +Sleep is a slow-transitioning neural process, and thus, the +data recorded from this process embeds abundant sequential +information. Capturing this sequential information has been +shown to be crucial for automatic sleep staging systems to +achieve good performance. In fact, the capacity of sequential +modelling has been the driving force behind existing deep- +learning-based sleep staging models, bringing the machine +scoring performance on par with that of human experts [1]. +Using recurrent neural networks (RNNs) (e.g., Long Short- +Term Memory (LSTM) [2]) or, more recently, the Transformer +architecture [3], these models are able to capture the temporal +H. Phan is with the School of Electronic Engineering and Computer +Science, Queen Mary University of London, London E1 4NS, UK and +the Alan Turing Institute, London NW1 2DB, UK. K. Lorenzen and K. +Mikkelsen are with the Department of Electrical and Computer Engineering, +Aarhus University, Aarhus 8200, Denmark. E. Heremans and M. De Vos are +with the Department of Electrical Engineering and with the Department of +Development and Regeneration, KU Leuven, 3001 Leuven, Belgium. O. Y. +Ch´en is with the School of Economics, Finance and Management, University +of Bristol, Bristol BS8 1TU, UK. M. C. Tran is with Nuffield Department +of Clinical Neurosciences, University of Oxford, Oxford OX3 9DU, UK. P. +Koch and A. Mertins are with the Institute for Signal Processing, University +of L¨ubeck, L¨ubeck 23562, Germany and with the German Research Center for +Artificial Intelligence (DFKI), L¨ubeck 23562, Germany. M. Baumert is with +School of Electrical and Electronic Engineering, The University of Adelaide, +Adelaide SA 5005, Australia. +∗Corresponding author: pquochuy@gmail.com +Table I: Overall accuracy of SeqSleepNet and SleepTrans- +former obtained on the SHSS database with a long sequence +length of {100, 200} compared to a typical values, 20 (Se- +qSleepNet) and 21 (SleepTransformer). ∗Note that this result +is slightly different from that reported for SeqSleepNet in [9] +as we did not exercise early stopping here. +Sequence length +20/21 +100 +200 +SeqSleepNet [4] +87.2∗ +87.4 (↑ 0.2) 87.5 (↑ 0.3) +SleepTransformer [7] +87.7 +86.6 (↓ 1.1) 86.3 (↓ 1.4) +dependency in a sequence of multiple consecutive epochs +of sleep data, resembling the way a human expert conducts +manual scoring. The sequence length (i.e., the number of +epochs in the sequence) was shown to be important. Many +different works [4]–[8] found a length around 20-30 epochs to +be effective and these have been de facto values for this hyper- +parameter. At least, using a longer sequence was reported to +lead to little to no performance gain at the cost of significantly +increased computational overhead [4], [5]. +We conducted some initial experiments with a large se- +quence length of {100, 200} to verify the above observation. +For this investigation, we employed two models, SeqSleep- +Net [4] and SleepTransformer [7], and the SHHS database +[10], [11], a large database consisting of recordings from +5,791 subjects. Both models conform to the state-of-the- +art sequence-to-sequence sleep staging framework [1] but +the former uses LSTM and the latter uses Transformer for +sequential modelling purposes. The obtained overall staging +accuracy in Table I indeed attest the negligible possible impact +of a long sequence length in case of SeqSleepNet, marginally +improving the accuracy by 0.2 − 0.3% when the sequence +length is 5 and 10 times longer than the typical value (i.e., +20 epochs). Even worse, an adverse effect is observed in the +case of SleepTransformer whose accuracy noticeably drops +> 1.0% when the sequence length increases from 21 to 100 +and 200 epochs. This is likely because the Transformer-based +architecture usually needs a lot more data to train, and thus, is +more prone to overfitting when the receptive field gets larger +with the increased sequence length. These results once again +consolidate what were observed in previous works [4], [5]. +However, in the particular context of sleep data, such a +negative effect of a long sequence length to the automatic +staging performance appears to be implausible. Typically, a +person goes through four to six sleep cycles per night. One + +2 +complete sleep cycle takes roughly 90 to 110 minutes (equiv- +alent to 180 to 220 epochs of 30 seconds each), transitioning +through Awake→N1→N2→N3→REM sequentially and the +time lasts in each stage is well-studied [12]. Furthermore, there +are temporal dynamic processes that underpin the sleep cycle. +Several phenomena can be observed from sleep’s physiological +signals, reflecting these dynamic processes. N2 sleep, for +example, at the beginning of a cycle is not the same as at +the end of the cycle. When looking at the cyclic alternating +patterns (CAP), for instance, there are more A phases before +REM sleep onset than after a REM period [13], [14]. Also, the +cycles themselves differ with more REM in the morning. All in +all, sleep cycles exhibit temporal sleep-transitioning structures +specific to the sleep process. The implication of this is that the +temporal interdependence in sleep data could be as long as a +whole cycle. For instance, intuitively, knowing that an epoch +is N1 should increase the likelihood of an epoch 25 minutes +after it to be N2, given the fact that N1 lasts between 1-5 +minutes and N2 lasts ≥ 25 minutes. +We argue that the negative effects of long sequences ob- +served so far in the literature are due to model deficiency. That +is, considering a long sequence as a “flat” sequence per se, the +typical sequential modelling method in existing models [1] is +incapable of handling long sequences, e.g., up to one whole +sleep cycle. We hypothesize that an appropriate method for +equipping a sleep staging system with long sequential mod- +elling capability would benefit its performance. The present +work introduces a new method that is able to model long +sequences (i.e. one sleep cycle or more) to achieve new state- +of-the-art performance, even with a single-EEG input. +II. MATERIALS +We employed four databases in this work. On the one hand, +the SHHS and SleepEDF databases are based on conventional +PSG setup from which scalp EEG derivations, C4-A1 for +SHHS and Fpz-Cz for SleepEDF, were derived (see Figure +1, top row). On the other hand, the ear-EEG and cEEGrid +databases are based on in-ear EEG setup [15], [16] (see Figure +1, middle row) and around-the-ear EEG setup [17], [18] (see +Figure 1, bottom row), respectively. +SHHS: This large database was gathered from multiple +centers as part of the clinical trial “Sleep Heart Health Study +(SHHS)”, ClinicalTrials.gov number NCT00005275 to study +the effect of sleep-disordered breathing on cardiovascular +diseases [10], [11]. It consists of two sets of PSG record- +ings, namely Visit 1 and Visit 2. Here, we employed Visit +1 consisting of 5,791 PSG recordings from 5,791 subjects, +aged 39-90. Following [19], we excluded those recordings +without the presence of all five sleep stages. As a result, 5,463 +PSG recordings were retained. The recordings were manually +scored following the R&K guidelines [20] where each 30- +second epoch was labelled as one of eight categories {W, +N1, N2, N3, N4, REM, MOVEMENT, UNKNOWN}. In our +experiments, N3 and N4 stages were merged and considered as +N3 collectively. MOVEMENT and UNKNOWN epochs were +discarded. We adopted C4-A1 EEG in the experiments. +SleepEDF: This is the Sleep Cassette subset of the Sleep- +EDF Expanded dataset [21], [22] (version 2013). It con- +Fpz +Cz +C4 +A1 +Figure 1: The EEG setups. +sists of 20 subjects (10 males and 10 females) aged 25-34. +Each subject had two consecutive day-night PSG recordings +recorded, except for the subject 13 whose one night’s data +was lost due to device failure, making a total of 39 PSG +recordings. This database was manually labelled according +to the R&K guideline [20] where each 30-second epoch was +labelled as one of eight categories {W, N1, N2, N3, N4, REM, +MOVEMENT, UNKNOWN}. Similar to SHHS, N3 and N4 +stages were merged and considered as N3 collectively while +MOVEMENT and UNKNOWN categories were excluded. We +adopted the Fpz-Cz EEG channel in the experiments. Adhering +to the common setting in literature, a recording was trimmed +starting from 30 minutes before to 30 minutes after its in-bed +part. +ear-EEG: This database constitutes ear-EEG recordings of +20 subjects recorded using the same ear-EEG equipment. Each +subject had four nights of recordings. Three recordings were +excluded after artefact rejection [16] and two other recordings +were excluded as their remaining lengths was less than 200 +epochs after artefact rejection. This resulted in 75 recordings in +total. The labels of the data were obtained via manual scoring +of the PSG recordings which were recorded concurrently to +the ear-EEG as a reference. Manual scoring was done by two +independent and experienced sleep technicians according to +the AASM guidelines [23] where each 30-second epoch is +labelled as one in five categories {Wake, N1, N2, N3, REM}. +As in [16], we used the labels from the scorer 1 as the ground +truth here. We adopted the bilateral ear-EEG derivation (i.e. +the average of the left ear electrodes relative to the average +of the right ear electrodes (see Figure 1, middle row, right +picture)) in the experiments. More details about the recording +setup and data preprocessing can be found in [15], [16]. +cEEGrid: This database [24], [25] was recorded at the +University of Surrey using a lightweight flex–printed electrode +strip, namely the cEEGrid array [17], [18], fitted behind the +ear, as illustrated in Figure 1 (bottom row, left and middle + +DB01 +02 +INIONNASION +pl +F +p2 +F7 +F8 +F3 +Fz +F4 +A1 +A2 +T3 +P3 +Pz +P4 +T513 +pictures). 20 subjects, aged 34.9±13.8 years, took part in +the data recording and one overnight cEEGrid recording was +recorded for each subject. Two recordings were lost due to +human error and six recordings were excluded because of +excessive artefacts and data missing. 12 remaining recordings +were retained and used in the experiments as in [26]. The +labels of the data were obtained via manual scoring of the +PSG recordings which were recorded concurrently as reference +for the cEEGrid data [25]. The FB(R) (“front versus back”) +derivation for the right ear (see Figure 1, bottom row, right +picture) which was the best derivation [24], was adopted for +the experiments. More details about the recording setup and +data preprocessing can be found in [24], [25]. +III. LONG SEQUENCE MODELLING WITH L-SEQSLEEPNET +Given a training set {Sn}N +n=1 of size N where Sn = +� +(S(n) +1 +, y(n) +1 +), . . . , (S(n) +L , y(n) +L ) +� +is the n-th sequence con- +sisting of L consecutive sleep epochs. S(n) +ℓ +and y(n) +ℓ +∈ +{0, 1}C represent the ℓ-th 30-second sleep epoch and its +one-hot encoding label in the n-th sequence, respectively. +Here, C = 5 as we are dealing with 5-stage sleep staging. +Similar to a sequence-to-sequence sleep staging model [1], +given a sequence (S1, S2, . . . , SL) as input, L-SeqSleepNet +aims to classify all the epochs in the input sequence at +once and produce the sequence of probability output vectors +(ˆy1, . . . , ˆyL), where ˆy(n) +ℓ +∈ [0, 1]C, 1 ≤ ℓ ≤ L, is for the ℓ-th +epoch. However, different from existing sequence-to-sequence +sleep staging models that consider short sequences (L between +20-30 epochs or 10-15 minutes equivalently), we are interested +in long sequences (e.g. L=200 or 100 minutes equivalently) +so that a sequence roughly covers an entire sleep cycle. +The architecture of L-SeqSleepNet is illustrated in Figure 2. +It receives the time-frequency input and has the epoch encod- +ing part inherited from SeqSleepNet [4] while the sequence +encoding part is devised to handle long sequences efficiently. +For completeness, we describe all of these components in order +in the following sections. +A. Input +The EEG signal of a 30-second epoch is converted into a +log-magnitude time-frequency image S with T =29 time steps +and F = 129 frequency bins. To that end, short-time Fourier +transform (STFT) is applied to the signal with a window +length of 2 seconds and 50% overlap. In addition, Hamming +window and 256-point fast Fourier transform (FFT) are used. +The obtained amplitude spectrum is then log-transformed to +result in the image S ∈ RT ×F . +B. Epoch encoding +The role of the epoch encoding component is to learn the +feature map, F(S) : S �→ x, in order to transform an input +epoch S into a high-level feature vector x for representation. +This is realized by a subnetwork which is shared across all +epochs in an input sequence. The subnetwork is composed +of (i) a learnable filterbank layer, (ii) a bidirectional Long +Short-Term Memory (BLSTM) [2], and (iii) a gated attention +layer. The learnable filterbank layer [27] consists of M filters +(M < F), being tasked to smooth and reduce the frequency +dimension from F to M bins. The resulting image ˜S of +size T × M is then treated as a sequence of T vectors +(i.e., T image columns), (˜s1, ˜s2, . . . , ˜sT ), where ˜st ∈ RM, +1 ≤ t ≤ T . In order to capture the sequential information at +the epoch level, this sequence is encoded by the BLSTM with +recurrent batch normalization [28], into a sequence of vectors +(˜x1, ˜x2, . . . , ˜xT ): +(˜x1, ˜x2, . . . , ˜xT ) = BLST Me(˜s1, ˜s2, . . . , ˜sT ). +(1) +Here, ˜xt ∈ RHe with He +2 +is the size of the hidden states in +BLST Me. We use the subscript e to indicate modelling at the +epoch level and distinguish it from other BLSTMs in Section +III-C. Afterwards, the gated attention layer [29] is learned to +produce attention weights (w1, w2, . . . , wT ) which are used +to combine the feature vectors (˜x1, ˜x2, . . . , ˜xT ) to derive the +embedding vector x ∈ RHe representing the input epoch S: +x = +T +� +t=1 +wt˜xt, +(2) +where +wt = +exp(uT +t a) +�T +i=1 exp(uT +i a) +, +(3) +ut = tanh(Wa˜xt + ba). +(4) +In above equations, Wa ∈ RA×He and ba ∈ RA are trainable +weight matrix and bias vector, respectively. a ∈ RA is the +trainable context vector and A is the so-called attention size. +After the epoch encoding subnetwork described in Section +III-B, the input sequence (S1, S2, . . . , SL) has been trans- +formed into the sequence of embeddings (x1, x2, . . . , xL). +C. Long sequence modelling +Encoding the sequential information in the sequence of +epoch-wise feature vectors (x1, x2, . . . , xL) has proved to be +the key behind the success of existing sequence-to-sequence +sleep staging models [1]. This has been commonly accom- +plished by a subnetwork with sequential modelling capacity, +such as RNN [4]–[6], [30], [31] or Transformer [7]. However, +we have shown earlier that this approach is inefficient to +handle long sequences. +Central to L-SeqSleepNet’s architecture is the subnetwork +(the big blue box in Figure 2) that is capable of long sequence +modelling. In intuition, the processing of this component is +composed of four steps indicated by the circled numbers in +the figure: folding, intra-subsequence sequential modelling, +inter-subsequence sequential modelling, and unfolding. We +firstly fold the long sequence of length L into B non- +overlapping subsequences of length K, where L = B × K. +Sequential modelling is then performed within each of the +subsequences (i.e. intra-subsequence sequential modelling), +followed by sequential modelling across the subsequences +(i.e. inter-subsequence sequential modelling). Eventually, the +subquences are unfolded to resume the long sequence of +original length L. + +4 +... +... +... +... +... +... +... +... +S 1 +S 2 +SL +˜s11 +˜s12 +˜s1T +... +˜s21 +˜s22 +˜s2T +... +˜sL1 +˜sL2 +˜sLT +R +R +R +... +R +R +R +... +R +R +R +... +... +... +... +w12 +w1T +w21 w22 +w2T +wL1 wL2 +wLT +w11 +x1 +x2 +xL +o1 +o2 +oL +ˆy1 +ˆy2 +ˆyL +fc +fc +fc +fc +fc +fc +... +B +K +1 +1 +x1 x2 +xL +... +xL-1 +o1 +o2 +oL +oL-1 +B +K +1 +1 +... +... +... +... +1 +2 +3 +4 +R +R +R +R +... +R +R +R +R +... +R +R +R +R +... +R +R +R +R +... +R +R +R +R +... +R +R +R +R +... +R +R +R +R +... +R +R +R +R +... +... +... +... +... +... +... +... +... +... +... +˜x11 +˜x12 +˜x1T +˜x21 +˜x22 +˜x2T +˜xL1 +˜xL2 +˜xLT +Figure 2: The architecture of L-SeqSleepNet. +Formally, assume that we have folded the sequence +(x1, x2, . . . , xL) into B non-overlapping subsequences of size +K: + + + + + +x(1) +1 +x(1) +2 +. . . +x(1) +K +x(2) +1 +x(2) +2 +. . . +x(2) +K +. . . +. . . +. . . +. . . +x(B) +1 +x(B) +2 +. . . +x(B) +K + + + + + . +(5) +Here, we use the subscript k, 1 ≤ k ≤ K, to indicate the +index of an element inside a subsequence and the superscript +b, 1 ≤ b ≤ B, to indicate the index of a subsequence. After +folding, the ℓ-th element in the original sequence will become +the k-th element in the b-th subsequence, where +b = +�ℓ − 1 +K +� ++ 1, +(6) +k = [(ℓ − 1) mod K] + 1. +(7) +Intra-subsequence sequential modelling (along horizontal +direction as illustrated in Figure 2) is carried out on a b-th +subsequence (x(b) +1 , x(b) +2 , . . . , x(b) +K ) using a BLSTM with recur- +rent batch normalization, transforming it into a subsequence +of output vectors (˜o(b) +1 , ˜o(b) +2 , . . . , ˜o(b) +K ): +(˜o(b) +1 , ˜o(b) +2 , . . . , ˜o(b) +K ) = BLST Mss(x(b) +1 , x(b) +2 , . . . , x(b) +K ), +(8) +where ˜o(b) +k +∈ RHss. Hss +2 +is the size of the hidden states in +BLST Mss. The subscript ss is used to indicate the modelling + +5 +at the subsequence level. The output vectors ˜o(b) +k +are then +linear transformed via a fully connected (fc) layer, followed +by layer normalization (LN) [32] and a residual connection: +¯o(b) +k += ˜o(b) +k ++ LN(Wss˜o(b) +k ++ bss). +(9) +Here, Wss ∈ RHss×Hss and bss ∈ RHss denote the trainable +weight matrix and bias vector of the fc layer, respectively. As +a result, we obtain the following B output subsequences: + + + + + +¯o(1) +1 +¯o(1) +2 +. . . +¯o(1) +K +¯o(2) +1 +¯o(2) +2 +. . . +¯o(2) +K +. . . +. . . +. . . +. . . +¯o(B) +1 +¯o(B) +2 +. . . +¯o(B) +K + + + + + . +(10) +Up to this point, each output vector ¯o(b) +k ∈ RHss in a b-th +subsequence is expected to contain the information of the +entire subsequence. +Inter-subsequence sequential modelling (along vertical di- +rection as illustrated in Figure 2) is then conducted at each +index k across all B subsequences using another BLSTM with +recurrent batch normalization: +(ˆo(1) +k , ˆo(2) +k , . . . , ˆo(B) +k +) = BLST Mws(¯o(1) +k , ¯o(2) +k , . . . , ¯o(B) +k +), +(11) +where ˆo(b) +k +∈ RHws. Similar to the intra-subsequence sequen- +tial modelling step, linear transformation via a fc layer, layer +normalization, and a residual connection are then applied: +o(b) +k += ˆo(b) +k ++ LN(Wwsˆo(b) +k ++ bws), +(12) +resulting in the following B output subsequences: + + + + + +o(1) +1 +o(1) +2 +. . . +o(1) +K +o(2) +1 +o(2) +2 +. . . +o(2) +K +. . . +. . . +. . . +. . . +o(B) +1 +o(B) +2 +. . . +o(B) +K + + + + + , +(13) +where o(b) +k +∈ RHws. In (12), Wws ∈ RHws×Hws and bws ∈ +RHws denote the trainable weight matrix and bias vector of +the fc layer, respectively. Hws +2 +is the size of the hidden states +in BLST Mws and we use the subscript ws to indicate the +modelling at the whole sequence level. Given that an output +vector ¯o(b) +k +contains the information of the entire b-th subse- +quence after the intra-subsequence sequential modelling step, +an output vector o(b) +k +is expected to contain the information +of all B subsequences after the inter-subsequence sequential +modelling step. In other words, o(b) +k +contains the information +of the whole original long sequence of length L. +Eventually, the output subsequences in (13) are un- +folded, resulting in a single long sequence of L elements +(o1, o2, . . . , oL). After unfolding, the k-th element in the b-th +subsequence, o(b) +k , in (13) will become the ℓ-th element oℓ in +the final output sequence, where +ℓ = (b − 1) × K + k. +(14) +D. Classification +For classification, the output vectors in the output sequence +(o1, o2, . . . , oL) are passed through two fc layers, each with +Nfc units and Rectified Linear Unit (ReLU) activation, before +being presented to an output layer with softmax activation to +produce the network predictions (ˆy1, ˆy2, . . . , ˆyL). Note that +these layers are shared across the time indices. +The network is trained to minimize the cross-entropy loss +averaged over the sequence length and the training data: +L(θ) = − +1 +N · L +N +� +n=1 +L +� +ℓ=1 +y(n) +ℓ +log ˆy(n) +ℓ ++ λ||θ||2 +2. +(15) +In (15), θ denotes the network parameters and λ is the hyper- +parameter of the ℓ2-norm regularization term. +IV. EXPERIMENTS +A. Experimental setup +The experimental setup using the four databases described +in Section II is summarized in Table II. Conforming to +majority of previous works in literature, we conducted leave- +one-subject-out cross validation (LOSO CV) on SleepEDF, +ear-EEG, and cEEGrid databases. For SHHS, we randomly +split the subjects into 70% for training and 30% for testing. +In each experiment, a number of subjects, specified in Table +II, were held out from the training set for validation purpose. +In particular, due to the small number of recordings in the +SleepEDF and cEEGrid databases, we repeated these experi- +ments 5 times and report the average performance. +B. Parameters +We experimented with the sequence length L = 200 epochs +(equivalent to 100 minutes of sleep data to roughly cover a +whole sleep cycle), the number of subsequences B = 10, and +the subsequence length K = 20. We also experimented with +other values for L, B, and K, and will discuss their influence +in Section IV-D3. Regarding the network architecture, we set +the number of filters M = 32, the attention size A = 64, the +size of BLSTM’s hidden states He +2 = Hss +2 += Hws +2 += 64, and +the size of the fc layers Nfc = 512. The hyper-parameter λ in +(15) was fixed to 10−4. In addition, a dropout rate of 0.1 was +applied to the LSTM cells and the fc layers during training. +The network was trained using Adam optimizer [33] with +a learning rate of 10−4, β1 = 0.9, β2 = 0.999, and ǫ = 10−7. +A minibatch size of 8 was used for training. The model was +validated on the validation set every 100 training steps for +the experiments on SHHS, SleepEDF, and ear-EEG. For the +smallest database cEEGrid, the validation was done every 25 +Table II: Summary of experimental setup. +Database +Num. of +subjects +Num. of +recordings +Experimental +setup +Held-out +valid. set +Repetition +SHHS +5,463 +5,463 +train/test: 0.7/0.3 +100 +1 +SleepEDF +20 +39 +LOSO CV +4 +5 +ear-EEG +20 +75 +LOSO CV +4 +1 +cEEGrid +12 +12 +LOSO CV +2 +5 + +6 +training steps. For the largest database SHHS, the model was +trained for fixed 5000 validation steps without early stopping. +Other than that, the model was trained for 10 training epochs +and stopped early after 50 validation steps (for SleepEDF and +ear-EEG) and 25 validation steps (for cEEGrid) without im- +provement on the validation accuracy. The model performing +best on the validation set was then retained to be evaluated on +the test data. +C. The baseline +We used SeqSleepNet presented in [4] as the main base- +line in the experiments. This network shares a similar in- +put format and epoch encoding component as the proposed +L-SeqSleepNet. However, different from L-SeqSleepNet, it +follows the typical “flat” sequential modelling approach as +many other works in literature, making it a natural baseline +to investigate the effects of the long sequential modelling +approach presented here. We compared L-SeqSleepNet and +the SeqSleepNet baseline under two initialization schemes: +(1) random initialization (i.e., training from scratch) and (2) +initialization with a pretrained network (i.e. finetuning for +transfer learning [26]). For the former, the networks were +trained from scratch as usual. For the latter, the networks +trained on SHHS were used as the pretrained models and +finetuned on the SleepEDF, ear-EEG, and cEEGrid databases. +In addition, in order to assess L-SeqSleepNet in terms +of sleep staging performance, we also compare it to other +methods in literature reporting results on the experimental +databases. +D. Experimental results +1) Overall sleep staging performance: A comprehensive +performance comparison between L-SeqSleepNet, the Se- +qSleepNet baseline, and prior works are shown in Table +III. On the one hand, it can be seen that L-SeqSleepNet +consistently outperforms the SeqSleepNet baseline over all +the experimental databases. With the random initialization +scheme, L-SeqSleepNet leads to overall accuracy gains of +1.2%, 0.7%, 0.7%, and 1.9% on SHHS, SleepEDF, ear-EEG, +and cEEGrid, respectively. In case of the pretraining-based +initialization, the gains on SleepEDF, ear-EEG, and cEEGrid +are even higher, reaching 1.0%, 1.9%, and 3.9%, respectively. +The wider gains obtained with respect to the pretraining-based +initialization shed some light on what is being transferred from +the source domain (i.e. SHHS) to the target domains (i.e., +SleepEDF, ear-EEG, and cEEGrid) via L-SeqSleepNet in these +transfer learning scenarios. More specifically, in addition to the +usual reuse of good feature representations [49], it is likely that +the diverse sleep cycle structure from the large cohort in the +source domain also contributes to the transferred knowledge +and gives rise to L-SeqSleepNet’s higher performance gains +compared to the SeqSleepNet baseline. +On the other hand, L-SeqSleepNet also results in better +performance than the current state-of-the-arts (where the direct +comparison is compatible) over all the databases. On the large +database SHHS, L-SeqSleepNet achieves an overall accuracy +of 88.4%, 0.7% and 0.8% higher than SleepTransformer [7] +and XSleepNet [9], respectively. This is particularly interesting +given that the SeqSleepNet-type architecture of L-SeqSleepNet +essentially constitutes only one half of the XSleepNet’s ar- +chitecture [9] and that L-SeqSleepNet has around 6.3×105 +parameters in total, roughly 9 times smaller than XSleepNet +which has 5.7 × 106 parameters. On the smaller databases +(i.e., SleepEDF, ear-EEG, and cEEGrid), a large margin of +performance is consistently seen between L-SeqSleepNet and +other counterparts. +2) The effects of whole-cycle sequence modelling: Using +the pretraining-based initialization scheme, we inspected the +effects of L-SeqSleepNet’s whole-cycle sequence modelling +to the model’s errors. From the confusion matrices and the +number of errors per sleep stage in Figure 3, it becomes clear +that across all the databases, compared to the SeqSleepNet +baseline, L-SeqSleepNet lowers the errors in all other sleep +stages, most noticeably in N3 and REM, at the small expense +of errors in N2. This implies that taking into account the +structure of sleep cycles helps to correct modelling errors that +are impossible to fix under the existing state-of-the-art (short) +context modelling approach. +Figure 4 visually represents the distribution of individual +errors produced by L-SeqSleepNet and the SeqSleepNet base- +line. It was found in [50] that even though the performance of +automatic sleep staging systems has been deemed sufficient +for clinical use these systems may generally struggle with +particular recordings, leading to exceptionally high individual +errors. The distribution of individual errors from both L- +SeqSleepNet and the baseline in the figure indeed reflects this +finding, particularly on SHHS. However, L-SeqSleepNet can +reduce these exceptionally high individual errors across all the +databases, and more strikingly on ear-EEG and cEEGrid. This +is particularly important from an application point of view +as bringing down these high individual errors would make +automatic sleep staging systems more acceptable in clinical +environments as well as require less human intervention for +manual editing or rescoring. +3) The influence of the sequence length: In this section, +we examined the influence of the sequence length to L- +SeqSleepNet and the SeqSleepNet baseline from two perspec- +tives: staging performance and computational time. Using the +large database SHHS in this examination, Table IV summa- +rizes the overall sleep staging performance and computational +time of the two models with different sequence lengths. +Firstly, as already mentioned in Section I, SeqSleepNet +with the “flat” sequential modelling approach is inefficient +in handling long sequences, resulting in marginal gains on +overall accuracy, for instance, 0.3% when the sequence length +increases to 200 from 20 epochs. In contrast, at the sequence +length of 200 epochs, the improvement on overall accuracy +achieved by L-SeqSleepNet over SeqSleepNet with L = 20 +reaches 1.2%, highlighting the effectiveness of the proposed +long sequential modelling approach. +Secondly, concerning L-SeqSleepNet itself, halving the se- +quence length to 100 epochs (50 minutes, roughly a half sleep +cycle) results in a drop of 0.3% on overall accuracy while +doubling it to 400 epochs (200 minutes, roughly two sleep +cycles) causes negligible consequence to the performance. + +7 +Table III: Performance obtained by L-SeqSleepNet, the SeqSleepNet baseline in comparison with previous works on the +experimental databases. Note that some results reported in previous works, marked by the ‡ superscript, are not compatible +for a direct comparison here due to the discrepancies in data split, the number of channel used, the number of subject used, +modelling tasks, etc. The ∗ superscript indicates the pretraining-based initialization scheme. +Database +System +Overall performance +Class-wise MF1 +Acc. +κ +MF1 +Sens. +Spec. +Wake +N1 +N1 +N3 +REM +SHHS +L-SeqSleepNet +88.4 +0.838 +81.4 +80.4 +96.7 +93.1 +51.1 +89.0 +84.9 +89.8 +SeqSleepNet [4] +87.2 +0.820 +80.2 +78.7 +96.3 +91.8 +49.1 +88.2 +83.5 +88.2 +SleepTransformer [7] +87.7 +0.828 +80.1 +78.7 +96.5 +92.2 +46.1 +88.3 +85.2 +88.6 +XSleepNet1 [9] +87.6 +0.826 +80.7 +79.7 +96.5 +91.6 +51.4 +88.5 +85.0 +88.4 +XSleepNet2 [9] +87.5 +0.826 +81.0 +80.4 +96.5 +92.0 +49.9 +88.3 +85.0 +88.2 +U-Sleep‡ [34] +− +− +80.0 +− +− +93.0 +51.0 +87.0 +76.0 +92.0 +Olesen et al.‡ [31] +87.1 +0.816 +78.8 +77.7 +96.3 +94.1 +47.8 +87.9 +74.3 +89.9 +CNN [19] +86.8 +0.810 +78.5 +− +95.0 +− +− +− +− +− +FCNN+RNN [9] +86.7 +0.813 +79.5 +78.1 +96.2 +91.1 +48.7 +88.0 +82.6 +87.1 +IITNet [6] +86.7 +0.810 +79.8 +− +− +− +− +− +− +− +AttnSleep‡ [35] +84.2 +0.780 +75.3 +− +− +86.7 +33.2 +87.1 +87.1 +82.1 +SleepEDF +L-SeqSleepNet∗ +88.6 ± 0.1 0.845 ± 0.001 82.9 ± 0.2 82.1 ± 0.1 96.9 ± 0.0 +94.1 ± 0.4 53.3 ± 1.2 89.7 ± 0.1 88.4 ± 0.3 88.9 ± 0.2 +L-SeqSleepNet +86.3 ± 0.2 0.813 ± 0.003 79.3 ± 0.4 78.8 ± 0.5 96.3 ± 0.1 +91.6 ± 0.4 45.3 ± 1.4 88.5 ± 0.3 86.2 ± 0.7 85.2 ± 0.2 +SeqSleepNet∗ [4] +87.6 ± 0.2 0.830 ± 0.002 81.8 ± 0.2 80.3 ± 0.3 96.6 ± 0.1 +92.7 ± 0.4 52.7 ± 0.7 88.9 ± 0.1 86.7 ± 0.2 87.8 ± 0.1 +SeqSleepNet [4] +85.6 ± 0.3 0.803 ± 0.004 78.6 ± 0.2 78.2 ± 0.1 96.2 ± 0.1 +91.2 ± 0.6 44.7 ± 0.8 88.0 ± 0.1 86.2 ± 0.2 83.0 ± 0.8 +SalientSleepNet‡ [36] +87.5 +− +83.0 +− +− +92.3 +56.2 +89.9 +87.2 +89.2 +TransSleep [37] +86.5 +− +82.5 +− +− +87.1 +60.8 +91.7 +85.5 +87.4 +XSleepNet2 [9] +86.3 +0.813 +80.6 +80.2 +96.4 +92.2 +51.8 +88.0 +86.8 +83.9 +XSleepNet1 [9] +86.0 +0.810 +80.0 +79.6 +96.3 +91.3 +49.5 +88.0 +86.9 +84.2 +SimpleSleepNet‡ [38] +− +− +80.5 +− +− +− +− +− +− +− +MNN‡ [39] +85.9 +− +80.5 +− +− +84.6 +56.3 +90.7 +84.8 +86.1 +Khalili & Asl [40] +85.4 +0.800 +79.3 +− +− +90.0 +46.6 +88.4 +86.1 +84.6 +TinySleepNet [41] +85.4 +0.800 +80.5 +− +− +90.1 +51.4 +88.5 +88.3 +84.3 +RobustSleepNet‡ [30] +− +− +79.1 +− +− +− +− +− +− +− +U-Sleep [34] +− +− +79.0 +− +− +93.0 +57.0 +86.0 +71.0 +88.0 +SleepFCN [42] +84.8 +0.780 +78.8 +− +− +89.6 +44.6 +89.1 +90.6 +80.3 +MRASleepNet [43] +84.5 +0.786 +78.9 +− +− +− +− +− +− +− +ResNetMHA [44] +84.3 +− +79.0 +− +− +90.2 +48.3 +87.8 +85.6 +83.3 +AttnSleep [35] +84.4 +0.790 +78.1 +− +− +89.7 +42.6 +88.8 +90.2 +79.0 +DeepSleepNet-Lite [45] +84.0 +0.780 +78.0 +− +− +87.1 +44.4 +87.9 +88.2 +82.4 +IITNet [6] +83.9 +0.780 +77.6 +− +− +− +− +− +− +− +DeepSleepNet [5] +82.0 +0.760 +76.9 +− +− +86.7 +45.5 +85.1 +83.3 +82.6 +FCNN+RNN [9] +81.8 +0.754 +75.6 +75.7 +95.3 +89.4 +44.1 +84.0 +84.0 +76.3 +SleepEEGNet [8] +81.5 +0.750 +76.6 +− +− +89.4 +44.4 +84.7 +84.6 +79.6 +earEEG +L-SeqSleepNet* +87.9 +0.829 +84.1 +83.1 +96.5 +92.7 +59.2 +89.8 +89.4 +89.2 +L-SeqSleepNet +83.7 +0.770 +79.4 +78.7 +95.3 +89.3 +52.3 +86.1 +87.4 +81.9 +SeqSleepNet* [4] +86.0 +0.801 +81.9 +80.3 +95.8 +90.7 +55.7 +88.2 +88.3 +86.5 +SeqSleepNet [4] +83.0 +0.759 +78.5 +77.3 +95.0 +89.0 +50.0 +85.7 +87.9 +79.8 +Ensemble [46] +− +0.780 +− +− +− +− +− +− +− +− +Random Forest‡ [15] +− +0.730 +− +− +− +− +− +− +− +− +cEEGrid +L-SeqSleepNet* +78.9 ± 0.6 0.703 ± 0.008 67.9 ± 0.5 67.0 ± 0.3 94.2 ± 0.1 +92.0 ± 0.7 25.8 ± 1.4 74.1 ± 0.6 79.7 ± 1.1 68.0 ± 1.5 +L-SeqSleepNet +72.3 ± 0.6 0.607 ± 0.008 57.3 ± 0.8 57.8 ± 0.9 92.4 ± 0.2 +89.9 ± 0.6 +7.5 ± 1.8 +65.8 ± 0.9 72.4 ± 1.8 50.8 ± 2.4 +SeqSleepNet* [4] +75.0 ± 0.4 0.647 ± 0.006 62.7 ± 0.7 61.3 ± 0.8 93.2 ± 0.1 +90.6 ± 0.5 23.1 ± 1.1 71.1 ± 0.5 72.1 ± 0.7 56.6 ± 3.1 +SeqSleepNet [4] +70.4 ± 1.5 0.578 ± 0.024 54.1 ± 1.5 55.0 ± 1.7 91.7 ± 0.5 +87.4 ± 1.9 +0.5 ± 0.5 +64.6 ± 1.7 69.3 ± 1.7 48.6 ± 5.0 +ADA pers‡ [47] +72.8 +0.618 +− +− +− +− +− +− +− +− +Feat. matching‡ [48] +71.3 +0.605 +− +− +− +− +− +− +− +− +Random Forest‡ [24] +70.0 +0.580 +− +− +− +− +− +− +− +− +DeepSleepNet* [26] +58.2 +0.391 +42.8 +− +− +74.9 +5.7 +47.0 +63.4 +23.3 +DeepSleepNet [26] +42.5 +0.195 +30.3 +− +− +57.6 +6.9 +23.1 +51.4 +12.3 +This suggests that the sequential information in one sleep cycle +is probably all we need for sleep staging. Interestingly, the +manner a long sequence is folded into subsequences seems +to matter. At L = 200, folding with B = 20 and K = 10 +(i.e., 20 subsequences, each of length 10 epochs) leads to +better performance than that with B = 10 and K = 20 (i.e., +10 subsequences, each of length 20 epochs). It is likely the +distance between two adjacent elements in the sequence during +inter-subsequence sequential modelling causes the difference. +In effect, this distance is 10 epochs in the former while it is +20 epochs in the latter. The double distance in the latter could +loosen the temporal dependency between the elements in the +sequence, worsening the performance consequentially. +Thirdly, given that the SeqSleepNet baseline’s training +time grows linearly as the sequence length increases, L- +SeqSleepNet’s training time only grows sub-linearly. For ex- +ample, SeqSleepNet at L = 200 requires 1,465 seconds for +1000 training steps, 4.5 times longer than itself at L = 20. +L-SeqSleepNet at L = 200, on the other hand, merely needs +around 450 seconds, just 1.4 times slower than SeqSleepNet +at L = 20 and around 3.3 times faster than SeqSleepNet at +L = 200. The reason is when a sequence of length L is folded +into B subsequences, each of length K, the number of time +steps engaged in sequential modelling is in fact reduced to +B + K which is much smaller than L. + +8 +407677 +91.5% +9287 +15.0% +9375 +1.4% +1031 +0.5% +2802 +1.2% +9754 +2.2% +28733 +46.4% +9323 +1.4% +9 +0.0% +2717 +1.1% +20587 +4.6% +17569 +28.4% +606035 +91.1% +38327 +17.2% +14301 +5.9% +2085 +0.5% +47 +0.1% +22737 +3.4% +182672 +82.1% +333 +0.1% +5524 +1.2% +6262 +10.1% +18038 +2.7% +531 +0.2% +221769 +91.7% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +406039 +91.1% +8911 +14.4% +15097 +2.3% +2648 +1.2% +6712 +2.8% +11928 +2.7% +28047 +45.3% +8535 +1.3% +12 +0.0% +3800 +1.6% +21072 +4.7% +19177 +31.0% +607275 +91.2% +43827 +19.7% +20606 +8.5% +1936 +0.4% +75 +0.1% +20434 +3.1% +175880 +79.0% +361 +0.1% +4652 +1.0% +5688 +9.2% +14167 +2.1% +203 +0.1% +210443 +87.0% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +0 +1 +2 +3 +4 +5 +6 +10 4 +11944 +95.4% +440 +15.8% +336 +1.9% +28 +0.5% +124 +1.6% +360 +2.9% +1328 +47.8% +355 +2.0% +1 +0.0% +162 +2.1% +99 +0.8% +642 +23.1% +15869 +90.2% +686 +12.2% +480 +6.2% +23 +0.2% +8 +0.3% +537 +3.1% +4911 +87.2% +3 +0.0% +98 +0.8% +361 +13.0% +503 +2.9% +3 +0.1% +6943 +90.0% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +11748 +93.8% +467 +16.8% +375 +2.1% +48 +0.9% +175 +2.3% +356 +2.8% +1292 +46.5% +303 +1.7% +0 +0.0% +168 +2.2% +279 +2.2% +655 +23.6% +16050 +91.2% +944 +16.8% +590 +7.7% +42 +0.3% +5 +0.2% +381 +2.2% +4637 +82.4% +3 +0.0% +99 +0.8% +360 +13.0% +489 +2.8% +1 +0.0% +6775 +87.9% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +0 +500 +1000 +1500 +2000 +Wake N1 +N2 +N3 REM +Wake N1 +N2 +N3 REM +# errors +# errors +L-SeqSleepNet +SeqSleepNet +L-SeqSleepNet’s confusion matrix +SeqSleepNet’s confusion matrix +Error comparison +[ SHHS ] +[ SleepEDF ] +L-SeqSleepNet’s confusion matrix +SeqSleepNet’s confusion matrix +Error comparison +L-SeqSleepNet +SeqSleepNet +407677 +91.5% +9287 +15.0% +9375 +1.4% +1031 +0.5% +2802 +1.2% +2% +9754 +2.2% +28733 +46.4% +9323 +1.4% +9 +0.0% +2717 +1.1% +1% +20587 +4.6% +17569 +28.4% +606035 +91.1% +38327 +17.2% +14301 +5. +5.9% +9% +2085 +0.5% +47 +0.1% +22737 +3.4% +182672 +82.1% +333 +0. +0.1% +1% +5524 +1.2% +6262 +10.1% +18038 +2.7% +531 +0.2% +221769 +91 +91.7% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +406039 +91.1% +8911 +14.4% +15097 +2.3% +2648 +1.2% +6712 +2.8% +8% +11928 +2.7% +28047 +45.3% +8535 +1.3% +12 +0.0% +3800 +1.6% +6% +21072 +4.7% +19177 +31.0% +607275 +91.2% +43827 +19.7% +20606 +8. +8.5% +5% +1936 +0.4% +75 +0.1% +20434 +3.1% +175880 +79.0% +361 +0. +0.1% +1% +4652 +1.0% +5688 +9.2% +14167 +2.1% +203 +0.1% +210443 +87 +87.0 +.0% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +0 +1 +2 +3 +4 +5 +6 +10 4 +Wake N1 +N2 +N3 REM +# errors +L-SeqSleepNet’s confusion matrix +SeqSleepNet’s confusion matrix +Error comparison +[ SHHS ] +L-SeqSleepNet +SeqSleepNet +6142 +92.3% +347 +7.2% +46 +0.2% +1 +0.0% +54 +0.4% +378 +5.7% +2501 +51.7% +534 +1.8% +19 +0.2% +180 +1.5% +34 +0.5% +1243 +25.7% +26313 +90.9% +1276 +10.7% +776 +6.3% +5 +0.1% +20 +0.4% +1165 +4.0% +10574 +89.1% +21 +0.2% +98 +1.5% +722 +14.9% +877 +3.0% +0 +0.0% +11251 +91.6% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +5932 +89.1% +308 +6.4% +125 +0.4% +2 +0.0% +55 +0.4% +463 +7.0% +2363 +48.9% +543 +1.9% +17 +0.1% +260 +2.1% +127 +1.9% +1506 +31.2% +26550 +91.8% +1626 +13.7% +1444 +11.8% +5 +0.1% +17 +0.4% +1002 +3.5% +10222 +86.1% +27 +0.2% +130 +2.0% +639 +13.2% +715 +2.5% +3 +0.0% +10496 +85.5% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +0 +500 +1000 +1500 +2000 +2500 +3000 +5238 +93.6% +218 +26.3% +253 +5.5% +40 +2.3% +38 +2.1% +90 +1.6% +166 +20.0% +175 +3.8% +1 +0.1% +25 +1.4% +186 +3.3% +356 +42.9% +3548 +76.9% +298 +17.3% +578 +31.4% +31 +0.6% +2 +0.2% +313 +6.8% +1373 +79.8% +6 +0.3% +53 +0.9% +87 +10.5% +324 +7.0% +8 +0.5% +1192 +64.8% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +5056 +90.3% +209 +25.2% +216 +4.7% +48 +2.8% +30 +1.6% +134 +2.4% +142 +17.1% +112 +2.4% +1 +0.1% +14 +0.8% +271 +4.8% +404 +48.8% +3626 +78.6% +444 +25.8% +838 +45.6% +57 +1.0% +0 +0.0% +326 +7.1% +1201 +69.8% +27 +1.5% +79 +1.4% +73 +8.8% +333 +7.2% +26 +1.5% +930 +50.6% +Wake +N1 +N2 +N3 +REM +Output +Wake +N1 +N2 +N3 +REM +Ground-truth +0 +200 +400 +600 +800 +1000 +1200 +Wake N1 +N2 +N3 REM +# errors +Wake N1 +N2 +N3 REM +# errors +L-SeqSleepNet +SeqSleepNet +L-SeqSleepNet’s confusion matrix +SeqSleepNet’s confusion matrix +Error comparison +[ ear-EEG ] +[ cEEGrid ] +L-SeqSleepNet’s confusion matrix +SeqSleepNet’s confusion matrix +Error comparison +L-SeqSleepNet +SeqSleepNet +Figure 3: Comparison between L-SeqSleepNet and SeqSleepNet. +V. DISCUSSION +On the ear-EEG database, where manual scoring from two +independent sleep technicians are available, it is peculiar to +compare the achieved performance to what was presented in +Mikkelsen et al. [16]. In particular, we see that the kappa +value of 0.829 obtained by L-SeqSleepNet is higher than the +inter-scorer agreement, despite the fact that L-SeqSleepNet +only used the bilateral ear-EEG derivation, and thus, did not +have access to any of the same electrode derivations as were +used in the manual scoring of this database. This indicates +that long time scale sleep information can compensate for +a decrease in “field of view” on the sensor level, which is +remarkable. This observation also suggests that consolidating +intrinsic information from a long context, like a whole sleep +cycle or more, is something an automated scorer can excel +at, and may do better than a human scorer who can only +consciously attend to a relatively short context (a few minutes +[23]) during manual scoring. + +9 +20 +30 +40 +50 +60 +70 +80 +90 +100 +60 +70 +80 +90 +100 +65 +70 +75 +80 +85 +90 +100 +95 +30 +40 +50 +60 +70 +80 +90 +100 +Accurarcy (%) +[ SHHS ] +[ SleepEDF ] +[ ear-EEG ] +[ cEEGrid ] +L-SeqSleepNet +SeqSleepNet +Figure 4: Distribution of individual accuracies produced by +L-SeqSleepNet and the SeqSleepNet baseline. +Table IV: The overall performance and the training time +produced by L-SeqSleepNet and the SeqSleepNet baseline +with different sequence lengths. Note that the training time +was measured for 1000 training steps on an NVIDIA Tesla +V100 GPU. +System +L (B × K) +Acc. +k +MF1 +Training +time (s) +L-SeqSleepNet +100 (10×10) +88.1 +0.833 +80.7 +354.1 +L-SeqSleepNet +200 (10×20) +88.2 +0.835 +81.4 +454.4 +L-SeqSleepNet +200 (20×10) +88.4 +0.837 +81.6 +449.0 +L-SeqSleepNet +400 (20×20) +88.4 +0.838 +81.4 +605.2 +SeqSleepNet +20 +87.2 +0.820 +80.2 +322.1 +SeqSleepNet +100 +87.4 +0.823 +80.1 +842.0 +SeqSleepNet +200 +87.5 +0.824 +80.4 +1465.6 +While the work here focuses on single-EEG input and +uses RNN as the backbone for sequential modelling, the +presented long sequential modelling method can be considered +as a generic method and can be used in replacement for the +“flat” sequential modelling method in the current state-of- +the-art sequence-to-sequence sleep staging framework [1]. It +can be readily integrated into existing works relying on this +framework to investigate the effects of whole-cycle sequential +modelling in multimodal fusion [30] and multi-view learning +[9] or to speed up the computation with recurrent-free archi- +tectures, such as Transformer [7]. +VI. CONCLUSIONS +We presented in this work a novel method for modelling +whole-cycle long sequences for automatic sleep staging. A +long sequence was first folded into multiple subsequences. +Intra-subsequence and inter-subsequence sequential modelling +were then performed before the subsequences were unfolded to +resume the size of the original sequence. We demonstrated that +the proposed approach overcomes the limitations of the exist- +ing sequential modelling approach in handling long sequences +and that taking the structural information of sleep cycles into +account consistently improved the sleep staging performance. +L-SeqSleepNet, the network with long sequential modelling +capacity we introduced, outperformed not only the baseline but +also the existing state-of-the-art methods across four distinct +databases with different EEG setups, including scalp EEG, +in-ear EEG, and around-the-ear EEG. We also empirically +showed that incorporating the logic of stage transition in +sleep cycles helped to reduce staging errors at epoch level +as well as exceptionally high individual errors at recording +level. Furthermore, the performance benefits came at just a +little cost of sub-linear growth in computational overhead. +ACKNOWLEDGMENT +This research received funding from the Flemish Govern- +ment (AI Research Program). 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Phan, “Automatic sleep staging for +the young and the old – evaluating age underrepresentation in deep +learning,” Sleep Medicine, 2023, (under revision). + diff --git a/g9E1T4oBgHgl3EQfzAXz/content/tmp_files/load_file.txt b/g9E1T4oBgHgl3EQfzAXz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a16648a8b9b647b52a2263c86c109bb933ab1a25 --- /dev/null +++ b/g9E1T4oBgHgl3EQfzAXz/content/tmp_files/load_file.txt @@ -0,0 +1,1610 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf,len=1609 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='03441v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='SP] 9 Jan 2023 1 L-SeqSleepNet: Whole-cycle Long Sequence Modelling for Automatic Sleep Staging Huy Phan∗, Kristian P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Lorenzen, Elisabeth Heremans, Oliver Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Ch´en, Minh C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Tran, Philipp Koch, Alfred Mertins, Mathias Baumert, Kaare B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Mikkelsen, and Maarten De Vos Abstract—Human sleep is cyclical with a period of approx- imately 90 minutes, implying long temporal dependency in the sleep data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Yet, exploring this long-term dependency when developing sleep staging models has remained untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In this work, we show that while encoding the logic of a whole sleep cycle is crucial to improve sleep staging performance, the sequential modelling approach in existing state-of-the-art deep learning models are inefficient for that purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We thus introduce a method for efficient long sequence modelling and propose a new deep learning model, L-SeqSleepNet, incorporating this method to take into account whole-cycle sleep information for sleep stag- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Evaluating L-SeqSleepNet on a set of four distinct databases of various sizes, we demonstrate state-of-the-art performance obtained by the model over three different EEG setups, including scalp EEG in conventional Polysomnography (PSG), in-ear EEG, and around-the-ear EEG (cEEGrid), even with a single-EEG channel input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Our analyses also show that L-SeqSleepNet is able to remedy the effect of N2 sleep (the major class in terms of classification) to bring down errors in other sleep stages and that the network largely reduces exceptionally high errors seen in many subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Finally, the computation time only grows at a sub-linear rate when the sequence length increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Index Terms—Automatic sleep staging, deep neural network, long sequence modelling, sequence-to-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' INTRODUCTION Sleep is a slow-transitioning neural process, and thus, the data recorded from this process embeds abundant sequential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Capturing this sequential information has been shown to be crucial for automatic sleep staging systems to achieve good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In fact, the capacity of sequential modelling has been the driving force behind existing deep- learning-based sleep staging models, bringing the machine scoring performance on par with that of human experts [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Using recurrent neural networks (RNNs) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', Long Short- Term Memory (LSTM) [2]) or, more recently, the Transformer architecture [3], these models are able to capture the temporal H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Phan is with the School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, UK and the Alan Turing Institute, London NW1 2DB, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Lorenzen and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Mikkelsen are with the Department of Electrical and Computer Engineering, Aarhus University, Aarhus 8200, Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Heremans and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' De Vos are with the Department of Electrical Engineering and with the Department of Development and Regeneration, KU Leuven, 3001 Leuven, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Ch´en is with the School of Economics, Finance and Management, University of Bristol, Bristol BS8 1TU, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Tran is with Nuffield Department of Clinical Neurosciences, University of Oxford, Oxford OX3 9DU, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Koch and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Mertins are with the Institute for Signal Processing, University of L¨ubeck, L¨ubeck 23562, Germany and with the German Research Center for Artificial Intelligence (DFKI), L¨ubeck 23562, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Baumert is with School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide SA 5005, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ∗Corresponding author: pquochuy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='com Table I: Overall accuracy of SeqSleepNet and SleepTrans- former obtained on the SHSS database with a long sequence length of {100, 200} compared to a typical values, 20 (Se- qSleepNet) and 21 (SleepTransformer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ∗Note that this result is slightly different from that reported for SeqSleepNet in [9] as we did not exercise early stopping here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Sequence length 20/21 100 200 SeqSleepNet [4] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2∗ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3) SleepTransformer [7] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 (↓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 (↓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4) dependency in a sequence of multiple consecutive epochs of sleep data, resembling the way a human expert conducts manual scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The sequence length (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', the number of epochs in the sequence) was shown to be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Many different works [4]–[8] found a length around 20-30 epochs to be effective and these have been de facto values for this hyper- parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' At least, using a longer sequence was reported to lead to little to no performance gain at the cost of significantly increased computational overhead [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We conducted some initial experiments with a large se- quence length of {100, 200} to verify the above observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' For this investigation, we employed two models, SeqSleep- Net [4] and SleepTransformer [7], and the SHHS database [10], [11], a large database consisting of recordings from 5,791 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Both models conform to the state-of-the- art sequence-to-sequence sleep staging framework [1] but the former uses LSTM and the latter uses Transformer for sequential modelling purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The obtained overall staging accuracy in Table I indeed attest the negligible possible impact of a long sequence length in case of SeqSleepNet, marginally improving the accuracy by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3% when the sequence length is 5 and 10 times longer than the typical value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', 20 epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Even worse, an adverse effect is observed in the case of SleepTransformer whose accuracy noticeably drops > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0% when the sequence length increases from 21 to 100 and 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This is likely because the Transformer-based architecture usually needs a lot more data to train, and thus, is more prone to overfitting when the receptive field gets larger with the increased sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' These results once again consolidate what were observed in previous works [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' However, in the particular context of sleep data, such a negative effect of a long sequence length to the automatic staging performance appears to be implausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Typically, a person goes through four to six sleep cycles per night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' One 2 complete sleep cycle takes roughly 90 to 110 minutes (equiv- alent to 180 to 220 epochs of 30 seconds each), transitioning through Awake→N1→N2→N3→REM sequentially and the time lasts in each stage is well-studied [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Furthermore, there are temporal dynamic processes that underpin the sleep cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Several phenomena can be observed from sleep’s physiological signals, reflecting these dynamic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' N2 sleep, for example, at the beginning of a cycle is not the same as at the end of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' When looking at the cyclic alternating patterns (CAP), for instance, there are more A phases before REM sleep onset than after a REM period [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Also, the cycles themselves differ with more REM in the morning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' All in all, sleep cycles exhibit temporal sleep-transitioning structures specific to the sleep process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The implication of this is that the temporal interdependence in sleep data could be as long as a whole cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' For instance, intuitively, knowing that an epoch is N1 should increase the likelihood of an epoch 25 minutes after it to be N2, given the fact that N1 lasts between 1-5 minutes and N2 lasts ≥ 25 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We argue that the negative effects of long sequences ob- served so far in the literature are due to model deficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' That is, considering a long sequence as a “flat” sequence per se, the typical sequential modelling method in existing models [1] is incapable of handling long sequences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', up to one whole sleep cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We hypothesize that an appropriate method for equipping a sleep staging system with long sequential mod- elling capability would benefit its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The present work introduces a new method that is able to model long sequences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' one sleep cycle or more) to achieve new state- of-the-art performance, even with a single-EEG input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' MATERIALS We employed four databases in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' On the one hand, the SHHS and SleepEDF databases are based on conventional PSG setup from which scalp EEG derivations, C4-A1 for SHHS and Fpz-Cz for SleepEDF, were derived (see Figure 1, top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' On the other hand, the ear-EEG and cEEGrid databases are based on in-ear EEG setup [15], [16] (see Figure 1, middle row) and around-the-ear EEG setup [17], [18] (see Figure 1, bottom row), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' SHHS: This large database was gathered from multiple centers as part of the clinical trial “Sleep Heart Health Study (SHHS)”, ClinicalTrials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='gov number NCT00005275 to study the effect of sleep-disordered breathing on cardiovascular diseases [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' It consists of two sets of PSG record- ings, namely Visit 1 and Visit 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Here, we employed Visit 1 consisting of 5,791 PSG recordings from 5,791 subjects, aged 39-90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Following [19], we excluded those recordings without the presence of all five sleep stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' As a result, 5,463 PSG recordings were retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The recordings were manually scored following the R&K guidelines [20] where each 30- second epoch was labelled as one of eight categories {W, N1, N2, N3, N4, REM, MOVEMENT, UNKNOWN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In our experiments, N3 and N4 stages were merged and considered as N3 collectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' MOVEMENT and UNKNOWN epochs were discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We adopted C4-A1 EEG in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' SleepEDF: This is the Sleep Cassette subset of the Sleep- EDF Expanded dataset [21], [22] (version 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' It con- Fpz Cz C4 A1 Figure 1: The EEG setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' sists of 20 subjects (10 males and 10 females) aged 25-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Each subject had two consecutive day-night PSG recordings recorded, except for the subject 13 whose one night’s data was lost due to device failure, making a total of 39 PSG recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This database was manually labelled according to the R&K guideline [20] where each 30-second epoch was labelled as one of eight categories {W, N1, N2, N3, N4, REM, MOVEMENT, UNKNOWN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Similar to SHHS, N3 and N4 stages were merged and considered as N3 collectively while MOVEMENT and UNKNOWN categories were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We adopted the Fpz-Cz EEG channel in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Adhering to the common setting in literature, a recording was trimmed starting from 30 minutes before to 30 minutes after its in-bed part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ear-EEG: This database constitutes ear-EEG recordings of 20 subjects recorded using the same ear-EEG equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Each subject had four nights of recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Three recordings were excluded after artefact rejection [16] and two other recordings were excluded as their remaining lengths was less than 200 epochs after artefact rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This resulted in 75 recordings in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The labels of the data were obtained via manual scoring of the PSG recordings which were recorded concurrently to the ear-EEG as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Manual scoring was done by two independent and experienced sleep technicians according to the AASM guidelines [23] where each 30-second epoch is labelled as one in five categories {Wake, N1, N2, N3, REM}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' As in [16], we used the labels from the scorer 1 as the ground truth here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We adopted the bilateral ear-EEG derivation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' the average of the left ear electrodes relative to the average of the right ear electrodes (see Figure 1, middle row, right picture)) in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' More details about the recording setup and data preprocessing can be found in [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' cEEGrid: This database [24], [25] was recorded at the University of Surrey using a lightweight flex–printed electrode strip, namely the cEEGrid array [17], [18], fitted behind the ear, as illustrated in Figure 1 (bottom row, left and middle DB01 02 INIONNASION pl F p2 F7 F8 F3 Fz F4 A1 A2 T3 P3 Pz P4 T513 pictures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' 20 subjects, aged 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 years, took part in the data recording and one overnight cEEGrid recording was recorded for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Two recordings were lost due to human error and six recordings were excluded because of excessive artefacts and data missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' 12 remaining recordings were retained and used in the experiments as in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The labels of the data were obtained via manual scoring of the PSG recordings which were recorded concurrently as reference for the cEEGrid data [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The FB(R) (“front versus back”) derivation for the right ear (see Figure 1, bottom row, right picture) which was the best derivation [24], was adopted for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' More details about the recording setup and data preprocessing can be found in [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' LONG SEQUENCE MODELLING WITH L-SEQSLEEPNET Given a training set {Sn}N n=1 of size N where Sn = � (S(n) 1 , y(n) 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , (S(n) L , y(n) L ) � is the n-th sequence con- sisting of L consecutive sleep epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' S(n) ℓ and y(n) ℓ ∈ {0, 1}C represent the ℓ-th 30-second sleep epoch and its one-hot encoding label in the n-th sequence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Here, C = 5 as we are dealing with 5-stage sleep staging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Similar to a sequence-to-sequence sleep staging model [1], given a sequence (S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , SL) as input, L-SeqSleepNet aims to classify all the epochs in the input sequence at once and produce the sequence of probability output vectors (ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˆyL), where ˆy(n) ℓ ∈ [0, 1]C, 1 ≤ ℓ ≤ L, is for the ℓ-th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' However, different from existing sequence-to-sequence sleep staging models that consider short sequences (L between 20-30 epochs or 10-15 minutes equivalently), we are interested in long sequences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' L=200 or 100 minutes equivalently) so that a sequence roughly covers an entire sleep cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The architecture of L-SeqSleepNet is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' It receives the time-frequency input and has the epoch encod- ing part inherited from SeqSleepNet [4] while the sequence encoding part is devised to handle long sequences efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' For completeness, we describe all of these components in order in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Input The EEG signal of a 30-second epoch is converted into a log-magnitude time-frequency image S with T =29 time steps and F = 129 frequency bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' To that end, short-time Fourier transform (STFT) is applied to the signal with a window length of 2 seconds and 50% overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In addition, Hamming window and 256-point fast Fourier transform (FFT) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The obtained amplitude spectrum is then log-transformed to result in the image S ∈ RT ×F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Epoch encoding The role of the epoch encoding component is to learn the feature map, F(S) : S �→ x, in order to transform an input epoch S into a high-level feature vector x for representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This is realized by a subnetwork which is shared across all epochs in an input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The subnetwork is composed of (i) a learnable filterbank layer, (ii) a bidirectional Long Short-Term Memory (BLSTM) [2], and (iii) a gated attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The learnable filterbank layer [27] consists of M filters (M < F), being tasked to smooth and reduce the frequency dimension from F to M bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The resulting image ˜S of size T × M is then treated as a sequence of T vectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', T image columns), (˜s1, ˜s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˜sT ), where ˜st ∈ RM, 1 ≤ t ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In order to capture the sequential information at the epoch level, this sequence is encoded by the BLSTM with recurrent batch normalization [28], into a sequence of vectors (˜x1, ˜x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˜xT ): (˜x1, ˜x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˜xT ) = BLST Me(˜s1, ˜s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˜sT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' (1) Here, ˜xt ∈ RHe with He 2 is the size of the hidden states in BLST Me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We use the subscript e to indicate modelling at the epoch level and distinguish it from other BLSTMs in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Afterwards, the gated attention layer [29] is learned to produce attention weights (w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , wT ) which are used to combine the feature vectors (˜x1, ˜x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˜xT ) to derive the embedding vector x ∈ RHe representing the input epoch S: x = T � t=1 wt˜xt, (2) where wt = exp(uT t a) �T i=1 exp(uT i a) , (3) ut = tanh(Wa˜xt + ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' (4) In above equations, Wa ∈ RA×He and ba ∈ RA are trainable weight matrix and bias vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' a ∈ RA is the trainable context vector and A is the so-called attention size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' After the epoch encoding subnetwork described in Section III-B, the input sequence (S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , SL) has been trans- formed into the sequence of embeddings (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , xL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Long sequence modelling Encoding the sequential information in the sequence of epoch-wise feature vectors (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , xL) has proved to be the key behind the success of existing sequence-to-sequence sleep staging models [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This has been commonly accom- plished by a subnetwork with sequential modelling capacity, such as RNN [4]–[6], [30], [31] or Transformer [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' However, we have shown earlier that this approach is inefficient to handle long sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Central to L-SeqSleepNet’s architecture is the subnetwork (the big blue box in Figure 2) that is capable of long sequence modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In intuition, the processing of this component is composed of four steps indicated by the circled numbers in the figure: folding, intra-subsequence sequential modelling, inter-subsequence sequential modelling, and unfolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We firstly fold the long sequence of length L into B non- overlapping subsequences of length K, where L = B × K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Sequential modelling is then performed within each of the subsequences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' intra-subsequence sequential modelling), followed by sequential modelling across the subsequences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' inter-subsequence sequential modelling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Eventually, the subquences are unfolded to resume the long sequence of original length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' S 1 S 2 SL ˜s11 ˜s12 ˜s1T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ˜s21 ˜s22 ˜s2T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ˜sL1 ˜sL2 ˜sLT R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' w12 w1T w21 w22 w2T wL1 wL2 wLT w11 x1 x2 xL o1 o2 oL ˆy1 ˆy2 ˆyL fc fc fc fc fc fc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' B K 1 1 x1 x2 xL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' xL-1 o1 o2 oL oL-1 B K 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' 1 2 3 4 R R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' R R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' R R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' R R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' R R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' R R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' R R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' R R R R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ˜x11 ˜x12 ˜x1T ˜x21 ˜x22 ˜x2T ˜xL1 ˜xL2 ˜xLT Figure 2: The architecture of L-SeqSleepNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Formally, assume that we have folded the sequence (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , xL) into B non-overlapping subsequences of size K: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed x(1) 1 x(1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' x(1) K x(2) 1 x(2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' x(2) K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' x(B) 1 x(B) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' x(B) K \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' (5) Here, we use the subscript k, 1 ≤ k ≤ K, to indicate the index of an element inside a subsequence and the superscript b, 1 ≤ b ≤ B, to indicate the index of a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' After folding, the ℓ-th element in the original sequence will become the k-th element in the b-th subsequence, where b = �ℓ − 1 K � + 1, (6) k = [(ℓ − 1) mod K] + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' (7) Intra-subsequence sequential modelling (along horizontal direction as illustrated in Figure 2) is carried out on a b-th subsequence (x(b) 1 , x(b) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , x(b) K ) using a BLSTM with recur- rent batch normalization, transforming it into a subsequence of output vectors (˜o(b) 1 , ˜o(b) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˜o(b) K ): (˜o(b) 1 , ˜o(b) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˜o(b) K ) = BLST Mss(x(b) 1 , x(b) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , x(b) K ), (8) where ˜o(b) k ∈ RHss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Hss 2 is the size of the hidden states in BLST Mss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The subscript ss is used to indicate the modelling 5 at the subsequence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The output vectors ˜o(b) k are then linear transformed via a fully connected (fc) layer, followed by layer normalization (LN) [32] and a residual connection: ¯o(b) k = ˜o(b) k + LN(Wss˜o(b) k + bss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' (9) Here, Wss ∈ RHss×Hss and bss ∈ RHss denote the trainable weight matrix and bias vector of the fc layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' As a result, we obtain the following B output subsequences: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed ¯o(1) 1 ¯o(1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ¯o(1) K ¯o(2) 1 ¯o(2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ¯o(2) K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ¯o(B) 1 ¯o(B) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ¯o(B) K \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' (10) Up to this point, each output vector ¯o(b) k ∈ RHss in a b-th subsequence is expected to contain the information of the entire subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Inter-subsequence sequential modelling (along vertical di- rection as illustrated in Figure 2) is then conducted at each index k across all B subsequences using another BLSTM with recurrent batch normalization: (ˆo(1) k , ˆo(2) k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˆo(B) k ) = BLST Mws(¯o(1) k , ¯o(2) k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ¯o(B) k ), (11) where ˆo(b) k ∈ RHws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Similar to the intra-subsequence sequen- tial modelling step, linear transformation via a fc layer, layer normalization, and a residual connection are then applied: o(b) k = ˆo(b) k + LN(Wwsˆo(b) k + bws), (12) resulting in the following B output subsequences: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed o(1) 1 o(1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' o(1) K o(2) 1 o(2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' o(2) K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' o(B) 1 o(B) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' o(B) K \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (13) where o(b) k ∈ RHws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In (12), Wws ∈ RHws×Hws and bws ∈ RHws denote the trainable weight matrix and bias vector of the fc layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Hws 2 is the size of the hidden states in BLST Mws and we use the subscript ws to indicate the modelling at the whole sequence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Given that an output vector ¯o(b) k contains the information of the entire b-th subse- quence after the intra-subsequence sequential modelling step, an output vector o(b) k is expected to contain the information of all B subsequences after the inter-subsequence sequential modelling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In other words, o(b) k contains the information of the whole original long sequence of length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Eventually, the output subsequences in (13) are un- folded, resulting in a single long sequence of L elements (o1, o2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , oL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' After unfolding, the k-th element in the b-th subsequence, o(b) k , in (13) will become the ℓ-th element oℓ in the final output sequence, where ℓ = (b − 1) × K + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' (14) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Classification For classification, the output vectors in the output sequence (o1, o2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , oL) are passed through two fc layers, each with Nfc units and Rectified Linear Unit (ReLU) activation, before being presented to an output layer with softmax activation to produce the network predictions (ˆy1, ˆy2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' , ˆyL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Note that these layers are shared across the time indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The network is trained to minimize the cross-entropy loss averaged over the sequence length and the training data: L(θ) = − 1 N · L N � n=1 L � ℓ=1 y(n) ℓ log ˆy(n) ℓ + λ||θ||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' (15) In (15), θ denotes the network parameters and λ is the hyper- parameter of the ℓ2-norm regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Experimental setup The experimental setup using the four databases described in Section II is summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Conforming to majority of previous works in literature, we conducted leave- one-subject-out cross validation (LOSO CV) on SleepEDF, ear-EEG, and cEEGrid databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' For SHHS, we randomly split the subjects into 70% for training and 30% for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In each experiment, a number of subjects, specified in Table II, were held out from the training set for validation purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In particular, due to the small number of recordings in the SleepEDF and cEEGrid databases, we repeated these experi- ments 5 times and report the average performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Parameters We experimented with the sequence length L = 200 epochs (equivalent to 100 minutes of sleep data to roughly cover a whole sleep cycle), the number of subsequences B = 10, and the subsequence length K = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We also experimented with other values for L, B, and K, and will discuss their influence in Section IV-D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Regarding the network architecture, we set the number of filters M = 32, the attention size A = 64, the size of BLSTM’s hidden states He 2 = Hss 2 = Hws 2 = 64, and the size of the fc layers Nfc = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The hyper-parameter λ in (15) was fixed to 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In addition, a dropout rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 was applied to the LSTM cells and the fc layers during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The network was trained using Adam optimizer [33] with a learning rate of 10−4, β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='999, and ǫ = 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' A minibatch size of 8 was used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The model was validated on the validation set every 100 training steps for the experiments on SHHS, SleepEDF, and ear-EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' For the smallest database cEEGrid, the validation was done every 25 Table II: Summary of experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Database Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' of subjects Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' of recordings Experimental setup Held-out valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' set Repetition SHHS 5,463 5,463 train/test: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 100 1 SleepEDF 20 39 LOSO CV 4 5 ear-EEG 20 75 LOSO CV 4 1 cEEGrid 12 12 LOSO CV 2 5 6 training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' For the largest database SHHS, the model was trained for fixed 5000 validation steps without early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Other than that, the model was trained for 10 training epochs and stopped early after 50 validation steps (for SleepEDF and ear-EEG) and 25 validation steps (for cEEGrid) without im- provement on the validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The model performing best on the validation set was then retained to be evaluated on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The baseline We used SeqSleepNet presented in [4] as the main base- line in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This network shares a similar in- put format and epoch encoding component as the proposed L-SeqSleepNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' However, different from L-SeqSleepNet, it follows the typical “flat” sequential modelling approach as many other works in literature, making it a natural baseline to investigate the effects of the long sequential modelling approach presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We compared L-SeqSleepNet and the SeqSleepNet baseline under two initialization schemes: (1) random initialization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', training from scratch) and (2) initialization with a pretrained network (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' finetuning for transfer learning [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' For the former, the networks were trained from scratch as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' For the latter, the networks trained on SHHS were used as the pretrained models and finetuned on the SleepEDF, ear-EEG, and cEEGrid databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In addition, in order to assess L-SeqSleepNet in terms of sleep staging performance, we also compare it to other methods in literature reporting results on the experimental databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Experimental results 1) Overall sleep staging performance: A comprehensive performance comparison between L-SeqSleepNet, the Se- qSleepNet baseline, and prior works are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' On the one hand, it can be seen that L-SeqSleepNet consistently outperforms the SeqSleepNet baseline over all the experimental databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' With the random initialization scheme, L-SeqSleepNet leads to overall accuracy gains of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7%, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9% on SHHS, SleepEDF, ear-EEG, and cEEGrid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In case of the pretraining-based initialization, the gains on SleepEDF, ear-EEG, and cEEGrid are even higher, reaching 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9%, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The wider gains obtained with respect to the pretraining-based initialization shed some light on what is being transferred from the source domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' SHHS) to the target domains (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', SleepEDF, ear-EEG, and cEEGrid) via L-SeqSleepNet in these transfer learning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' More specifically, in addition to the usual reuse of good feature representations [49], it is likely that the diverse sleep cycle structure from the large cohort in the source domain also contributes to the transferred knowledge and gives rise to L-SeqSleepNet’s higher performance gains compared to the SeqSleepNet baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' On the other hand, L-SeqSleepNet also results in better performance than the current state-of-the-arts (where the direct comparison is compatible) over all the databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' On the large database SHHS, L-SeqSleepNet achieves an overall accuracy of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8% higher than SleepTransformer [7] and XSleepNet [9], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This is particularly interesting given that the SeqSleepNet-type architecture of L-SeqSleepNet essentially constitutes only one half of the XSleepNet’s ar- chitecture [9] and that L-SeqSleepNet has around 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3×105 parameters in total, roughly 9 times smaller than XSleepNet which has 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 × 106 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' On the smaller databases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', SleepEDF, ear-EEG, and cEEGrid), a large margin of performance is consistently seen between L-SeqSleepNet and other counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' 2) The effects of whole-cycle sequence modelling: Using the pretraining-based initialization scheme, we inspected the effects of L-SeqSleepNet’s whole-cycle sequence modelling to the model’s errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' From the confusion matrices and the number of errors per sleep stage in Figure 3, it becomes clear that across all the databases, compared to the SeqSleepNet baseline, L-SeqSleepNet lowers the errors in all other sleep stages, most noticeably in N3 and REM, at the small expense of errors in N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This implies that taking into account the structure of sleep cycles helps to correct modelling errors that are impossible to fix under the existing state-of-the-art (short) context modelling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Figure 4 visually represents the distribution of individual errors produced by L-SeqSleepNet and the SeqSleepNet base- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' It was found in [50] that even though the performance of automatic sleep staging systems has been deemed sufficient for clinical use these systems may generally struggle with particular recordings, leading to exceptionally high individual errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The distribution of individual errors from both L- SeqSleepNet and the baseline in the figure indeed reflects this finding, particularly on SHHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' However, L-SeqSleepNet can reduce these exceptionally high individual errors across all the databases, and more strikingly on ear-EEG and cEEGrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This is particularly important from an application point of view as bringing down these high individual errors would make automatic sleep staging systems more acceptable in clinical environments as well as require less human intervention for manual editing or rescoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' 3) The influence of the sequence length: In this section, we examined the influence of the sequence length to L- SeqSleepNet and the SeqSleepNet baseline from two perspec- tives: staging performance and computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Using the large database SHHS in this examination, Table IV summa- rizes the overall sleep staging performance and computational time of the two models with different sequence lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Firstly, as already mentioned in Section I, SeqSleepNet with the “flat” sequential modelling approach is inefficient in handling long sequences, resulting in marginal gains on overall accuracy, for instance, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3% when the sequence length increases to 200 from 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In contrast, at the sequence length of 200 epochs, the improvement on overall accuracy achieved by L-SeqSleepNet over SeqSleepNet with L = 20 reaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2%, highlighting the effectiveness of the proposed long sequential modelling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Secondly, concerning L-SeqSleepNet itself, halving the se- quence length to 100 epochs (50 minutes, roughly a half sleep cycle) results in a drop of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3% on overall accuracy while doubling it to 400 epochs (200 minutes, roughly two sleep cycles) causes negligible consequence to the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' 7 Table III: Performance obtained by L-SeqSleepNet, the SeqSleepNet baseline in comparison with previous works on the experimental databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Note that some results reported in previous works, marked by the ‡ superscript, are not compatible for a direct comparison here due to the discrepancies in data split, the number of channel used, the number of subject used, modelling tasks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The ∗ superscript indicates the pretraining-based initialization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Database System Overall performance Class-wise MF1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' κ MF1 Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Wake N1 N1 N3 REM SHHS L-SeqSleepNet 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='838 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 SeqSleepNet [4] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='820 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 SleepTransformer [7] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='828 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 XSleepNet1 [9] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='826 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 XSleepNet2 [9] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='826 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 U-Sleep‡ [34] − − 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 − − 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 Olesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='‡ [31] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='816 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 CNN [19] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='810 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 − 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 − − − − − FCNN+RNN [9] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='813 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 IITNet [6] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='810 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 − − − − − − − AttnSleep‡ [35] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='780 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 − − 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 SleepEDF L-SeqSleepNet∗ 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='845 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='001 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 L-SeqSleepNet 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='813 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='003 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 SeqSleepNet∗ [4] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='830 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='002 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 SeqSleepNet [4] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='803 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='004 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 SalientSleepNet‡ [36] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 − 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 − − 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 TransSleep [37] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 − 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 − − 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 XSleepNet2 [9] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='813 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 XSleepNet1 [9] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='810 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 SimpleSleepNet‡ [38] − − 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 − − − − − − − MNN‡ [39] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 − 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 − − 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 Khalili & Asl [40] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='800 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 − − 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 TinySleepNet [41] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='800 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 − − 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 RobustSleepNet‡ [30] − − 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 − − − − − − − U-Sleep [34] − − 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 − − 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 SleepFCN [42] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='780 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 − − 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 MRASleepNet [43] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='786 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 − − − − − − − ResNetMHA [44] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 − 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 − − 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 AttnSleep [35] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 IITNet [6] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='780 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 − − − − − − − DeepSleepNet [5] 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='760 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 − − 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 FCNN+RNN [9] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='754 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 76.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 SeqSleepNet* [4] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='801 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 SeqSleepNet [4] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='759 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 77.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='780 − − − − − − − − Random Forest‡ [15] − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='730 − − − − − − − − cEEGrid L-SeqSleepNet* 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='703 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='008 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 L-SeqSleepNet 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='607 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='008 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 SeqSleepNet* [4] 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='647 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='006 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 SeqSleepNet [4] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='578 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='024 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 ADA pers‡ [47] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='618 − − − − − − − − Feat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' matching‡ [48] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='605 − − − − − − − − Random Forest‡ [24] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='580 − − − − − − − − DeepSleepNet* [26] 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='391 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8 − − 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 DeepSleepNet [26] 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='195 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 − − 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 This suggests that the sequential information in one sleep cycle is probably all we need for sleep staging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Interestingly, the manner a long sequence is folded into subsequences seems to matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' At L = 200, folding with B = 20 and K = 10 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', 20 subsequences, each of length 10 epochs) leads to better performance than that with B = 10 and K = 20 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=', 10 subsequences, each of length 20 epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' It is likely the distance between two adjacent elements in the sequence during inter-subsequence sequential modelling causes the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In effect, this distance is 10 epochs in the former while it is 20 epochs in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The double distance in the latter could loosen the temporal dependency between the elements in the sequence, worsening the performance consequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Thirdly, given that the SeqSleepNet baseline’s training time grows linearly as the sequence length increases, L- SeqSleepNet’s training time only grows sub-linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' For ex- ample, SeqSleepNet at L = 200 requires 1,465 seconds for 1000 training steps, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 times longer than itself at L = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' L-SeqSleepNet at L = 200, on the other hand, merely needs around 450 seconds, just 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 times slower than SeqSleepNet at L = 20 and around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='3 times faster than SeqSleepNet at L = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' The reason is when a sequence of length L is folded into B subsequences, each of length K, the number of time steps engaged in sequential modelling is in fact reduced to B + K which is much smaller than L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5% 79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4% 73 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='8% 333 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2% 26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5% 930 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6% Wake N1 N2 N3 REM Output Wake N1 N2 N3 REM Ground-truth 0 200 400 600 800 1000 1200 Wake N1 N2 N3 REM # errors Wake N1 N2 N3 REM # errors L-SeqSleepNet SeqSleepNet L-SeqSleepNet’s confusion matrix SeqSleepNet’s confusion matrix Error comparison [ ear-EEG ] [ cEEGrid ] L-SeqSleepNet’s confusion matrix SeqSleepNet’s confusion matrix Error comparison L-SeqSleepNet SeqSleepNet Figure 3: Comparison between L-SeqSleepNet and SeqSleepNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' DISCUSSION On the ear-EEG database, where manual scoring from two independent sleep technicians are available, it is peculiar to compare the achieved performance to what was presented in Mikkelsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' In particular, we see that the kappa value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='829 obtained by L-SeqSleepNet is higher than the inter-scorer agreement, despite the fact that L-SeqSleepNet only used the bilateral ear-EEG derivation, and thus, did not have access to any of the same electrode derivations as were used in the manual scoring of this database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This indicates that long time scale sleep information can compensate for a decrease in “field of view” on the sensor level, which is remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' This observation also suggests that consolidating intrinsic information from a long context, like a whole sleep cycle or more, is something an automated scorer can excel at, and may do better than a human scorer who can only consciously attend to a relatively short context (a few minutes [23]) during manual scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' 9 20 30 40 50 60 70 80 90 100 60 70 80 90 100 65 70 75 80 85 90 100 95 30 40 50 60 70 80 90 100 Accurarcy (%) [ SHHS ] [ SleepEDF ] [ ear-EEG ] [ cEEGrid ] L-SeqSleepNet SeqSleepNet Figure 4: Distribution of individual accuracies produced by L-SeqSleepNet and the SeqSleepNet baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Table IV: The overall performance and the training time produced by L-SeqSleepNet and the SeqSleepNet baseline with different sequence lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Note that the training time was measured for 1000 training steps on an NVIDIA Tesla V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' System L (B × K) Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' k MF1 Training time (s) L-SeqSleepNet 100 (10×10) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='833 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='7 354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 L-SeqSleepNet 200 (10×20) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='835 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 L-SeqSleepNet 200 (20×10) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='837 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 L-SeqSleepNet 400 (20×20) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='838 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 SeqSleepNet 20 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='820 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='2 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 SeqSleepNet 100 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='823 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='1 842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='0 SeqSleepNet 200 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='824 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='4 1465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='6 While the work here focuses on single-EEG input and uses RNN as the backbone for sequential modelling, the presented long sequential modelling method can be considered as a generic method and can be used in replacement for the “flat” sequential modelling method in the current state-of- the-art sequence-to-sequence sleep staging framework [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' It can be readily integrated into existing works relying on this framework to investigate the effects of whole-cycle sequential modelling in multimodal fusion [30] and multi-view learning [9] or to speed up the computation with recurrent-free archi- tectures, such as Transformer [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' CONCLUSIONS We presented in this work a novel method for modelling whole-cycle long sequences for automatic sleep staging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' A long sequence was first folded into multiple subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Intra-subsequence and inter-subsequence sequential modelling were then performed before the subsequences were unfolded to resume the size of the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We demonstrated that the proposed approach overcomes the limitations of the exist- ing sequential modelling approach in handling long sequences and that taking the structural information of sleep cycles into account consistently improved the sleep staging performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' L-SeqSleepNet, the network with long sequential modelling capacity we introduced, outperformed not only the baseline but also the existing state-of-the-art methods across four distinct databases with different EEG setups, including scalp EEG, in-ear EEG, and around-the-ear EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' We also empirically showed that incorporating the logic of stage transition in sleep cycles helped to reduce staging errors at epoch level as well as exceptionally high individual errors at recording level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Furthermore, the performance benefits came at just a little cost of sub-linear growth in computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' ACKNOWLEDGMENT This research received funding from the Flemish Govern- ment (AI Research Program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Maarten De Vos is affiliated to Leuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content='AI - KU Leuven institute for AI, B-3000, Leuven, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Phan is supported by a Turing Fellowship under the EPSRC grant EP/N510129/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' REFERENCES [1] H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Neyshabur, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Sedghi, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Zhang, “What is being transferred in transfer learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=',” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' NeurIPS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Baumert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Hartmann, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} +page_content=' Phan, “Automatic sleep staging for the young and the old – evaluating age underrepresentation in deep learning,” Sleep Medicine, 2023, (under revision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf'} diff --git a/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf b/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..71d43447e87a769724c118650b03bda3ba45a9a9 Binary files /dev/null and b/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf differ diff --git a/jNFPT4oBgHgl3EQf0zUa/content/tmp_files/2301.13180v1.pdf.txt b/jNFPT4oBgHgl3EQf0zUa/content/tmp_files/2301.13180v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..736571b7d93595ec6765d7237e2e1826aa4db091 --- /dev/null +++ b/jNFPT4oBgHgl3EQf0zUa/content/tmp_files/2301.13180v1.pdf.txt @@ -0,0 +1,387 @@ +arXiv:2301.13180v1 [gr-qc] 30 Jan 2023 +Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 +1 +Massive Gravity and Lorentz Symmetry +R. Potting1,2 +1Departamento de F´ısica, Faculdade de Ciˆencias e Tecnologia, +Universidade do Algarve, 8005-139 Faro, Portugal +2Centro de Astrof´ısica e Gravita¸c˜ao, Instituto Superior T´ecnico, +Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisbon, Portugal +We consider Lorentz-symmetry properties of the ghost-free massive gravity the- +ory proposed by de Rham, Gabadadze, and Tolley. In particular, we present +potentially observable effects in gravitational-wave propagation and in New- +ton’s law, including Lorentz-violating signals. +1. The de Rham–Tolley–Gabadadze action +One of the motivations for considering massive gravity is the possibility +that modifications of General Relativity over large distances may yield a +solution to the cosmological constant problem. A ghost-free theory of non- +interacting gravitons was constructed in 1939 by Fierz and Pauli.1 However, +attempts to generalize it to the nonlinear level failed during decades, with +the work of Boulware and Deser showing that generically such theories will +suffer from ghost instabilities.2 Recently, however, de Rham, Gabadadze, +and Tolley (dRGT) showed that there exists a non-linear extension of Fierz– +Pauli massive gravity that does not suffer from ghosts.3 It can be shown +that their model can be maximally extended to the action:4 +S = 1 +2κ +� +d4x√−g +� +R − 2m2 +4 +� +n=0 +βnen(X) +� +, +(1) +where R is the Ricci scalar, βi are free parameters and κ = 8πG. The +4 × 4 matrix Xµν equals ( +� +g−1f)µν, where, in addition to the usual phys- +ical metric gµν, one assumes a given, nondynamical “fiducial” background +metric fµν. The invariant polynomials en(X) are defined through the re- +lation det(1 + λX) = �4 +n=0 λnen(X), yielding e0(X) = 1, e1(X) = [X], +e2(X) = 1 +2([X]2 − [X2]), . . . . + +Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 +2 +By expressing the metric in ADM form,5 using as the dynamical vari- +ables the spatial metric together with lapse and shift variables it can be +shown that the equations of motion arising from the action (1) yield five +local degrees of freedom, corresponding to the helicity states of a massive +spin-2 particle. +Crucial in the counting is the presence of the so-called +Hamiltonian constraint. +2. Spacetime symmetries +In this talk I will report on recent work with Alan Kosteleck´y6 in which we +studied the role of Lorentz symmetry in dRGT massive gravity. Defining the +vierbein ea +µ as usual through gµν = ηabea +µeb +ν, one can identify local Lorentz +transformations as well as diffeomorphisms. As was explained in the talk +by Alan Kosteleck´y at this conference, suitable local Lorentz transforma- +tions and diffeomorphisms can be combined to yield the so-called manifold +Lorentz transformations, defined by +xµ → (Λ−1)µνxν, +gµν → (Λ−1)ρµ(Λ−1)σνgρσ, +eµa → (Λ−1)ρµΛabeρb, +fµν → fµν, +(2) +for spacetime-independent Lorentz transformations Λ. These are the ana- +logues in approximately Minkowski spacetime of global Lorentz transfor- +mations in Minkowski spacetime. +The dRGT action is invariant under +manifold Lorentz transformations if fµν ∝ ηµν, but is otherwise Lorentz +violating7 due to the presence of the background fµν. +In Ref. [6] a careful study was done of the static solutions of the dRGT +potential, for flat fiducial metrics. It was shown that the four-parameter +potential has a highly nontrivial structure of extrema and saddle points, +depending on the values of the parameters. +Stability of these solutions +was investigated by using the technique of bordered Hessians. The surface +generated by the Hamiltonian constraint and the positions of the solutions +on its connected sheets were used to establish global and absolute stability +properties. We concluded that extrema of the potential are invariant un- +der manifold Lorentz transformations, while the saddle point solutions are +Lorentz violating, with maximally four broken Lorentz generators. +3. Linearized massive gravity +The action (1) yields the equations of motion +Gµν + m2 +2 +3 +� +n=0 +(−1)nβn +� +gµαY α +(n)ν + gναY α +(n)µ +� += κTµν, +(3) + +Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 +3 +where Y(n)(X) = �n +k=0(−1)kXn−kek(X). +Writing gµν = ηµν + hµν and +fµν = ηµν + δfµν it follows that, to first order in hµν and δfµν, +X ≈ +1 + 1 +2η−1δf − 1 +2η−1h + 1 +8η−1δf η−1h − 3 +8η−1h η−1δf. +(4) +The equations of motion (3) become +GL +µν + m2 +2 +3 +� +n=0 +βn +� +2 +�3 +n +�� +ηµν + hµν +� ++ +� +2 +n − 1 +� � +hµν − δfµν − ηµν +� +h − δf +�� ++ +� +1 +2 +� +1 +n − 2 +� ++ +� +2 +n − 1 +�� � +δf +� +hµν + 1 +2 +� +1 +n − 2 +�� +h +� +δfµν +− 1 +2 +� +1 +n − 1 +�� +δf η−1h +� +ηµν − 1 +2 +� +1 +n − 2 +�� +δf +� � +h +� +ηµν +− +� +3 +4 +� +2 +n − 1 +� +− 1 +2 +� +1 +n − 1 +�� � +h η−1 δf + δf η−1 h +� +µν +� += κTµν, +(5) +where GL +µν is the linearized Einstein tensor and [X] = ηµνXµν. It is usual +to require that hµν = δfµν = Tµν = 0 satifies the equations of motion (5), +yielding the constraint β0 + 3β1 + 3β2 + β3 = 0. Moreover, we normalize +the mass m such that β1 + 2β2 + β3 = 1. +Consider now a nontrivial fiducial background metric δfµν ̸= 0, it fol- +lows that spacetime now has nonzero curvature in the absence of matter! +In order to simplify the further analysis we will assume a special constant +background energy–momentum tensor +κTµν = −m2 +2 (δfµν − ηµν[δf]) . +(6) +For this special choice, hµν = 0 solves the equations of motion. +4. Gravitational waves +In order to investigate the propagation of gravitational waves, we define +the Fourier transform of hµν through hµν(x) = (2π)−4 � +d4p e−ip·x˜hµν(p). +For δfµν = 0, ˜hµν satisfies the conditions ˜hµαpα = 0 and ˜hµµ = 0, thus +yielding 10 − 5 = 5 propagating modes. These can be identified with the +helicity eigenstates of a massive spin-two field, ˜h(n) +µν , with n = 0, ±1, ±2. +All modes satisfy the massive dispersion relation (p2 + m2)˜h(n) +µν = 0. +For nonzero δfµν the situation becomes more complicated. The equa- +tions of motion (5) can be cast in the form +(p2 + m2)˜hµν = c2m2 +2 +Sµν αβ˜hαβ +(7) + +Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 +4 +where c2 = � +n βn +�� +1 +n−1 +� +− 3 +2 +� 2 +n−1 +�� +. +The quantities Sµν αβ are tensor +coefficients depending on the momentum pµ and on δfµν. It is convenient +to expand them in a momentum-dependent orthonormal basis spanned by +pµ and three other, spacelike vectors. It is then straightforward to work out +the expressions Sµν αβ˜h(n) +αβ for any helicity mode n, which are well-defined +linear combinations of the five helicity modes. +Equation (7) can be solved by constructing the eigenstates of Sµν αβ. For +general δf, we find a “pentarefringence” effect: each of these eigenstates +solves Eq. (7) with a (slightly) different dispersion relation. These pentare- +fringence effects are momentum (and direction) dependent. The modes can +be sub- or superluminal, a result typical of Lorentz-violating theories. For +details, see Ref. [6]. +5. Corrections to Newton’s law +Next we study the effects of the extra terms in the equation of motion (5) on +Newton’s law. Writing the momentum-space linearized modified Einstein +equation as ˜Oµν αβ˜hαβ = 0, the corresponding propagator is defined to +satisfy ˜Dµνστ ˜Oστ αβ = δα +(µδβ +ν). At first order in δf it has the form +˜Dµν αβ = +1 +p2 + m2 +� +δα +(µδβ +ν) − 1 +3ηµνηαβ + 2 +m2 p(µp(αδβ) +ν) +− +1 +3m2 +� +pµpνηαβ + ηµνpαpβ − 2 +m2 pµpνpαpβ +�� +− +m2 +(p2 + m2)2 +� +ρ1 δf α +(µδβ +ν) + ρ2 δfµνηαβ + ρ4 pµpνδf αβ + . . . +� +, +(8) +where ρi = ρi(p2) are momentum-dependent scalars (parentheses in the +indices indicate symmetrization). For given energy–momentum tensor the +linearized solution for the metric can then be expressed as +hµν(x) = 2κ +� +d4p +(2π)4 e−ip·xDµναβ ˜Tαβ(p), +(9) +where T µν(x) = (2π)−4 � +d4p e−ip·x ˜T µν(p). +Consider now the gravitational potential energy between two stationary +point masses with energy–momentum tensors +T µν +1 (x) = M1 δµ +0 δν +0 δ3(⃗x), +T µν +2 (x) = M2 δµ +0 δν +0 δ3(⃗x − ⃗r). +(10) +At linear order in hµν the matter Lagrangian corresponding to a given +energy–momentum tensor is Lm ≈ − 1 +2hµνTµν. From this it follows that + +Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 +5 +the potential energy corresponding to the energy–momentum tensors (10) +is given by U(⃗r) = +� +d3x hµν +1 (⃗x)T2,µν(⃗x) = M2 h1,00(⃗r) . Using Eqs. (8) +and (9) we obtain +U(⃗r) = −GM1M2e−mr +9r +� +24 − δf00 +� +(4c2 + 9)mr + 8c2 + 8 +� +− δfi +i +� +(4c2 − 9)mr + 2c2 + 4 + 2 +mr + +4 +m2r2 +� +− δfijxixj +r2 +� +2c2mr + 2c2 − 4 − 6 +mr − +12 +m2r2 +�� ++ 2κM1M2 +9m2 +δ3(⃗r) +� +δf00 − 2 +3δfi +i� +. +(11) +The exponential suppression factor e−mr is as expected due to the massive +graviton. The term independent of δfµν is scaled by 4/3 relative to the +gravitational potential in General Relativity, in concordance with the van +Dam–Veltman–Zakharov discontinuity. Moreover, note that U(⃗r) acquires +terms that generally violate rotational invariance. +Acknowledgments +This work was supported in part by the Portuguese Funda¸c˜ao para +a Ciˆencia e a Tecnologia under grants SFRH/BSAB/150324/2019 and +UID/FIS/00099/2019, and by the Indiana University Center for Spacetime +Symmetries. +References +1. M. Fierz, Helv. Phys. Acta 12, 3 (1939); M. Fierz and W. Pauli, Proc. Roy. +Soc. (London) A 173, 211 (1939). +2. D.G. Boulware and S. Deser, Phys. Rev. D 6, 3368 (1972). +3. C. de Rham, G. Gabadadze, and A.J. Tolley, Phys. Rev. Lett. 106, 231101 +(2011); Phys. Lett. B 711, 190 (2012). +4. S.F. Hassan and R.A. Rosen, JHEP 04, 123 (2012); Phys. Rev. Lett. 108, +041101 (2012); S.F. Hassan, R.A. Rosen and A. Schmidt-May, JHEP 02, 026 +(2012). +5. R. Arnowitt, S. Deser, and C. Misner, Phys. Rev. 116, 1322 (1959). +6. V.A. Kosteleck´y and R. Potting, Phys. Rev. D 104, 104046 (2021). +7. V.A. Kosteleck´y, Phys. Rev. D 69, 105009 (2004). +8. H. van Dam and M. Veltman, Nucl. Phys. 22, 397 (1970). +9. V.I. Zakharov, JETP Lett. (Sov. Phys.), 12, 312 (1970). + diff --git a/jNFPT4oBgHgl3EQf0zUa/content/tmp_files/load_file.txt b/jNFPT4oBgHgl3EQf0zUa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..530a6cf92e0d79466ccf921f708b77001ef88a1e --- /dev/null +++ b/jNFPT4oBgHgl3EQf0zUa/content/tmp_files/load_file.txt @@ -0,0 +1,153 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf,len=152 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content='13180v1 [gr-qc] 30 Jan 2023 Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 1 Massive Gravity and Lorentz Symmetry R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Potting1,2 1Departamento de F´ısica, Faculdade de Ciˆencias e Tecnologia, Universidade do Algarve, 8005-139 Faro, Portugal 2Centro de Astrof´ısica e Gravita¸c˜ao, Instituto Superior T´ecnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisbon, Portugal We consider Lorentz-symmetry properties of the ghost-free massive gravity the- ory proposed by de Rham, Gabadadze, and Tolley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' In particular, we present potentially observable effects in gravitational-wave propagation and in New- ton’s law, including Lorentz-violating signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' The de Rham–Tolley–Gabadadze action One of the motivations for considering massive gravity is the possibility that modifications of General Relativity over large distances may yield a solution to the cosmological constant problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' A ghost-free theory of non- interacting gravitons was constructed in 1939 by Fierz and Pauli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content='1 However, attempts to generalize it to the nonlinear level failed during decades, with the work of Boulware and Deser showing that generically such theories will suffer from ghost instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content='2 Recently, however, de Rham, Gabadadze, and Tolley (dRGT) showed that there exists a non-linear extension of Fierz– Pauli massive gravity that does not suffer from ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content='3 It can be shown that their model can be maximally extended to the action:4 S = 1 2κ � d4x√−g � R − 2m2 4 � n=0 βnen(X) � , (1) where R is the Ricci scalar, βi are free parameters and κ = 8πG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' The 4 × 4 matrix Xµν equals ( � g−1f)µν, where, in addition to the usual phys- ical metric gµν, one assumes a given, nondynamical “fiducial” background metric fµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' The invariant polynomials en(X) are defined through the re- lation det(1 + λX) = �4 n=0 λnen(X), yielding e0(X) = 1, e1(X) = [X], e2(X) = 1 2([X]2 − [X2]), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 2 By expressing the metric in ADM form,5 using as the dynamical vari- ables the spatial metric together with lapse and shift variables it can be shown that the equations of motion arising from the action (1) yield five local degrees of freedom, corresponding to the helicity states of a massive spin-2 particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Crucial in the counting is the presence of the so-called Hamiltonian constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Spacetime symmetries In this talk I will report on recent work with Alan Kosteleck´y6 in which we studied the role of Lorentz symmetry in dRGT massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Defining the vierbein ea µ as usual through gµν = ηabea µeb ν, one can identify local Lorentz transformations as well as diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' As was explained in the talk by Alan Kosteleck´y at this conference, suitable local Lorentz transforma- tions and diffeomorphisms can be combined to yield the so-called manifold Lorentz transformations, defined by xµ → (Λ−1)µνxν, gµν → (Λ−1)ρµ(Λ−1)σνgρσ, eµa → (Λ−1)ρµΛabeρb, fµν → fµν, (2) for spacetime-independent Lorentz transformations Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' These are the ana- logues in approximately Minkowski spacetime of global Lorentz transfor- mations in Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' The dRGT action is invariant under manifold Lorentz transformations if fµν ∝ ηµν, but is otherwise Lorentz violating7 due to the presence of the background fµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' [6] a careful study was done of the static solutions of the dRGT potential, for flat fiducial metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' It was shown that the four-parameter potential has a highly nontrivial structure of extrema and saddle points, depending on the values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Stability of these solutions was investigated by using the technique of bordered Hessians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' The surface generated by the Hamiltonian constraint and the positions of the solutions on its connected sheets were used to establish global and absolute stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' We concluded that extrema of the potential are invariant un- der manifold Lorentz transformations, while the saddle point solutions are Lorentz violating, with maximally four broken Lorentz generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Linearized massive gravity The action (1) yields the equations of motion Gµν + m2 2 3 � n=0 (−1)nβn � gµαY α (n)ν + gναY α (n)µ � = κTµν, (3) Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 3 where Y(n)(X) = �n k=0(−1)kXn−kek(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Writing gµν = ηµν + hµν and fµν = ηµν + δfµν it follows that, to first order in hµν and δfµν, X ≈ 1 + 1 2η−1δf − 1 2η−1h + 1 8η−1δf η−1h − 3 8η−1h η−1δf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' (4) The equations of motion (3) become GL µν + m2 2 3 � n=0 βn � 2 �3 n �� ηµν + hµν � + � 2 n − 1 � � hµν − δfµν − ηµν � h − δf �� + � 1 2 � 1 n − 2 � + � 2 n − 1 �� � δf � hµν + 1 2 � 1 n − 2 �� h � δfµν − 1 2 � 1 n − 1 �� δf η−1h � ηµν − 1 2 � 1 n − 2 �� δf � � h � ηµν − � 3 4 � 2 n − 1 � − 1 2 � 1 n − 1 �� � h η−1 δf + δf η−1 h � µν � = κTµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' (5) where GL µν is the linearized Einstein tensor and [X] = ηµνXµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' It is usual to require that hµν = δfµν = Tµν = 0 satifies the equations of motion (5), yielding the constraint β0 + 3β1 + 3β2 + β3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Moreover, we normalize the mass m such that β1 + 2β2 + β3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Consider now a nontrivial fiducial background metric δfµν ̸= 0, it fol- lows that spacetime now has nonzero curvature in the absence of matter!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' In order to simplify the further analysis we will assume a special constant background energy–momentum tensor κTµν = −m2 2 (δfµν − ηµν[δf]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' (6) For this special choice, hµν = 0 solves the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Gravitational waves In order to investigate the propagation of gravitational waves, we define the Fourier transform of hµν through hµν(x) = (2π)−4 � d4p e−ip·x˜hµν(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' For δfµν = 0, ˜hµν satisfies the conditions ˜hµαpα = 0 and ˜hµµ = 0, thus yielding 10 − 5 = 5 propagating modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' These can be identified with the helicity eigenstates of a massive spin-two field, ˜h(n) µν , with n = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' All modes satisfy the massive dispersion relation (p2 + m2)˜h(n) µν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' For nonzero δfµν the situation becomes more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' The equa- tions of motion (5) can be cast in the form (p2 + m2)˜hµν = c2m2 2 Sµν αβ˜hαβ (7) Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 4 where c2 = � n βn �� 1 n−1 � − 3 2 � 2 n−1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' The quantities Sµν αβ are tensor coefficients depending on the momentum pµ and on δfµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' It is convenient to expand them in a momentum-dependent orthonormal basis spanned by pµ and three other, spacelike vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' It is then straightforward to work out the expressions Sµν αβ˜h(n) αβ for any helicity mode n, which are well-defined linear combinations of the five helicity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Equation (7) can be solved by constructing the eigenstates of Sµν αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' For general δf, we find a “pentarefringence” effect: each of these eigenstates solves Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' (7) with a (slightly) different dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' These pentare- fringence effects are momentum (and direction) dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' The modes can be sub- or superluminal, a result typical of Lorentz-violating theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' For details, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Corrections to Newton’s law Next we study the effects of the extra terms in the equation of motion (5) on Newton’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Writing the momentum-space linearized modified Einstein equation as ˜Oµν αβ˜hαβ = 0, the corresponding propagator is defined to satisfy ˜Dµνστ ˜Oστ αβ = δα (µδβ ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' At first order in δf it has the form ˜Dµν αβ = 1 p2 + m2 � δα (µδβ ν) − 1 3ηµνηαβ + 2 m2 p(µp(αδβ) ν) − 1 3m2 � pµpνηαβ + ηµνpαpβ − 2 m2 pµpνpαpβ �� − m2 (p2 + m2)2 � ρ1 δf α (µδβ ν) + ρ2 δfµνηαβ + ρ4 pµpνδf αβ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' � , (8) where ρi = ρi(p2) are momentum-dependent scalars (parentheses in the indices indicate symmetrization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' For given energy–momentum tensor the linearized solution for the metric can then be expressed as hµν(x) = 2κ � d4p (2π)4 e−ip·xDµναβ ˜Tαβ(p), (9) where T µν(x) = (2π)−4 � d4p e−ip·x ˜T µν(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Consider now the gravitational potential energy between two stationary point masses with energy–momentum tensors T µν 1 (x) = M1 δµ 0 δν 0 δ3(⃗x), T µν 2 (x) = M2 δµ 0 δν 0 δ3(⃗x − ⃗r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' (10) At linear order in hµν the matter Lagrangian corresponding to a given energy–momentum tensor is Lm ≈ − 1 2hµνTµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' From this it follows that Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022 5 the potential energy corresponding to the energy–momentum tensors (10) is given by U(⃗r) = � d3x hµν 1 (⃗x)T2,µν(⃗x) = M2 h1,00(⃗r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' (8) and (9) we obtain U(⃗r) = −GM1M2e−mr 9r � 24 − δf00 � (4c2 + 9)mr + 8c2 + 8 � − δfi i � (4c2 − 9)mr + 2c2 + 4 + 2 mr + 4 m2r2 � − δfijxixj r2 � 2c2mr + 2c2 − 4 − 6 mr − 12 m2r2 �� + 2κM1M2 9m2 δ3(⃗r) � δf00 − 2 3δfi i� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' (11) The exponential suppression factor e−mr is as expected due to the massive graviton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' The term independent of δfµν is scaled by 4/3 relative to the gravitational potential in General Relativity, in concordance with the van Dam–Veltman–Zakharov discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Moreover, note that U(⃗r) acquires terms that generally violate rotational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFPT4oBgHgl3EQf0zUa/content/2301.13180v1.pdf'} +page_content=' Acknowledgments This work was supported in part by the Portuguese Funda¸c˜ao para a Ciˆencia e a Tecnologia under grants SFRH/BSAB/150324/2019 and UID/FIS/00099/2019, and by the Indiana University Center for Spacetime Symmetries.' metadata={'source': 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b/ndE5T4oBgHgl3EQfjg9a/content/tmp_files/2301.05656v1.pdf.txt @@ -0,0 +1,1819 @@ +arXiv:2301.05656v1 [math.NT] 13 Jan 2023 +COUNTING (SKEW-)RECIPROCAL LITTLEWOOD POLYNOMIALS +WITH SQUARE DISCRIMINANT +DAVID HOKKEN +Abstract. A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in {±1}. +We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood +polynomials with square discriminant. This relates to a bounded-height analogue of the Van der Waerden +conjecture on Galois groups of random polynomials. +1. Introduction +Background and main result. Let 푓 be a monic polynomial of degree 푛 with integer coefficients that +are at most 퐻 in absolute value. In 1934, Van der Waerden [26] presented an elementary proof that 푓 is +almost surely ohne Affekt: the Galois group 퐺 푓 of 푓 over Q is the symmetric group 푆푛 with probability +tending to 1 as 퐻 goes to infinity. Two years later, he posed a conjecture [27, p. 139] on the probability +that 푓 does not have maximal Galois group, which states +Prob(퐺 푓 ≠ 푆푛) ∼ Prob( 푓 is reducible) +(1.1) +as 퐻 goes to infinity. Last year, Bhargava [5] established the breakthrough result +Prob(퐺 푓 ≠ 푆푛) ∼ Prob( 푓 is reducible) + Prob(퐺 푓 = 퐴푛) ≍ 퐻−1 +where 퐴푛 denotes the alternating group on 푛 letters. This is a weak form of the Van der Waerden +conjecture. Since 푓 is reducible with probability ≍ 퐻−1 if 푛 > 2 (see [26, 8]), the remaining task to +obtain (1.1) consists of showing that Prob(퐺 푓 = 퐴푛) = 표(퐻−1); Bary-Soroker, Ben-Porath and Matei +[1] conjecture the much stronger bound Prob(퐺 푓 = 퐴푛) = 푂(퐻−푛/2+휖 ) when 푛 ⩾ 4. +The height 퐻 of the polynomial 푓 in the above setup tends to infinity, whereas the degree 푛 stays +fixed. This approach to random polynomials is called the large box model. In the restricted coefficient +model, the height 퐻 — or any specific set N of coefficients of 푓 — is fixed, and it is the degree that +tends to infinity. Recent years have seen a surge of interest in questions about Galois groups in this +setting as well [2, 3, 6, 7, 11, 19]. For example, if N consists of 35 consecutive integers, Bary-Soroker, +Koukoulopoulos and Kozma [2] show that 퐺 푓 is 푆푛 or the alternating group 퐴푛 with probability tending +to 1. Conditionally on the Riemann Hypothesis for a family of Dedekind zeta functions, Breuillard +and Varjú [7] show a similar result for more general distributions of the coefficients of 푓 . Typically, +probabilistic methods are used to establish high transitivity of 퐺 푓 by reducing 푓 modulo various primes. +This leaves only 퐴푛 and 푆푛 as possible Galois groups, but as these are respectively (푛 − 2)- and 푛- +transitive, it is hard to distinguish them based on this property. In other words, the alternating group has +a special role in the restricted coefficient model as well. Generally, it is believed that 퐴푛 should occur +with probability tending to 0 as 푛 tends to infinity [3]. +A property that distinguishes 퐴푛 from 푆푛 as Galois group 퐺 푓 of a separable polynomial 푓 is the +following: 퐺 푓 is contained in 퐴푛 if and only if the discriminant Δ( 푓 ) of 푓 is a (necessarily nonzero) +square. Hence +Prob(퐺 푓 = 퐴푛) ⩽ Prob(Δ( 푓 ) = □ ≠ 0). +Date: January 16, 2023. +2020 Mathematics Subject Classification. Primary: 11C08, 11R32, 11R09, 05A16. Secondary: 11P21. +Key words and phrases. Littlewood polynomials, square discriminant, Galois theory, asymptotic enumeration, lattice points. +Many thanks to Gunther Cornelissen, Mar Curcó Iranzo, and Berend Ringeling for helpful conversations and feedback on +earlier versions of this manuscript. This publication is part of the project Littlewood polynomials with square discriminant +(OCENW.M20.233), financed by the Dutch Research Council (NWO). +1 + +This paper studies the probability that the discriminant of the monic polynomial 푓 is a square when the +coefficients of 푓 are independently and uniformly selected from {±1}. Such polynomials are often called +Littlewood polynomials. These are extremal examples of polynomials with restricted coefficients: all +Littlewood polynomials in degree 푛 coincide over F2, whereas they form a sparse (that is, exponentially +small in 푛) subset of the degree-푛 monic polynomials in F푝[푋] for any prime 푝 > 2. Furthermore, since +they are of height 1, the results mentioned in the first paragraph cannot be made effective in any way. +The state-of-the-art result concerning the Galois theory of random Littlewood polynomials is that at least +a fraction of 0.00068 of the Littlewood polynomials of degree 푛, with 푛 ⩾ 10104.9, is irreducible (see [2, +Theorem 3.5]). +Following Littlewood [13], denote the collection of Littlewood polynomials of degree 푛 by F푛; let +Sq푛 ⊂ F푛 consist of those with square discriminant. Furthermore, call 푓 reciprocal if 푓 (푋) = 푋푛 푓 (푋−1) +and skew-reciprocal if 푓 (푋) = (−1)푛(푛−1)/2푋푛 푓 (−푋−1) (the latter appear e.g. in [18, 10] in connection +to questions about the flatness of Littlewood polynomials on the unit circle). Denote by 푅푛, 푆푛 ⊂ F푛 the +sets of Littlewood polynomials of degree 푛 that have square discriminant and are reciprocal, respectively +skew-reciprocal. Our main result concerns the size of 푅푛 and 푆푛 as 푛 tends to infinity. +Theorem 1.1. The sets 푅8푛, 푆8푛, 푅8푛−2, and 푆8푛−2 are all of size ≍ 16푛 log 푛/√푛. More precisely: +(a) lim +푛→∞ +|푅8푛| +16푛 log 푛/√푛 = +Γ( 1 +4)2 +4 +√ +2휋3 = 0.0749 . . . , +lim +푛→∞ +|푆8푛| +16푛 log 푛/√푛 = +1 +2휋3/2 = 0.0897 . . . ; +(b) |푅8푛−2| ∼ 1 +2|푅8푛| and |푆8푛−2| ∼ 1 +2|푆8푛|. +The limits in Theorem 1.1 are approached extremely slowly. For example, when 푛 = 1011, the fraction +|푅8푛|/(16푛 log 푛/√푛) is 0.099 . . .. This is (at least in part) due to large contributions of order ≍ 16푛/√푛 +to |푅8푛| and |푆8푛| coming from error terms in lattice point counts that we use. +As observed in [3, §4], any 푓 ∈ F2푛 of even degree is separable, because 푓 coincides modulo 2 +with the separable polynomial (푋2푛+1 − 1)/(푋 − 1). Furthermore, the roots of a reciprocal polynomial +푓 come in pairs {훼, 훼−1}; if 푓 is skew-reciprocal, they come in pairs {훼, −훼−1}. The separability of 푓 +implies that 훼 and ±훼−1 are distinct. As a result, the Galois group of 푓 is contained in the wreath product +퐶2 ≀ 푆푛/2, see [25]. Recall that the wreath product of two groups 퐺 and 퐻 ⩽ 푆푛, denoted 퐺 ≀ 퐻, is the +semidirect product 퐺푛 ⋊ 퐻 where 퐻 acts on the 푛 copies of 퐺 by permuting the coordinates. Theorem +1.1 therefore leads to the following corollary. +Corollary 1.2. Let 푓 be a (skew-)reciprocal Littlewood polynomial of degree 푛 ≡ 0, 6 mod 8. As 푛 → ∞, +we have +Prob(Δ( 푓 ) = □ ≠ 0) ≍ log 푛 +√푛 +and +Prob(퐺 푓 ⩽ (퐶2 ≀ 푆푛/2) ∩ 퐴푛) ≫ log 푛 +√푛 . +The set Sq푛 is empty whenever 푛 ≡ 2, 4 mod 8, which is the reason to leave out these degrees in the +above statements. In §7, we expound the proof sketch for this fact provided in [3, §4]. In the same +section we also make some remarks on the case of odd 푛. +Reciprocals and skew-reciprocals are decomposable: a polynomial 푓 is reciprocal if it is of the form +푓 (푋) = 푋푛/2푔(푋 + 푋−1), and skew-reciprocal if it is of the form 푓 (푋) = 푋푛/2푔(푋 − 푋−1) for some +polynomial 푔. The group (퐶2 ≀ 푆푛/2) ∩ 퐴푛 is much smaller than 퐴푛 — of index 1 · 3 · 5 · . . . · (푛 − 1) to be +precise — and the sizes of 푅푛 and 푆푛 compared to |F푛| = 2푛 decrease exponentially in 푛. Nevertheless, +back in the large box model, the best known bound on the probability that the discriminant of 푓 is a square +also come from decomposable polynomials with the very same Galois group: Bary-Soroker, Ben-Porath +and Matei [1, Theorem 1.3] show for all even 푛 ⩾ 6 that +Prob(Δ( 푓 ) = □) ≫ 퐻−(푛+1)/2 +2 + +as 퐻 tends to infinity by applying an explicit version of Hilbert’s irreducibility theorem to polynomials +of the form 푓 (푋) = 푔(푋2). No Littlewood polynomial of the form 푓 (푋) = 푔(푋2) exists, and it appears +that (skew-)reciprocal polynomials are ‘the next best thing’. +Outline. In the setting of Littlewood polynomials, reducing modulo primes or applying probabilistic +methods seems difficult. Instead, the proofs in this paper combine counting arguments to derive explicit +combinatorial expressions for the objects of study with lattice point counts in certain geometric regions +and asymptotics of binomial coefficients. +In §2 we derive combinatorial expressions for |푅8푛| and the three other sets under consideration, see +Proposition 2.1 and Proposition 2.3. In each case, we obtain a sum that extends over certain tuples related +to Pythagorean triples; these come from a square discriminant criterion for (skew-)reciprocal polynomials +given in Lemma 2.2. This criterion can in theory be used to find similar expressions when Littlewood +polynomials are replaced by polynomials with coefficients in any fixed set N . Auxiliary results to study +the asymptotics of these combinatorial expressions, as well as an analysis of the Pythagorean triples, are +contained in §3. The latter essentially boils down to counting lattice points with parity and coprimality +conditions inside elliptic (for the reciprocals) or parabolic (for the skew-reciprocals) hyperboloids. These +results are then combined in §4 and §5, where the lattice point regions are split into three suitably chosen +parts. This makes it possible to evaluate the combinatorial expressions from §2 asymptotically by using +integral estimates. The proof of Theorem 1.1 is finally given in §6. We end with some observations +about the set Sq푛 in the case 푛 � 0, 6 mod 8 in §7. +Notation. The expression 푓 ≪ 푔 as well as 푔 ≫ 푓 and 푓 = 푂(푔) all mean there exists a positive constant +퐶 such that 푓 (푛) ⩽ 퐶푔(푛) for all sufficiently large values of 푛 (all asymptotics in this paper will be in +푛). The notation 푓 ≍ 푔 is shorthand for 푔 ≪ 푓 ≪ 푔. The functions 푓 and 푔 are said to be asymptotically +equal, denoted 푓 ∼ 푔, if the fraction 푓 (푛)/푔(푛) tends to 1 as 푛 tends to infinity. In particular, 푓 ∼ 푔 +implies 푓 ≍ 푔. Lastly, the notation 푓 = 표(푔) is used when the fraction 푓 (푛)/푔(푛) tends to 0 as 푛 tends +to infinity. +We write 푖 for an index and i ∈ C for the imaginary unit and adopt the convention that �푛 +푘 +� = 0 if 푘 > 푛. +2. A counting proposition +In this section, we prove the following expression for |푅8푛| in terms of binomial coefficients. +Proposition 2.1. The number of reciprocal Littlewood polynomials of degree 8푛 with (nonvanishing) +square discriminant equals +|푅8푛| = 22푛 +�2푛 +푛 +� ++ 2 +� � +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 +푛 + 1 +4 (푘푟2 + 푘푠2 + (−1) +푘+1 +2 ) +� +(2.1) +where the sum extends over all tuples (푘, 푟, 푠) such that 푘 > 0 is odd and 푟 > 푠 > 0 are coprime and of +opposite parity (i.e., 푟 is odd if and only if 푠 is even). +Similar expressions for |푅8푛−2|, |푆8푛| and |푆8푛−2| are given in Proposition 2.3. The first term in (2.1) +is ≍ 16푛/√푛 as a consequence of the well-known asymptotic expression �2푛 +푛 +� ∼ 4푛/√휋푛 for the central +binomial coefficient [22, §5.4]. Theorem 1.1 claims that this falls short by a factor logarithmic in 푛 of +the true growth rate. +The proof of Proposition 2.1 is based on the following square discriminant criterion. +Lemma 2.2. Let 푓 ∈ Q[푋] be a separable polynomial of degree 2푛. Suppose 푓 is reciprocal. Then the +discriminant of 푓 is a square if and only if (−1)푛 푓 (1) 푓 (−1) is a square. Similarly, if 푓 is skew-reciprocal, +then its discriminant is a square if and only if the integer 푓 (i) 푓 (−i) is a square. +Proof. In the case of reciprocal polynomials, this criterion is well-known and recorded in the literature in +several places, see e.g. [9, p. 85]. With a similar proof, here we show the criterion for skew-reciprocals. +Write 푎푛 for the leading coefficient of 푓 . If 푓 is not monic, then Δ( 푓 ) = 푎2푛−2 +푛 +Δ( 푓 /푎푛). Since 푎2푛−2 +푛 +is a square, we may assume without loss of generality that 푓 is in fact monic. Since 푓 is separable, it has +3 + +2푛 distinct roots. These come in pairs 훼푖, 훼푛+푖 = −훼−1 +푖 +for 푖 = 1, . . . , 푛. Hence +Δ( 푓 ) = +� +1⩽푖< 푗⩽푛 +� +(훼푖 − 훼 푗)(훼푖 + 훼−1 +푗 )(−훼−1 +푖 ++ 훼−1 +푗 )(−훼−1 +푖 +− 훼 푗) +�2 � +1⩽ 푗⩽푛 +(훼 푗 + 훼−1 +푗 )2. +The first of the two products above is the square of an integer, since +� +1⩽푖< 푗⩽푛 +(훼푖 − 훼 푗)(훼푖 + 훼−1 +푗 )(−훼−1 +푖 ++ 훼−1 +푗 )(−훼−1 +푖 +− 훼 푗) = +� +1⩽푖< 푗⩽푛 +−(훼푖 − 훼−1 +푖 +− 훼 푗 + 훼−1 +푗 )2 +is a symmetric expression in the roots of 푓 . The other product can be expanded as +� +1⩽ 푗⩽푛 +(훼 푗 + 훼−1 +푗 )2 = +� +1⩽ 푗⩽푛 +(i + 훼 푗)(i + 훼−1 +푗 )(i − 훼 푗)(i − 훼−1 +푗 ) = 푓 (i) 푓 (−i) +as claimed. +□ +To count (skew-)reciprocal polynomials with square discriminant, we recall that any polynomial 푓 +can be written as the sum 푓 (푋) = 푓e(푋2) + 푋 푓o(푋2) of its even and odd parts. Therefore +푓 (1) 푓 (−1) = ( 푓e(1) + 푓o(1))( 푓e(1) − 푓o(1)) = 푓e(1)2 − 푓o(1)2 +(2.2) +and +푓 (i) 푓 (−i) = ( 푓e(i2) + i 푓o(i2))( 푓e((−i)2) − i 푓o((−i)2)) = 푓e(−1)2 + 푓o(−1)2. +(2.3) +If 푓 is a Littlewood polynomial and we want these expressions to be squares (or minus a square – see +Lemma 2.2), we can count the possible choices of coefficients of 푓e and 푓o giving rise to (possibly +degenerate) Pythagorean triples. This is key in the proof of Proposition 2.1. +Proof of Proposition 2.1. Consider a not-necessarily monic reciprocal Littlewood polynomial +푓 = 푎4푛푋8푛 + · · · + 푎1푋4푛+1 + 푎0푋4푛 + 푎1푋4푛−1 + · · · + 푎4푛−1푋 + 푎4푛 +of degree 8푛; since 푓 has square discriminant if and only if − 푓 has square discriminant, we must divide by +2 whatever final expression we obtain to establish the count of monic reciprocal Littlewood polynomials +with square discriminant. Set +푐 := 푓e(1) = 푎0 + 2(푎2 + 푎4 + · · · + 푎4푛), +푏 := 푓o(1) = 2(푎1 + 푎3 + · · · + 푎4푛−1). +By Lemma 2.2 and (2.2), we need to pick the 푎푖 such that 푐2 − 푏2 is a square, say equal to 푎2. In the �2푛 +푛 +� +cases that exactly half of the odd-index coefficients 푎1, 푎3, . . . , 푎4푛−1 are equal to 1 and thus 푏 = 0, we +find that any choice of the coefficients 푎0, 푎2, . . . , 푎4푛 will make 푓 a Littlewood polynomial with square +discriminant. There are in total 22푛+1�2푛 +푛 +� such polynomials. After dividing by two, this is the first term +in (2.1). +Now suppose 푏 is nonzero. Recall that if 푎2 + 푏2 = 푐2 is a Pythagorean triple and 푎, 푏 and 푐 are +positive, then there are unique positive integers 푘, 푟 and 푠 such that 푐 = 푘(푟2 + 푠2), 푏 = 2푘푟푠, and +푎 = 푘(푟2 − 푠2), and 푟 > 푠 and the numbers 푟 and 푠 are coprime and of opposite parity. Since 푐 is odd +by definition, we must add the condition that 푘 be odd. This gives the summation condition in (2.1). +The prefactor of 2 before the sum arises because we treat each of the four triples (푎, ±푏, ±푐) separately +– we care if 푐2 − 푏2 is a square, so the sign of 푎 doesn’t matter; but the polynomials corresponding to the +four tuples (±푏, ±푐) are genuinely different. We conclude that the final expression must be multiplied +by 4/2 = 2. +It remains to show that the second summand in (2.1) is correct. That is, we must count all choices of +the 푎푖 that lead to the equalities 푐 = 푘(푟2 + 푠2) and 푏 = 2푘푟푠. Notice that +푎2 + 푎4 + · · · + 푎4푛 = 푐 − 푎0 +2 += 푘(푟2 + 푠2) − 푎0 +2 +. +(2.4) +4 + +Since all 푎푖 lie in {±1}, the left-hand side in (2.4) is even. As 푟2+푠2 ≡ 1 mod 4, we find that 푎0 ≡ 푘 mod 4. +Hence 푎0 = (−1) +푘−1 +2 . Therefore a total of 푛 + (푘(푟2 + 푠2) + (−1) +푘+1 +2 )/4 of the even-index coefficients +푎2, 푎4, . . . , 푎4푛 must be equal to 1. This yields +� +2푛 +푛 + 1 +4 (푘푟2 + 푘푠2 + (−1) +푘+1 +2 ) +� +options for the even-index coefficients. Similarly, there are 2푛 choices to be made for the odd-index +coefficients 푎1, 푎3, . . . , 푎4푛−1; since the sum of the latter equals 푏/2 = 푘푟푠, we find that 푛 + 푘푟푠/2 of +the odd-index coefficients must be equal to 1. So we have in total � +2푛 +푛+푘푟푠/2 +� options for the odd-index +coefficients. This gives +� +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 +푛 + 1 +4 (푘푟2 + 푘푠2 + (−1) +푘+1 +2 ) +� +combinations in total, which is the summand in (2.1). +□ +It is clear that the proof method can in principle be applied to derive a combinatorial expression for +the number of square-discriminant (skew-)reciprocal polynomials of given degree with coefficients in +any fixed set N . For |푅8푛−2|, |푆8푛| and |푆8푛−2|, we obtain the following expressions. +Lemma 2.3. We have +|푅8푛−2| = 22푛−1 +�2푛 +푛 +� ++ 2 +� � +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 − 1 +푛 + 1 +4 (푘푟2 + 푘푠2 + (−1) +푘−1 +2 − 2) +� +, +(2.5) +|푆8푛| = 22푛 +�2푛 +푛 +� ++ 2 +� � +2푛 +푛 + 1 +2 푘푟푠 +�� +2푛 +푛 + 1 +4 (푘푟2 − 푘푠2 + (−1) +푘+1 +2 +푠) +� +, +(2.6) +|푆8푛−2| = 22푛−1 +�2푛 +푛 +� ++ 2 +� � +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 − 1 +푛 + 1 +4 (푘푟2 − 푘푠2 + (−1) +푘−1 +2 +푠 − 2) +� +, +(2.7) +where in each case the sum extends over all tuples (푘, 푟, 푠) such that 푘 > 0 is odd and 푟 > 푠 > 0 are +coprime and of opposite parity. +□ +3. Auxiliary results +In this section, we gather lemmas that we will use to evaluate the binomial sums derived in §2. +General. In the next proposition, we record useful information about the asymptotics of binomial +coefficients that are (almost) central, see [22, §5.4]. +Proposition 3.1. We have �2푛 +푛 +� ∼ 4푛/√휋푛. Furthermore, if 푘 is of order 표(푛2/3), then +� 2푛 +푛 + 푘 +� +∼ +�2푛 +푛 +� +푒− 푘2 +푛 . +The following corollary of [17, Theorem 2] is a variant of Möbius inversion that we will use. +Lemma 3.2. Let 퐹 and 푓 be real-valued functions defined on R⩾1 and related through +퐹(푛) = +� +1⩽푑⩽푛 +푑 odd +푓 (푛/푑). +(More explicitly, the sum extends over all odd integers between 1 and 푛, not just the odd divisors of 푛.) +Then +푓 (푛) = +� +1⩽푑⩽푛 +푑 odd +휇(푑)퐹(푛/푑). +5 + +Reciprocals. We define the circle sector 퐶휃 (푛) for 0 ⩽ 휃 ⩽ 휋/4 as +퐶휃 (푛) = {(푥, 푦) ∈ R2 +>0 | 푥2 + 푦2 < 푛2, 푦 ⩽ 푥 tan(휃)}, +i.e. the part of the circle of radius 푛 centered at the origin in R2 that lies in the upper-right quadrant and +is bounded by 푦 = 0 and 푦 = 푥 tan(휃). +Lemma 3.3. Denote by 퐹휃 (푛) the number of integral, opposite-parity lattice points in the circle sector +퐶휃 (푛). Let 푓휃 (푛) be the number of such that are also coprime. Then +푓휃 (푛) = +� +1⩽푑⩽푛 +푑 odd +휇(푑)퐹휃 (푛/푑). +Proof. If (푥, 푦) is an integral, opposite-parity lattice point in the circle sector 퐶휃 (푛) with greatest +common divisor 푑, then (푥/푑, 푦/푑) is a primitive, integral, opposite-parity lattice point in the circle +sector 퐶휃 (푛/푑). The opposite holds as well. Noting that a pair of opposite-parity integers that are both +at most 푛 must have odd greatest common divisor at most 푛, we find +퐹휃 (푛) = +� +1⩽푑⩽푛 +푑 odd +푓휃 (푛/푑). +Lemma 3.2 gives the desired result. +□ +The next lemma shows that 퐹휃 (푛) and 푓휃 (푛) are linear in 휃. This is a key result that helps to overcome +the main difficulty in evaluating (2.1), see Lemma 4.2. +Lemma 3.4. The following asymptotics for 퐹휃 and 푓휃 hold as 푛 goes to infinity: +(a) 퐹휃 (푛) ∼ 휃푛2/4. +(b) 푓휃 (푛) ∼ 2휃푛2/휋2. +Proof. Part (b) follows after combining part (a) with Lemma 3.3 and +� +푑⩾1 +푑 odd +휇(푑) +푑2 += +� +푑⩾1 +휇(푑) +푑2 +− +� +푑⩾1 +푑 even +휇(푑) +푑2 += +� +푑⩾1 +휇(푑) +푑2 +− +� +푑⩾1 +휇(2푑) +4푑2 += +� +푑⩾1 +휇(푑) +푑2 ++ 1 +4 +� +푑⩾1 +푑 odd +휇(푑) +푑2 +by multiplicativity of the Möbius function, so that +� +푑⩾1 +푑 odd +휇(푑) +푑2 += 4 +3 +� +푑⩾1 +휇(푑) +푑2 += 8 +휋2 , +see [14, Corollary 1.10]. +For part (a), we start by distributing the lattice points in 퐶휃 (푛) over four subsets depending on the +parity of each of the coordinates. Denote by 퐹00 the number of lattice points in 퐶휃 (푛) whose coordinates +are both even, by 퐹01 the number of those whose 푥-coordinate is even and 푦-coordinate is odd, and +similarly for 퐹10 and 퐹11. For each even number 푥0, the number of lattice points (푥0, 푦) with odd 푦 +exceeds those with even 푦 by at most one. Since 푥0 lies between 1 and 푛, we find that 퐹00 + 푛/2 ⩾ 퐹01. +Similarly, we deduce 퐹10 + (푛+1)/2 ⩾ 퐹11 and 퐹00 +푛 sin(휃)/2 ⩾ 퐹10. Therefore the difference between +any two of the sets 퐹00, 퐹01, 퐹10, and 퐹11 is of order 푛. On the other hand, the quantity 퐹00 equals the +number of total lattice points in 퐶휃 (푛/2). This is asymptotically equal to the area of 퐶휃 (푛/2), which is +휃푛2/8, see e.g. [12, Chapter 1.1]. As 퐹00, 퐹01, 퐹10, and 퐹11 differ by a term of order 푛 at most, they are +equal asymptotically. Therefore 퐹휃 (푛) = 퐹01 + 퐹10 ∼ 휃푛2/4. +□ +Consider the set +퐵푛 = +� +(푘, 푟, 푠) ∈ Z3 +���� +푘 > 0 and odd, 푟 > 푠 > 0 coprime and +of opposite parity, and 푘(푟2 + 푠2) ⩽ 푛 +� +. +Since the inequality 푘(푟2 + 푠2) ⩾ 2푘푟푠 holds for all positive integers 푘, 푟 and 푠, the set 퐵5푛 certainly +contains all tuples (푘, 푟, 푠) over which the sum in (2.1) extends. The next lemma concerns the asymptotic +size of 퐵푛 and thus shows how the number of summands in (2.1) depends on 푛 as 푛 tends to infinity. +6 + +Proposition 3.5. The set 퐵푛 is of size asymptotically equal to +1 +4휋 푛 log 푛. +Proof. Take 휃 = 휋/4 in Lemma 3.4. Then +|퐵푛| = +� +1⩽푑⩽푛 +푑 odd +푓휃 +��푛 +푑 +� +. +(3.1) +Calculating the sum up to 푑 = 푛+ := 푛/log log 푛, we find +� +1⩽푑⩽푛+ +푑 odd +푓휃 +�� +푛 +푑 +� +∼ 2휃푛 +휋2 +� +1⩽푑⩽푛+ +푑 odd +1 +푑 ∼ 휃푛 log 푛 +휋2 += 푛 log 푛 +4휋 +where the last asymptotic equality follows since log(푛/log log 푛) ∼ log 푛. +This gives the claimed +asymptotic size of 퐵푛. The remaining terms of the sum in (3.1), where 푑 > 푛/log log 푛, are bounded by +� +푛+<푑⩽푛 +푑 odd +푓휃 +��푛 +푑 +� +< 푛 푓휃 +�� +푛 +푛/log log 푛 +� +∼ 푛 log log 푛 +2휋 +and thus do not contribute to the asymptotic size of 퐵푛. +□ +Let (푎, 푏, 푐) be a Pythagorean triple, i.e. 푎2 + 푏2 = 푐2. Proposition 3.8 implies that the number of +Pythagorean triples with hypotenuse less than 푛 and of opposite parity (considering the triples (푎, 푏, 푐) +and (푏, 푎, 푐) to be the same) is asymptotic to +1 +4휋 푛 log 푛; an error term of the form 푐0푛 for an explicit +constant 푐0 appears to have been given first by Sierpiński in [21, Eq. (7)], with further improvements due +to Stronina [23] and Nowak & Recknagel [16]. +Skew-reciprocals. Here, we establish results analogous to those in §3, but for skew-reciprocal polyno- +mials. The tuples (푘, 푟, 푠) over which the sum in (2.6) extends are contained in 퐷5푛, where +퐷푛 = +� +(푘, 푟, 푠) ∈ Z3 +���� +푘 > 0 and odd, 푟 > 푠 > 0 coprime and of +opposite parity, and 푘(푟2 − 푠2) ⩽ 푛 and 2푘푟푠 ⩽ 푛 +� +. +Neither of 푘(푟2 − 푠2) and 2푘푟푠 dominates the other for every choice of positive integers 푘, 푟 and 푠 with +푟 > 푠; indeed, the inequality 푘(푟2 − 푠2) > 2푘푟푠 holds if and only if ( +√ +2 − 1)푟 > 푠. Hence both of the +inequalities (푟2 − 푠2) ⩽ 푛 and 2푘푟푠 ⩽ 푛 are required in the definition of 퐷푛. +Set 훼 = artanh( +√ +2 − 1) = log +� +1 + +√ +2; this is the inverse hyperbolic tangent of the angle between the +푟-axis and the line from the origin to the intersection point of the hyperbolas 푟2 − 푠2 = 푛 and 2푟푠 = 푛. +Define the hyperbolic sectors 퐻휃 (푛) and 퐻∗ +휃 (푛) for 0 < 휃 ⩽ 훼 as +퐻휃 (푛) = {(푥1, 푦1) ∈ R2 +>0 | 푥2 +1 − 푦2 +1 < 푛2, 푦1 ⩽ tanh(휃)푥1}, +(3.2) +퐻∗ +휃 (푛) = {(푥2, 푦2) ∈ R2 +>0 | 2푥2푦2 < 푛2, 푦2 < 푥2 ⩽ 푒2휃 푦2}. +(3.3) +Note that 푒2휃 = (1 + tanh(휃))/(1 − tanh(휃)), and that both tanh(훼) and 푒2훼 are equal to +√ +2 − 1. +As in the reciprocal case, the reason to consider these sectors is that their areas scale linearly in 휃. +Lemma 3.6. The areas of 퐻휃 (푛) and of 퐻∗ +휃 (푛) are each equal to 휃푛2/2. +Proof. The linear transformation sending 푥2 ↦→ (푥1 + 푦1)/ +√ +2 and 푦2 ↦→ (푥1 − 푦1)/ +√ +2 maps 퐻∗ +휃 (푛) to +퐻휃 (푛) and has determinant 1. Thus 퐻∗ +휃 (푛) and 퐻휃 (푛) have equal area. The area of 퐻휃 (푛) is 푛2 times +as large as that of the region bounded by the hyperbola 푥2 +1 − 푦2 +1 = 1, the axis 푦1 = 0, and the ray through +the origin and the point (cosh(휃), sinh(휃)). But that is simply 휃/2. +□ +We summarise the analogues of Lemma 3.3 and Lemma 3.4 in the following lemma. +Lemma 3.7. Denote by 퐺 휃 (푛) (resp. 퐺∗ +휃 (푛)) the number of integral, opposite-parity lattice points in +퐻휃 (푛) (resp. 퐻∗ +휃 (푛)), and by 푔휃 (푛) (resp. 푔∗ +휃 (푛)) the number of such that are also coprime. Then the +following hold: +(a) 푔(푛) = � 휇(푑)퐺(푛/푑) where the sum extends over all odd 1 ⩽ 푑 ⩽ 푛, and similarly for 푔∗ +휃. +7 + +(b) 퐺 휃 (푛) ∼ 퐺∗ +휃 (푛) ∼ 휃푛2/4. +(c) 푔휃 (푛) ∼ 푔∗ +휃 (푛) ∼ 2휃푛2/휋2. +Proof sketch. All proofs are analogous to those of the mentioned lemmas, where 퐻휃 (푛) (respectively +퐻∗ +휃 (푛)) plays the role of 퐶휃 (푛). That the asymptotic expressions for 푓휃, 푔휃, and 푔∗ +휃 are all equal comes +from the fact that the circle sector 퐶휃 and the hyperbolic sectors 퐻휃 and 퐻∗ +휃 all have equal area, see +Lemma 3.6. +□ +The next result is the analogue of Proposition 3.5. +Proposition 3.8. The set 퐷푛 is of size asymptotically equal to 2훼 +휋2 푛 log 푛. +Proof. Note that +|퐷푛| = +� +1⩽푑⩽푛 +푑 odd +푔훼 +��푛 +푑 +� ++ 푔∗ +훼 +��푛 +푑 +� +. +(3.4) +Writing 푛+ = 푛/log log 푛 and reasoning as in the proof of Proposition 3.5 that the terms in the sum with +푑 > 푛+ do not contribute, we find with help of Lemma 3.7(c) that +|퐷푛| ∼ +� +1⩽푑⩽푛+ +푑 odd +푔훼 +��푛 +푑 +� ++ 푔∗ +훼 +��푛 +푑 +� +∼ 4훼푛 +휋2 +� +1⩽푑⩽푛+ +푑 odd +1 +푑 ∼ 2훼푛 log 푛 +휋2 +as claimed. +□ +Let (푎, 푏, 푐) be a Pythagorean triple, i.e. 푎2 + 푏2 = 푐2. Proposition 3.8 implies that the number of +Pythagorean triples with legs less than 푛 and of opposite parity (considering the triples (푎, 푏, 푐) and +(푏, 푎, 푐) to be the same) is asymptotic to 2훼 +휋2 푛 log 푛; see [4, Cor. 2] for an improved error term. +4. The reciprocals +In this section, we build up towards the proof of the part of Theorem 1.1 that concerns reciprocals. For +the proof, we break up the sum in (2.1) into several pieces. Fix a (large) integer 푁 and set 휖 = 푁−1 and +푚 = 5 +� +푛 log 푛 (the number 5 is a convenient choice, but could be replaced by any real number greater +than 2 +√ +2). Write +Σ1 = +� +(푘,푟,푠)∈퐵푁√푛 +� +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 +푛 + 1 +4 (푘푟2 + 푘푠2 + (−1) +푘+1 +2 ) +� +, +Σ2 = +� +(푘,푟,푠)∈퐵푚\퐵푁√푛 +� +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 +푛 + 1 +4 (푘푟2 + 푘푠2 + (−1) +푘+1 +2 ) +� +, +and define Σ3 through 2Σ3 = |푅8푛| − 22푛�2푛 +푛 +� − 2Σ1 − 2Σ2. Figure 1 shows how the domain consisting of +lattice points over which the sum in (2.1) extends is divided into parts associated with the sums Σ1, Σ2 +and Σ3. The following subsections go into the asymptotics of each of these terms, showing that Σ1 is the +dominant term. To obtain an exact expression for the main term in the asymptotics of Σ1, precise control +over both binomial coefficients in its summand is needed. In contrast, to show that Σ2 is negligible in +comparison, we only need to control one binomial coefficient precisely, and for Σ3 it suffices to estimate +both binomial coefficients appearing in the summand by the maximal value they can obtain. Proposition +3.1 and Proposition 3.5 are key in this. +We often use elementary estimates of sums by integrals without reference; proofs for any such estimate +may be found in [22, Theorems 4.1 and 4.2]. +8 + +푠 +푟 +Σ3 +� +5푛 +푘 +Σ2 +� +푚 +푘 +Σ1 +� +푁 √푛 +푘 +푟 = 푠 +휋 +4 +Figure 1. Slice of the domain containing 퐵5푛 at a fixed 푘, showing the subdomains +related to the sums Σ푖 with 푖 = 1, 2, 3. The full domain (with 푘 varying) is part of the +interior of an elliptic paraboloid. +The sums Σ2 and Σ3. In this subsection, we show that the term 22푛�2푛 +푛 +� and the sums Σ2 and Σ3 each +have negligible contribution in comparison to 16푛 log 푛/√푛 when 휖 tends to zero. First, recall that we +have already seen in the introduction that 22푛�2푛 +푛 +� ≍ 16푛/√푛. The sum Σ3 satisfies +Σ3 ⩽ |퐵5푛| +�2푛 +푛 +� � +2푛 +푛 + +� +푛 log 푛 +� +≍ |퐵5푛| +�2푛 +푛 +�2 +푒− 푛 log 푛 +푛 +≍ 16푛 log 푛 +푛 +(4.1) +by Proposition 3.5 and Proposition 3.1. In conclusion, both 22푛�2푛 +푛 +� and Σ3 are of order 표(16푛 log 푛/√푛). +Lemma 4.1. The sum Σ2 satisfies +lim +휖 →0 lim +푛→∞ +Σ2 +16푛 log 푛/√푛 = 0. +Proof. Defining 퐶 = 퐶(푘,푟, 푠) = 4푘2푟2푠2 + (푘(푟2 + 푠2) + (−1) +푘+1 +2 )2, Proposition 3.1 implies that +Σ2 ∼ +�2푛 +푛 +�2 +� +(푘,푟,푠)∈퐵푚\퐵푁√푛 +푒− 퐶 +16푛 . +Since 퐶 > 푘2(푟2 + 푠2)2 for all positive integers 푘, 푟, and 푠, the sum on the right-hand side is bounded +from above by +� +1⩽푟⩽√푚 +2⩽푠⩽√푚 +푘> 푁√푛 +푟2+푠2 +푒− +1 +16푛 푘2(푟2+푠2)2 ⩽ +∫ √푚 +1 +∫ √푚 +0 +� +1 + +∫ ∞ +푁√푛 +푟2+푠2 +푒− +1 +16푛 푘2(푟2+푠2)2 d푘 +� +d푟 d푠 +by applying elementary estimates for sums by integrals. Pulling out the 1 from the middle integral and +evaluating the innermost integral yields +� +(푘,푟,푠)∈퐵푚\퐵푁√푛 +푒− 퐶 +16푛 < 푚 + 2√휋푛(1 − erf(푁/4)) +∫ √푚 +1 +∫ √푚 +0 +1 +푟2 + 푠2 d푟 d푠, +where erf(푥) = 2휋−1/2 ∫ 푥 +0 푒−푡2 d푡 is the error function. Switching to polar coordinates with 푅2 = 푟2 + 푠2, +the remaining double integral is bounded by +∫ √푚 +1 +∫ √푚 +0 +1 +푟2 + 푠2 d푟 d푠 < 휋 +2 +∫ √ +2푚 +1 +1 +푅 d푅 = 휋 +4 log 2푚 ≍ log 푛. +9 + +Thus, as �2푛 +푛 +�2 ≍ 16푛/푛, the sum Σ2 is asymptotically at most +(1 − erf(푁/4)) 16푛 log 푛 +√푛 +up to a multiplicative constant independent of 푛 and 푁. As erf(푥) goes to 1 as 푥 tends to ∞, this yields +the claimed limit. +□ +The sum Σ1. To obtain a precise estimate of Σ1, we need to control both binomial coefficients in the +summand of (2.1) simultaneously. This is achieved by dividing the domain over which the sum extends +in boxes as follows. Let 1 ⩽ 푖 ⩽ 푁 and 1 ⩽ 푗 ⩽ 푁2. Write 휃푖 = 푖휖휋/4 and consider the inequalities +( 푗 − 1)휖√푛 < 푘(푟2 + 푠2) ⩽ 푗휖√푛, +(4.2) +tan(휃푖−1) < 푠/푟 ⩽ tan(휃푖); +(4.3) +this is a region enclosed between two circles and two lines. For fixed positive 푘, the inequalities (4.2) +and (4.3) divide the right-upper quadrant of the disk 푟2 + 푠2 ⩽ 푁√푛/푘 in boxes of equal area. Define the +set 푇푖 푗 as +푇푖 푗 = +� +(푘, 푟, 푠) ∈ Z3 +���� +푘 > 0 and odd, 푟 > 푠 > 0 coprime and of opposite +parity, and (푘, 푟, 푠) satisfies (4.2) and (4.3) +� +. +The lattice point sets 푇푖 푗 are asymptotically equal in size as well. +Lemma 4.2. As 푛 tends to infinity, we have +��푇푖 푗 +�� ∼ 휖2 +8휋 +√푛 log 푛. +In particular, the size of 푇푖 푗 does not depend on 푖 and 푗. +Proof. Write 푎 = +� +푗휖√푛/푘 and 푏 = +� +( 푗 − 1)휖√푛/푘. For fixed positive 푘, the number of integral, +coprime, opposite-parity lattice points (푟, 푠) in the box bounded by the inequalities (4.2) and (4.3) equals +푞(푖, 푗) := � 푓휃푖 (푎) − 푓휃푖−1 (푎)� − � 푓휃푖 (푏) − 푓휃푖−1 (푏)� . +(4.4) +For 푘 ⩽ 푛+ := √푛/log log 푛, the quantity 푞 satisfies the asymptotic equality +푞(푖, 푗) ∼ 2 +휋2 (휃푖 − 휃푖−1)(푎2 − 푏2) = 휖2 +2휋 +√푛 +푘 +by Lemma 3.4 (notice that this is just the area of the box multiplied by 4/휋2). When 푘 > 푛+, the bound +푞(푖, 푗) ≪ log log 푛 holds as each of the four terms on the right-hand side of (4.4) are at most of this +order. Therefore +� +1⩽푘⩽푛+ +푘 odd +푞(푖, 푗) ∼ 휖2 +2휋 +√푛 +� +1⩽푘⩽푛+ +푘 odd +1 +푘 ∼ 휖2 +8휋 +√푛 log 푛 +and +� +푛+<푘⩽√푛 +푘 odd +푞(푖, 푗) ≪ √푛 log log 푛, +which implies +��푇푖 푗 +�� = +� +1⩽푘⩽√푛 +푘 odd +푞(푖, 푗) ∼ 휖2 +8휋 +√푛 log 푛 +as claimed. +□ +Multiplying (4.3) through by 푟 and using both of the resulting inequalities, some rewriting of (4.2) +leads to +1 +4푚(푖, 휖)( 푗 − 1)휖√푛 < 1 +2푘푟푠 < 1 +4 푀(푖, 휖) 푗휖√푛, +where +푚(푖, 휖) = 2 tan(휃푖−1) cos2(휃푖) +and +푀(푖, 휖) = 2 tan(휃푖) cos2(휃푖−1). +Note that 푚(푖, 휖) is increasing on the interval [−1, 푁] and 푀(푖, 휖) is increasing on [−1, 푁 + 1]. +10 + +Lemma 4.3. The sum Σ1 satisfies +lim +휖 →0 lim +푛→∞ +Σ1 +16푛 log 푛/√푛 = +1 +4휋3/2 +∫ 1 +0 +1 +� +1 + sin2(휋푡/2) +d푡. +(4.5) +Proof. We give an upper and a lower bound that converge to the same value as 휖 tends to 0. For the +upper bound, note that +Σ1 = +� +1⩽푖⩽푁 +1⩽ 푗⩽푁 2 +� +(푘,푟,푠)∈푇푖 푗 +� +2푛 +푛 + 1 +2 푘푟푠 +�� +2푛 +푛 + 1 +4 (푘푟2 + 푘푠2 + (−1)(푘+1)/2) +� +(4.6) +⩽ +�2푛 +푛 +�2 � +1⩽푖⩽푁 +|푇푖1| + +� +1⩽푖⩽푁 +2⩽ 푗⩽푁 2 +��푇푖 푗 +�� +� +2푛 +푛 + 1 +4푚(푖, 휖)( 푗 − 1)휖√푛 +� � +2푛 +푛 + 1 +4 ( 푗 − 1)휖√푛 − 1 +� +. +The first sum in the last line, where 푗 = 1 is fixed, has negligible contribution as 휖 → 0. In addition, +the asymptotics of the last binomial coefficient is not altered by changing 푛 + 1 +4 ( 푗 − 1)휖√푛 − 1 to +푛+ 1 +4 ( 푗 −1)휖√푛. Combined with Proposition 3.1 and Lemma 4.2, the sum Σ1 is therefore asymptotically +no larger than +휖2 +8휋2 +16푛 log 푛 +√푛 +푁 +� +푖=1 +푁 2 +� +푗=2 +푒− 1 +16 ( 푗−1)2 휖 2(1+푚(푖,휖 )2). +(4.7) +The inner sum in (4.7) is smaller than +∫ ∞ +1 +푒− 1 +16 ( 푗−1)2 휖 2(1+푚(푖,휖 )2) d푗 = +2√휋 +휖 +� +1 + 푚(푖, 휖)2 . +Plugging this into (4.7), moving out all constants from the sum but keeping all 휖’s in it shows that it +remains to evaluate +푁 +� +푖=1 +휖 +� +1 + 푚(푖, 휖)2 . +Again, we employ an integral estimate (using that 푚(푖, 휖) is increasing on the interval [0, 푁]) to bound +the last sum from above by +∫ +푁 +0 +휖 +� +1 + 푚(푖, 휖)2 d푖 = +∫ 1 +0 +1 +� +1 + 4 tan2 +� +(푥−휖 ) 휋 +4 +� +cos4 � 푥 휋 +4 +� d푥 +after the substitution 푥 = 푖휖. As 휖 tends to 0, this becomes the integral shown in (4.5). +Now we prove that the asymptotic lower bound is the same. Starting from (4.6), notice that this can +be bounded from below by +� +1⩽푖⩽푁 +1⩽ 푗⩽푁 2 +��푇푖 푗 +�� +� +2푛 +푛 + 1 +4 푀(푖, 휖) 푗휖√푛 +� � +2푛 +푛 + 1 +4 푗휖√푛 + 1 +� +. +Again, dropping the +1 in the last binomial coefficient, this sum is asymptotically at least +휖2 +8휋2 +16푛 log 푛 +√푛 +푁 +� +푖=1 +푁 2 +� +푗=1 +푒− 1 +16 푗2휖 2(1+푀 (푖,휖 )2). +The inner sum is at least +∫ +푁 2 +1 +푒− 1 +16 푗2휖 2(1+푀 (푖,휖 )2) d푗 = +2√휋 +휖 +� +1 + 푀(푖, 휖)2 +� +erf +�� +1 + 푀(푖, 휖)2 +4휖 +� +− erf +� +휖 +� +(1 + 푀(푖, 휖)2) +4 +�� +. +11 + +Since the error function is monotonously increasing, and 푀(푖, 휖) is monotonously increasing on [1, 푁] +as well, the term involving the error functions is at least +erf +� 1 +4휖 +� +− erf +� +휖 +√ +3 +4 +� +which tends to 1 as 휖 tends to 0. We are left with the sum +푁 +� +푖=1 +휖 +� +1 + 푀(푖, 휖)2 +which is bounded from below by +∫ +푁 +1 +휖 +� +1 + 푀(푖, 휖)2 d푖 = +∫ 1 +휖 +1 +� +1 + 4 tan2( 푥 휋 +4 ) cos4( (푥−휖 ) 휋 +4 +) +d푥 +where 푥 = 푖휖. In the limit 휖 → 0 this becomes the integral on the right-hand side in (4.5). +□ +5. The skew-reciprocals +As in the reciprocal case, fix some (large) integer 푁, define 휖 = 푁−1 and 푚 = 5 +� +푛 log 푛, and write +|푆8푛| = 22푛 +�2푛 +푛 +� ++ 2Σ′ +1 + 2Σ′ +2 + 2Σ′ +3 +where +Σ′ +1 = +� +(푘,푟,푠)∈퐷푁√푛 +� +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 +푛 + 1 +4 (푘푟2 − 푘푠2 + (−1)푠+ 푘+1 +2 ) +� +, +Σ′ +2 = +� +(푘,푟,푠)∈퐷푚\퐷푁√푛 +� +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 +푛 + 1 +4 (푘푟2 − 푘푠2 + (−1)푠+ 푘+1 +2 ) +� +. +With methods very similar to the ones employed in the reciprocal case, in the double limit as first 푛 and +then 푁 tends to infinity, each of 22푛�2푛 +푛 +� and the sums Σ′ +2 and Σ′ +3 are negligible compared to 16푛 log 푛/√푛. +Here, we focus on the evaluation of Σ′ +1. +Let 1 ⩽ 푖 ⩽ 푁 and 1 ⩽ 푗 ⩽ 푁2. Recall that 훼 is the constant log +� +1 + +√ +2. Write 휃푖 = 푖휖훼 and +consider the inequalities +( 푗 − 1)휖√푛 < 푘(푟2 − 푠2) ⩽ 푗휖√푛, +(5.1) +tanh(휃푖−1) < 푠/푟 ⩽ tanh(휃푖); +(5.2) +this is a region enclosed between two hyperbolas and two lines. The quantity tanh(휃) varies between 0 +and tanh(훼) = +√ +2 − 1 as 휃 varies between 0 and 훼. Therefore, fixing 푘, the regions described by the +inequalities (5.1) and (5.2) partition 퐻훼( +� +푗휖√푛/푘) (with 퐻훼 as in (3.2)). Similarly, the inequalities +( 푗 − 1)휖√푛 < 2푘푟푠 ⩽ 푗휖√푛, +(5.3) +푒−2휃푖 < 푠/푟 ⩽ 푒−2휃푖−1 +(5.4) +partition 퐻∗ +훼( +� +푗휖√푛/푘). Define the set +푇 ′ +푖 푗 = +� +(푘, 푟, 푠) ∈ Z3 +���� +푘 > 0 and odd, 푟 > 푠 > 0 coprime and of opposite +parity, and (푘, 푟, 푠) satisfies (5.1) and (5.2) +� +, +and let 푇 ′∗ +푖 푗 be the similar set of tuples that satisfy (5.3) and (5.4) instead. +Lemma 5.1. As 푛 tends to infinity, we have +|푇 ′ +푖 푗| ∼ |푇 ′∗ +푖 푗 | ∼ 훼휖2 +2휋2 +√푛 log 푛. +In particular, the size of 푇 ′ +푖 푗 does not depend on 푖 and 푗. +12 + +Proof. We argue as in Lemma 4.2. Write 푎 = +� +푗휖√푛/푘 and 푏 = +� +( 푗 − 1)휖√푛/푘. For fixed positive +푘, the number of integral, coprime, opposite-parity lattice points in the box bounded by the inequalities +(5.1) and (5.2) equals +푞(푖, 푗) := �푔휃푖 (푎) − 푔휃푖−1 (푎)� − �푔휃푖 (푏) − 푔휃푖−1 (푏)� . +(5.5) +For 푘 ⩽ 푛+ := √푛/log log 푛, we deduce the asymptotic equality +푞(푖, 푗) ∼ 2 +휋2 (휃푖 − 휃푖−1)(푎2 − 푏2) = 2훼휖2 +휋2 +√푛 +푘 +by Lemma 3.7. When 푘 > 푛+, the bound 푞(푖, 푗) ≪ log log 푛 holds as each of the four terms on the +right-hand side in (5.5) are at most of this order. By an argument entirely similar to the one in Lemma +4.2, we find +|푇 ′ +푖 푗| = +� +1⩽푘⩽√푛 +푘 odd +푞(푖, 푗) ∼ +� +1⩽푘⩽푛+ +푘 odd +푞(푖, 푗) ∼ 2훼휖2 +휋2 +√푛 +� +1⩽푘⩽푛+ +푘 odd +1 +푘 ∼ 훼휖2 +2휋2 +√푛 log 푛, +as claimed. The same argument gives the result for 푇 ′∗ +푖 푗 . +□ +Write +푚′(푖, 휖) = sinh(2휃푖−1) +and +푀 ′(푖, 휖) = sinh(2휃푖). +Manipulating the inequalities (5.1) and (5.2) leads to +1 +4푚′(푖, 휖)( 푗 − 1)휖√푛 < 1 +2푘푟푠 ⩽ 1 +4 푀 ′(푖, 휖) 푗휖√푛, +whereas the inequalities (5.3) and (5.4) yield +1 +4푚′(푖, 휖)( 푗 − 1)휖√푛 < 1 +4푘(푟2 − 푠2) ⩽ 1 +4 푀 ′(푖, 휖) 푗휖√푛 +for the same functions 푚′ and 푀 ′. Write +Σ = +� +1⩽푖⩽푁 +1⩽ 푗⩽푁 2 +� +(푘,푟,푠)∈푇 ′ +푖 푗 +� +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 +푛 + 1 +4 (푘푟2 − 푘푠2 + (−1)푠+ 푘+1 +2 ) +� +(5.6) +and +Σ∗ = +� +1⩽푖⩽푁 +1⩽ 푗⩽푁 2 +� +(푘,푟,푠)∈푇 ′∗ +푖 푗 +� +2푛 +푛 + 1 +2 푘푟푠 +� � +2푛 +푛 + 1 +4 (푘푟2 − 푘푠2 + (−1)푠+ 푘+1 +2 ) +� +, +so that Σ′ +1 = Σ + Σ∗. +Lemma 5.2. Each of the sums Σ and Σ∗ can be asymptotically bounded from above by +훼휖2 +2휋3 +16푛 log 푛 +√푛 +푁 +� +푖=1 +푁 2 +� +푗=1 +푒− 1 +16 ( 푗−1)2 휖 2(1+푚′(푖,휖 )2) +(5.7) +and from below by +훼휖2 +2휋3 +16푛 log 푛 +√푛 +푁 +� +푖=1 +푁 2 +� +푗=1 +푒− 1 +16 푗2 휖 2(1+푀′(푖,휖 )2). +(5.8) +In particular, Σ′ +1 is asymptotically equal to 2Σ. +Proof. We give an upper and a lower bound that converge to the same value as 휖 tends to 0. For the +upper bound, note that +Σ ⩽ +�2푛 +푛 +�2 � +1⩽푖⩽푁 +|푇 ′ +푖1| + +� +1⩽푖⩽푁 +2⩽ 푗⩽푁 2 +|푇 ′ +푖 푗| +� +2푛 +푛 + 1 +4푚′(푖, 휖)( 푗 − 1)휖√푛 +� � +2푛 +푛 + 1 +4 ( 푗 − 1)휖√푛 − 1 +� +. +The −1 appearing in the last binomial coefficient can simply be ignored, because it doesn’t affect the +asymptotics in 푛 of that binomial coefficient. In addition, we see that the first term in the last line will +13 + +have negligible contribution as 휖 → 0. The asymptotics for almost central binomial coefficients given in +Proposition 3.1 and for |푇 ′ +푖 푗| of Lemma 5.1 show the last sum is asymptotically no larger than the sum in +(5.7). +For the lower bound, we observe +Σ ⩾ +� +1⩽푖⩽푁 +1⩽ 푗⩽푁 2 +|푇 ′ +푖 푗| +� +2푛 +푛 + 1 +4 푀 ′(푖, 휖) 푗휖√푛 +� � +2푛 +푛 + 1 +4 푗휖√푛 + 1 +� +starting from (5.6). Again, dropping the +1 in the last binomial coefficient, this sum is asymptotically at +least the sum in (5.8). After replacing 푇 ′ +푖 푗 by 푇 ′∗ +푖 푗 , the same argument holds for Σ∗. +□ +We are now in the position to obtain our main result for Σ′ +1. +Lemma 5.3. With 훼 = log +� +1 + +√ +2, the sum Σ′ +1 satisfies +lim +휖 →0 lim +푛→∞ +Σ′ +1 +16푛 log 푛/√푛 = 2훼 +휋5/2 +∫ 1 +0 +1 +� +1 + sinh2(2훼푡) +d푡. +(5.9) +Proof. We show that the sums (5.7) and (5.8) are asymptotically equal. This implies that 1 +2Σ′ +1 and (5.7) +are asymptotically equal. We start with the upper bound. The inner sum in (5.7) is smaller than +∫ ∞ +1 +푒− 1 +16 ( 푗−1)2 휖 2(1+푚′(푖,휖 )2) d푗 = +2√휋 +휖 +� +1 + 푚′(푖, 휖)2 . +Plugging this into (5.7), moving out all constants from the sum but keeping all 휖’s in it shows that it +remains to evaluate +푁 +� +푖=1 +휖 +� +1 + 푚′(푖, 휖)2 . +Again, we employ an integral estimate (using that 푚′(푖, 휖) is increasing on the interval [0, 푁]) to bound +the last sum from above by +∫ +푁 +0 +휖 +� +1 + 푚′(푖, 휖)2 d푖 = +∫ 1 +0 +1 +� +1 + sinh2(2(푡 − 휖)훼) +d푡 +after the substitution 푡 = 푖휖. As 휖 tends to 0, this becomes the integral shown in (5.9). +For the lower bound, the inner sum in (5.8) is at least +∫ +푁 2 +1 +푒− 1 +16 푗2휖 2(1+푀 (푖,휖 )2) d푗 = +2√휋 +휖 +� +1 + 푀(푖, 휖)2 +� +erf +�� +1 + 푀(푖, 휖)2 +4휖 +� +− erf +� +휖 +� +(1 + 푀(푖, 휖)2) +4 +�� +. +Since the error function is monotonously increasing, and 푀(푖, 휖) is monotonously increasing on [1, 푁] +as well, the term involving the error functions is at least +erf +� 1 +4휖 +� +− erf +� 휖 +2 +√ +2 +� +which tends to 1 as 휖 tends to 0. We are left with the sum +푁 +� +푖=1 +휖 +� +1 + 푀(푖, 휖)2 +which is bounded from below by +∫ +푁 +1 +휖 +� +1 + 푀(푖, 휖)2 d푖 = +∫ 1 +휖 +1 +� +1 + sinh2(2훼푥) +d푡 +where 푡 = 푖휖. In the limit 휖 → 0 this again becomes the integral on the right-hand side in (5.9). +□ +14 + +6. Proof of Theorem 1.1 +We are ready to prove Theorem 1.1. We first prove part (a) and then part (b). +Proof of Theorem 1.1(a). Whereas Σ1 and Σ2 depend on 휖, the total sum |푅8푛| does not. In particular, +lim +푛→∞ +|푅8푛| +16푛 log 푛/√푛 = lim +휖 →0 lim +푛→∞ +|푅8푛| +16푛 log 푛/√푛. +The last double limit can be split in several pieces using that |푅8푛| = 22푛�2푛 +푛 +� + 2Σ1 + 2Σ2 + 2Σ3. In +particular, (4.1), Lemma 4.1, and Lemma 4.3 show that +lim +휖 →0 lim +푛→∞ +|푅8푛| +16푛 log 푛/√푛 = lim +휖 →0 lim +푛→∞ +2Σ1 +16푛 log 푛/√푛 = +1 +2휋3/2 +∫ 1 +0 +1 +� +1 + sin2(휋푡/2) +d푡. +To evaluate the integral, substitute 푥 = sin4(휋푡/2) so that 2휋 d푡 = 푥−3/4(1 − √푥)−1/2 d푥. Hence +∫ 1 +0 +1 +� +1 + sin2(휋푡/2) +d푡 = 1 +2휋 +∫ 1 +0 +푥−3/4(1 − 푥)−1/2 d푥 = 1 +2휋 퐵 +�1 +4, 1 +2 +� += Γ( 1 +4)Γ( 1 +2) +2휋Γ( 3 +4) +(6.1) +where 퐵 is the beta function, which satisfies 퐵(푚, 푛) = Γ(푚)Γ(푛)/Γ(푚 + 푛). +Legendre’s duplica- +tion formula for the gamma function yields Γ(1/2) = Γ(1/4)Γ(3/4)/ +√ +2휋, showing that (6.1) equals +Γ( 1 +4)2/ +√ +8휋3. This gives the desired result. +The skew-reciprocal case is entirely similar. With the same steps, we deduce +lim +푛→∞ +|푆8푛| +16푛 log 푛/√푛 = lim +휖 →0 lim +푛→∞ +2Σ′ +1 +16푛 log 푛/√푛 = 4훼 +휋5/2 +∫ 1 +0 +1 +� +1 + sinh2(2훼푡) +d푡 +where again 훼 = log +� +1 + +√ +2. +To evaluate the integral, substituting 푥 = sinh(2훼푡) yields d푥 = +2훼 cosh(2훼푡) d푡 = 2훼 +√ +푥2 + 1 d푡. Therefore +∫ 1 +0 +1 +� +1 + sinh2(2훼푡) +d푡 = 1 +2훼 +∫ 1 +0 +1 +푥2 + 1 d푥 = 1 +2훼 (arctan(1) − arctan(0)) = 휋 +8훼, +as claimed. +□ +Proof of Theorem 1.1(b). We prove the result for the reciprocal polynomials; an analogous argument +works for the skew-reciprocals as well. Write 푛0 = 1 +4 (푘푟2 + 푘푠2 + (−1) +푘+1 +2 ). The second binomial +coefficient in (2.5) equals �2푛 − 1 +푛 + 푛0 +� += +�1 +2 − 푛0 +2푛 +� � 2푛 +푛 + 푛0 +� +if 푘 ≡ 1 mod 4, +� 2푛 − 1 +푛 + 푛0 − 1 +� += +�1 +2 + 푛0 +2푛 +� � 2푛 +푛 + 푛0 +� +if 푘 ≡ 3 mod 4; +these identities also hold when 푛0 = 푛. Therefore +|푅8푛−2| = 1 +2 |푅8푛| + Σ, +where +Σ = 1 +2푛 +� +(푘,푟,푠)∈퐵5푛 +1 +4 (1 + (−1) +푘+1 +2 푘(푟2 + 푠2)) +� +2푛 +푛 + 1 +2 푘푟푠 +� � 2푛 +푛 + 푛0 +� +. +Write +푉푡 = 1 +2푛 +� +(푘,푟,푠)∈퐵푡 +1 +4 (1 + 푘(푟2 + 푠2)) +� +2푛 +푛 + 1 +2 푘푟푠 +� � 2푛 +푛 + 푛0 +� +. +Then 푉5푛 is at least as big as Σ in absolute value. Estimating in each case the term +1 +4 (1 + 푘(푟2 + 푠2)) +15 + +by the maximum value it can possibly attain, we see that 푉5푛 −푉푚 is asymptotically at most Σ3, whereas +푉푚 is asymptotically at most +� +log 푛/푛(Σ1 + Σ2) (both up to a multiplicative constant). Both of these are +negligible compared to |푅8푛|. +□ +7. Square discriminants in other degrees +In this section, we discuss Littlewood polynomials with square discriminant in degree 푛 � 0, 6 mod 8. +The following surprising result, attributed to Alexei Entin in [3, §4], shows that such polynomials do not +even exist in even degree 푛 ≡ 2, 4 mod 8. +Lemma 7.1 (Entin). Let 푛 ≡ 2, 4 mod 8 be a positive integer. Then no Littlewood polynomial of degree +푛 has square discriminant. +Proof. Suppose that 푛 is even and 푓 ∈ F푛. Set 푝푛(푋) = (푋푛+1 − 1)/(푋 − 1) and note that 푓 and +푝푛 coincide modulo 2. Since 푋푛+1 − 1 and its derivative are coprime modulo 2, the polynomial 푝푛 is +separable over F2. Thus 푝푛 is separable over the 2-adic field Q2 as well by Hensel’s lemma. The splitting +field of 푝푛 over Q2, which is the cyclotomic extension Q2(휁)/Q2 where 휁 is a primitive 푛 + 1-th root of +unity, is an unramified extension of Q2 because 2 and 푛 + 1 are coprime, see [15, Proposition II.7.12]. +Writing 퐺( 푓 /퐾) for the Galois group of 푓 over a field 퐾, this implies that 퐺(푝푛/Q2) is isomorphic to +퐺(푝푛/F2) = 퐺( 푓 /F2) ⩽ 퐺( 푓 /Q). The discriminant of 푝푛 is a square in Z2 if and only if it is 1 mod 8. +A resultant calculation shows that Δ(푝푛) = (−1) +푛(푛−1) +2 +(푛 + 1)푛−1, which is congruent to 5 mod 8 if +푛 ≡ 2, 4 mod 8 (and congruent to 1 mod 8 otherwise). Therefore 푓 cannot have square discriminant over +Q. +□ +In the case of odd-degree Littlewood polynomials, the situation is different. +Call a degree-푛 +polynomial 푓 nearly reciprocal if it is of the form 푓 (푋) = ±푋푛 푓 (푋−1), nearly skew-reciprocal if +푓 (푋) = ±푋푛 푓 (−푋−1). We give some examples: +• Littlewood polynomials with vanishing square discriminant exist in any odd degree. Indeed, +the nearly reciprocal polynomial given by +(푋푛+1 − 1)(푋푛 + 푋푛−1 + · · · + 푋 + 1) = (푋 − 1)(푋푛 + 푋푛−1 + · · · + 푋 + 1)2 ∈ F2푛+1 +has a multiple factor and thus its discriminant vanishes. +• An odd-degree Littlewood polynomial with vanishing square discriminant is not necessarily +nearly (skew-)reciprocal, or the product of such. Indeed, the polynomial +(푋 + 1)2(푋2 − 푋 + 1)(푋7 − 푋5 + 푋4 − 푋3 + 푋2 + 1) +has vanishing discriminant, but the Galois group of its splitting field is 퐶2 × 푆7. +• A computer experiment shows that all Littlewood polynomials of odd degree ⩽ 29 with nonva- +nishing square discriminant have a cyclotomic factor; in fact, each such polynomial is divisible +by 푋 + 1 or 푋 − 1. Does there exist an odd-degree Littlewood polynomial without cyclotomic +factors that has square discriminant? (If not, this would imply for example that no irreducible +Littlewood polynomial of odd degree 푛 has Galois group contained in 퐴푛.) +A related question, raised by Peled, Sen and Zeitouni [20, §7], is whether Littlewood polyno- +mials with a repeated non-cyclotomic factor exist. This was recently answered in the affirmative +in response to a question on MathOverflow [24]: Peter Taylor found the polynomial +(푋18 + 푋16 + 2푋15 + 2푋13 + 푋12 + 2푋11 + 3푋10 + 3푋8 + 2푋7 + 푋6 + 2푋5 + 2푋3 + 1) +× (푋2 + 1)(푋 − 1)(푋3 + 푋2 − 1)2 +along with three other examples. +References +1. L. Bary-Soroker, O. Ben-Porath, and V. Matei, Probabilistic Galois Theory – The Square Discriminant Case, preprint +arXiv:2207.12493, 15 pp., 2022. +2. L. Bary-Soroker, D. Koukoulopoulos, and G. Kozma, Irreducibility of random polynomials: general measures, preprint +arXiv:2007.14567, 64 pp., 2020. +3. L. Bary-Soroker and G. Kozma, Irreducible polynomials of bounded height, Duke Math. J. 169 (2020), 579–598. +16 + +4. M. Benito and J. L. Varona, Pythagorean triangles with legs less than 푛, J. Comput. Appl. Math. 143 (2002), 117–126. +5. M. Bhargava, Galois groups of random integer polynomials and Van der Waerden’s Conjecture, preprint arXiv:2111.06507, +33 pp., 2021. +6. C. Borst, E. Boyd, C. Brekken, S. Solberg, M. M. Wood, and P. M. Wood, Irreducibility of random polynomials, Exp. +Math. 27 (2018), 498–506. +7. E. Breuillard and P. Varjú, Irreducibility of random polynomials of large degree, Acta Math. 223 (2019), 195–249. +8. R. Chela, Reducible polynomials, J. London Math. Soc. 38 (1963), 183–188.‘ +9. A. Dubickas, Salem numbers as Mahler measures of nonreciprocal units, Acta Arith. 176 (2016), 81–88. +10. T. Erdélyi, Do Flat Skew-Reciprocal Littlewood Polynomials Exist?, Constr. Approx. 56 (2022), 537–554. +11. S. V. Konyagin, On the number of irreducible polynomials with 0, 1 coefficients, Acta Arith. 88 (1999), 333–350. +12. E. Krätzel, Lattice points, Mathematics and its Applications (East European Series) 33, Kluwer Academic Publishers +Group, Dordrecht, 1988. +13. J. E. Littlewood, On polynomials �푛 ±푧푚, �푛 푒훼푚푖푧푚, 푧 = 푒휃푖, J. London Math. Soc. 41 (1966), 367–276. +14. H. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced +Mathematics 97, Cambridge University Press, Cambridge, 2006. +15. J. Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften 322, Springer-Verlag, Berlin, +1999. +16. W. G. Nowak and W. Recknagel, The distribution of Pythagorean triples and a three-dimensional divisor problem, Math. +J. Okayama Univ. 31 (1989), 213–220. +17. J. E. Nymann, On the probability that 푘 positive integers are relatively prime II, J. Number Theory 7 (1975), 406–412. +18. A. Odlyzko, Search for ultraflat polynomials with plus and minus one coefficients, in: Connections in Discrete Mathematics, +Cambridge Univ. Press, Cambridge, 2018, pp. 39–55. +19. S. O’Rourke and P. M. Wood, Low-degree factors of random polynomials, J. Theoret. Probab. 32 (2019), 1076–1104. +20. R. Peled, A. Sen and O. Zeitouni, Double roots of random Littlewood polynomials, Israel J. Math. 213 (2016), 55–77. +21. W. Sierpiński, O sumowaniu szeregu �푛⩽푏 +푛>푎 휏(푛) 푓 (푛), gdzie 휏(푛) oznacza liczbę rozkładów liczby 푛 na sumę kwadratów +dwóch liczb całkowitych [Polish; On the summation of the series �푛⩽푏 +푛>푎 휏(푛) 푓 (푛), where 휏(푛) denotes the number of ways +to write 푛 as the sum of squares of two integers], Prace Mat.-Fiz. 18 (1907), 1–59. French in: Oeuvres choisies, Tome I, +PWN—Éditions Scientifiques de Pologne, Warsaw, 1974, 109–154. +22. J. Spencer with L. Florescu, Asymptopia, Student Mathematical Library 71, American Mathematical Society, Providence, +RI, 2014. +23. M. I. Stronina, Integral points on circular cones, Izv. Vysš. Učebn. Zaved. Matematika 8 (1969), 112-116. +24. P. Taylor, Answer to question “Multiple roots of polynomials with coefficients ±1”. Question posted by user Taras Banakh +on MathOverflow, https://mathoverflow.net/questions/424408/, 2022. +25. P. Viana and P. M. Veloso, Galois theory of reciprocal polynomials, Amer. Math. Monthly 109 (2002), 466-471. +26. B. L. van der Waerden, Die Seltenheit der Gleichungen mit Affekt, Math. Ann. 109 (1934), 13–16. +27. B. L. van der Waerden, Die Seltenheit der reduziblen Gleichungen und der Gleichungen mit Affekt, Monatsh. Math. Phys. +43 (1936), 133–147. +Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland +Email address: d.p.t.hokken@uu.nl +17 + diff --git a/ndE5T4oBgHgl3EQfjg9a/content/tmp_files/load_file.txt b/ndE5T4oBgHgl3EQfjg9a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bdf5cbf364dda2c7248bb84186106ee2a91f071e --- /dev/null +++ b/ndE5T4oBgHgl3EQfjg9a/content/tmp_files/load_file.txt @@ -0,0 +1,807 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf,len=806 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='05656v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='NT] 13 Jan 2023 COUNTING (SKEW-)RECIPROCAL LITTLEWOOD POLYNOMIALS WITH SQUARE DISCRIMINANT DAVID HOKKEN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood polynomials with square discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This relates to a bounded-height analogue of the Van der Waerden conjecture on Galois groups of random polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Introduction Background and main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Let 푓 be a monic polynomial of degree 푛 with integer coefficients that are at most 퐻 in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In 1934, Van der Waerden [26] presented an elementary proof that 푓 is almost surely ohne Affekt: the Galois group 퐺 푓 of 푓 over Q is the symmetric group 푆푛 with probability tending to 1 as 퐻 goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Two years later, he posed a conjecture [27, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 139] on the probability that 푓 does not have maximal Galois group, which states Prob(퐺 푓 ≠ 푆푛) ∼ Prob( 푓 is reducible) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) as 퐻 goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Last year, Bhargava [5] established the breakthrough result Prob(퐺 푓 ≠ 푆푛) ∼ Prob( 푓 is reducible) + Prob(퐺 푓 = 퐴푛) ≍ 퐻−1 where 퐴푛 denotes the alternating group on 푛 letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This is a weak form of the Van der Waerden conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Since 푓 is reducible with probability ≍ 퐻−1 if 푛 > 2 (see [26, 8]), the remaining task to obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) consists of showing that Prob(퐺 푓 = 퐴푛) = 표(퐻−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Bary-Soroker, Ben-Porath and Matei [1] conjecture the much stronger bound Prob(퐺 푓 = 퐴푛) = 푂(퐻−푛/2+휖 ) when 푛 ⩾ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The height 퐻 of the polynomial 푓 in the above setup tends to infinity, whereas the degree 푛 stays fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This approach to random polynomials is called the large box model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In the restricted coefficient model, the height 퐻 — or any specific set N of coefficients of 푓 — is fixed, and it is the degree that tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Recent years have seen a surge of interest in questions about Galois groups in this setting as well [2, 3, 6, 7, 11, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For example, if N consists of 35 consecutive integers, Bary-Soroker, Koukoulopoulos and Kozma [2] show that 퐺 푓 is 푆푛 or the alternating group 퐴푛 with probability tending to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Conditionally on the Riemann Hypothesis for a family of Dedekind zeta functions, Breuillard and Varjú [7] show a similar result for more general distributions of the coefficients of 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Typically, probabilistic methods are used to establish high transitivity of 퐺 푓 by reducing 푓 modulo various primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This leaves only 퐴푛 and 푆푛 as possible Galois groups, but as these are respectively (푛 − 2)- and 푛- transitive, it is hard to distinguish them based on this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In other words, the alternating group has a special role in the restricted coefficient model as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Generally, it is believed that 퐴푛 should occur with probability tending to 0 as 푛 tends to infinity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' A property that distinguishes 퐴푛 from 푆푛 as Galois group 퐺 푓 of a separable polynomial 푓 is the following: 퐺 푓 is contained in 퐴푛 if and only if the discriminant Δ( 푓 ) of 푓 is a (necessarily nonzero) square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Hence Prob(퐺 푓 = 퐴푛) ⩽ Prob(Δ( 푓 ) = □ ≠ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Primary: 11C08, 11R32, 11R09, 05A16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Secondary: 11P21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Littlewood polynomials, square discriminant, Galois theory, asymptotic enumeration, lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Many thanks to Gunther Cornelissen, Mar Curcó Iranzo, and Berend Ringeling for helpful conversations and feedback on earlier versions of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This publication is part of the project Littlewood polynomials with square discriminant (OCENW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='M20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='233), financed by the Dutch Research Council (NWO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 1 This paper studies the probability that the discriminant of the monic polynomial 푓 is a square when the coefficients of 푓 are independently and uniformly selected from {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Such polynomials are often called Littlewood polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' These are extremal examples of polynomials with restricted coefficients: all Littlewood polynomials in degree 푛 coincide over F2, whereas they form a sparse (that is, exponentially small in 푛) subset of the degree-푛 monic polynomials in F푝[푋] for any prime 푝 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Furthermore, since they are of height 1, the results mentioned in the first paragraph cannot be made effective in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The state-of-the-art result concerning the Galois theory of random Littlewood polynomials is that at least a fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='00068 of the Littlewood polynomials of degree 푛, with 푛 ⩾ 10104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='9, is irreducible (see [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Following Littlewood [13], denote the collection of Littlewood polynomials of degree 푛 by F푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' let Sq푛 ⊂ F푛 consist of those with square discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Furthermore, call 푓 reciprocal if 푓 (푋) = 푋푛 푓 (푋−1) and skew-reciprocal if 푓 (푋) = (−1)푛(푛−1)/2푋푛 푓 (−푋−1) (the latter appear e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' in [18, 10] in connection to questions about the flatness of Littlewood polynomials on the unit circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Denote by 푅푛, 푆푛 ⊂ F푛 the sets of Littlewood polynomials of degree 푛 that have square discriminant and are reciprocal, respectively skew-reciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Our main result concerns the size of 푅푛 and 푆푛 as 푛 tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The sets 푅8푛, 푆8푛, 푅8푛−2, and 푆8푛−2 are all of size ≍ 16푛 log 푛/√푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' More precisely: (a) lim 푛→∞ |푅8푛| 16푛 log 푛/√푛 = Γ( 1 4)2 4 √ 2휋3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='0749 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' , lim 푛→∞ |푆8푛| 16푛 log 푛/√푛 = 1 2휋3/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='0897 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (b) |푅8푛−2| ∼ 1 2|푅8푛| and |푆8푛−2| ∼ 1 2|푆8푛|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The limits in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 are approached extremely slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For example, when 푛 = 1011, the fraction |푅8푛|/(16푛 log 푛/√푛) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='099 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='. This is (at least in part) due to large contributions of order ≍ 16푛/√푛 to |푅8푛| and |푆8푛| coming from error terms in lattice point counts that we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As observed in [3, §4], any 푓 ∈ F2푛 of even degree is separable, because 푓 coincides modulo 2 with the separable polynomial (푋2푛+1 − 1)/(푋 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Furthermore, the roots of a reciprocal polynomial 푓 come in pairs {훼, 훼−1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' if 푓 is skew-reciprocal, they come in pairs {훼, −훼−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The separability of 푓 implies that 훼 and ±훼−1 are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As a result, the Galois group of 푓 is contained in the wreath product 퐶2 ≀ 푆푛/2, see [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Recall that the wreath product of two groups 퐺 and 퐻 ⩽ 푆푛, denoted 퐺 ≀ 퐻, is the semidirect product 퐺푛 ⋊ 퐻 where 퐻 acts on the 푛 copies of 퐺 by permuting the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 therefore leads to the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Let 푓 be a (skew-)reciprocal Littlewood polynomial of degree 푛 ≡ 0, 6 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As 푛 → ∞, we have Prob(Δ( 푓 ) = □ ≠ 0) ≍ log 푛 √푛 and Prob(퐺 푓 ⩽ (퐶2 ≀ 푆푛/2) ∩ 퐴푛) ≫ log 푛 √푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The set Sq푛 is empty whenever 푛 ≡ 2, 4 mod 8, which is the reason to leave out these degrees in the above statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In §7, we expound the proof sketch for this fact provided in [3, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In the same section we also make some remarks on the case of odd 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Reciprocals and skew-reciprocals are decomposable: a polynomial 푓 is reciprocal if it is of the form 푓 (푋) = 푋푛/2푔(푋 + 푋−1), and skew-reciprocal if it is of the form 푓 (푋) = 푋푛/2푔(푋 − 푋−1) for some polynomial 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The group (퐶2 ≀ 푆푛/2) ∩ 퐴푛 is much smaller than 퐴푛 — of index 1 · 3 · 5 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' · (푛 − 1) to be precise — and the sizes of 푅푛 and 푆푛 compared to |F푛| = 2푛 decrease exponentially in 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Nevertheless, back in the large box model, the best known bound on the probability that the discriminant of 푓 is a square also come from decomposable polynomials with the very same Galois group: Bary-Soroker, Ben-Porath and Matei [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3] show for all even 푛 ⩾ 6 that Prob(Δ( 푓 ) = □) ≫ 퐻−(푛+1)/2 2 as 퐻 tends to infinity by applying an explicit version of Hilbert’s irreducibility theorem to polynomials of the form 푓 (푋) = 푔(푋2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' No Littlewood polynomial of the form 푓 (푋) = 푔(푋2) exists, and it appears that (skew-)reciprocal polynomials are ‘the next best thing’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In the setting of Littlewood polynomials, reducing modulo primes or applying probabilistic methods seems difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Instead, the proofs in this paper combine counting arguments to derive explicit combinatorial expressions for the objects of study with lattice point counts in certain geometric regions and asymptotics of binomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In §2 we derive combinatorial expressions for |푅8푛| and the three other sets under consideration, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In each case, we obtain a sum that extends over certain tuples related to Pythagorean triples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' these come from a square discriminant criterion for (skew-)reciprocal polynomials given in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This criterion can in theory be used to find similar expressions when Littlewood polynomials are replaced by polynomials with coefficients in any fixed set N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Auxiliary results to study the asymptotics of these combinatorial expressions, as well as an analysis of the Pythagorean triples, are contained in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The latter essentially boils down to counting lattice points with parity and coprimality conditions inside elliptic (for the reciprocals) or parabolic (for the skew-reciprocals) hyperboloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' These results are then combined in §4 and §5, where the lattice point regions are split into three suitably chosen parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This makes it possible to evaluate the combinatorial expressions from §2 asymptotically by using integral estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 is finally given in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We end with some observations about the set Sq푛 in the case 푛 � 0, 6 mod 8 in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The expression 푓 ≪ 푔 as well as 푔 ≫ 푓 and 푓 = 푂(푔) all mean there exists a positive constant 퐶 such that 푓 (푛) ⩽ 퐶푔(푛) for all sufficiently large values of 푛 (all asymptotics in this paper will be in 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The notation 푓 ≍ 푔 is shorthand for 푔 ≪ 푓 ≪ 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The functions 푓 and 푔 are said to be asymptotically equal, denoted 푓 ∼ 푔, if the fraction 푓 (푛)/푔(푛) tends to 1 as 푛 tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In particular, 푓 ∼ 푔 implies 푓 ≍ 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lastly, the notation 푓 = 표(푔) is used when the fraction 푓 (푛)/푔(푛) tends to 0 as 푛 tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We write 푖 for an index and i ∈ C for the imaginary unit and adopt the convention that �푛 푘 � = 0 if 푘 > 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' A counting proposition In this section, we prove the following expression for |푅8푛| in terms of binomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The number of reciprocal Littlewood polynomials of degree 8푛 with (nonvanishing) square discriminant equals |푅8푛| = 22푛 �2푛 푛 � + 2 � � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 1 4 (푘푟2 + 푘푠2 + (−1) 푘+1 2 ) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) where the sum extends over all tuples (푘, 푟, 푠) such that 푘 > 0 is odd and 푟 > 푠 > 0 are coprime and of opposite parity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=', 푟 is odd if and only if 푠 is even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Similar expressions for |푅8푛−2|, |푆8푛| and |푆8푛−2| are given in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The first term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) is ≍ 16푛/√푛 as a consequence of the well-known asymptotic expression �2푛 푛 � ∼ 4푛/√휋푛 for the central binomial coefficient [22, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 claims that this falls short by a factor logarithmic in 푛 of the true growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 is based on the following square discriminant criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Let 푓 ∈ Q[푋] be a separable polynomial of degree 2푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Suppose 푓 is reciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Then the discriminant of 푓 is a square if and only if (−1)푛 푓 (1) 푓 (−1) is a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Similarly, if 푓 is skew-reciprocal, then its discriminant is a square if and only if the integer 푓 (i) 푓 (−i) is a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In the case of reciprocal polynomials, this criterion is well-known and recorded in the literature in several places, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' With a similar proof, here we show the criterion for skew-reciprocals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Write 푎푛 for the leading coefficient of 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' If 푓 is not monic, then Δ( 푓 ) = 푎2푛−2 푛 Δ( 푓 /푎푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Since 푎2푛−2 푛 is a square, we may assume without loss of generality that 푓 is in fact monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Since 푓 is separable, it has 3 2푛 distinct roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' These come in pairs 훼푖, 훼푛+푖 = −훼−1 푖 for 푖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' , 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Hence Δ( 푓 ) = � 1⩽푖< 푗⩽푛 � (훼푖 − 훼 푗)(훼푖 + 훼−1 푗 )(−훼−1 푖 + 훼−1 푗 )(−훼−1 푖 − 훼 푗) �2 � 1⩽ 푗⩽푛 (훼 푗 + 훼−1 푗 )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The first of the two products above is the square of an integer, since � 1⩽푖< 푗⩽푛 (훼푖 − 훼 푗)(훼푖 + 훼−1 푗 )(−훼−1 푖 + 훼−1 푗 )(−훼−1 푖 − 훼 푗) = � 1⩽푖< 푗⩽푛 −(훼푖 − 훼−1 푖 − 훼 푗 + 훼−1 푗 )2 is a symmetric expression in the roots of 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The other product can be expanded as � 1⩽ 푗⩽푛 (훼 푗 + 훼−1 푗 )2 = � 1⩽ 푗⩽푛 (i + 훼 푗)(i + 훼−1 푗 )(i − 훼 푗)(i − 훼−1 푗 ) = 푓 (i) 푓 (−i) as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ To count (skew-)reciprocal polynomials with square discriminant, we recall that any polynomial 푓 can be written as the sum 푓 (푋) = 푓e(푋2) + 푋 푓o(푋2) of its even and odd parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Therefore 푓 (1) 푓 (−1) = ( 푓e(1) + 푓o(1))( 푓e(1) − 푓o(1)) = 푓e(1)2 − 푓o(1)2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) and 푓 (i) 푓 (−i) = ( 푓e(i2) + i 푓o(i2))( 푓e((−i)2) − i 푓o((−i)2)) = 푓e(−1)2 + 푓o(−1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) If 푓 is a Littlewood polynomial and we want these expressions to be squares (or minus a square – see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2), we can count the possible choices of coefficients of 푓e and 푓o giving rise to (possibly degenerate) Pythagorean triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This is key in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Consider a not-necessarily monic reciprocal Littlewood polynomial 푓 = 푎4푛푋8푛 + · · · + 푎1푋4푛+1 + 푎0푋4푛 + 푎1푋4푛−1 + · · · + 푎4푛−1푋 + 푎4푛 of degree 8푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' since 푓 has square discriminant if and only if − 푓 has square discriminant, we must divide by 2 whatever final expression we obtain to establish the count of monic reciprocal Littlewood polynomials with square discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Set 푐 := 푓e(1) = 푎0 + 2(푎2 + 푎4 + · · · + 푎4푛), 푏 := 푓o(1) = 2(푎1 + 푎3 + · · · + 푎4푛−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2), we need to pick the 푎푖 such that 푐2 − 푏2 is a square, say equal to 푎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In the �2푛 푛 � cases that exactly half of the odd-index coefficients 푎1, 푎3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' , 푎4푛−1 are equal to 1 and thus 푏 = 0, we find that any choice of the coefficients 푎0, 푎2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' , 푎4푛 will make 푓 a Littlewood polynomial with square discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' There are in total 22푛+1�2푛 푛 � such polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' After dividing by two, this is the first term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Now suppose 푏 is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Recall that if 푎2 + 푏2 = 푐2 is a Pythagorean triple and 푎, 푏 and 푐 are positive, then there are unique positive integers 푘, 푟 and 푠 such that 푐 = 푘(푟2 + 푠2), 푏 = 2푘푟푠, and 푎 = 푘(푟2 − 푠2), and 푟 > 푠 and the numbers 푟 and 푠 are coprime and of opposite parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Since 푐 is odd by definition, we must add the condition that 푘 be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This gives the summation condition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The prefactor of 2 before the sum arises because we treat each of the four triples (푎, ±푏, ±푐) separately – we care if 푐2 − 푏2 is a square, so the sign of 푎 doesn’t matter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' but the polynomials corresponding to the four tuples (±푏, ±푐) are genuinely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We conclude that the final expression must be multiplied by 4/2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' It remains to show that the second summand in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' That is, we must count all choices of the 푎푖 that lead to the equalities 푐 = 푘(푟2 + 푠2) and 푏 = 2푘푟푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Notice that 푎2 + 푎4 + · · · + 푎4푛 = 푐 − 푎0 2 = 푘(푟2 + 푠2) − 푎0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4) 4 Since all 푎푖 lie in {±1}, the left-hand side in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As 푟2+푠2 ≡ 1 mod 4, we find that 푎0 ≡ 푘 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Hence 푎0 = (−1) 푘−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Therefore a total of 푛 + (푘(푟2 + 푠2) + (−1) 푘+1 2 )/4 of the even-index coefficients 푎2, 푎4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' , 푎4푛 must be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This yields � 2푛 푛 + 1 4 (푘푟2 + 푘푠2 + (−1) 푘+1 2 ) � options for the even-index coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Similarly, there are 2푛 choices to be made for the odd-index coefficients 푎1, 푎3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' , 푎4푛−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' since the sum of the latter equals 푏/2 = 푘푟푠, we find that 푛 + 푘푟푠/2 of the odd-index coefficients must be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' So we have in total � 2푛 푛+푘푟푠/2 � options for the odd-index coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This gives � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 1 4 (푘푟2 + 푘푠2 + (−1) 푘+1 2 ) � combinations in total, which is the summand in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ It is clear that the proof method can in principle be applied to derive a combinatorial expression for the number of square-discriminant (skew-)reciprocal polynomials of given degree with coefficients in any fixed set N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For |푅8푛−2|, |푆8푛| and |푆8푛−2|, we obtain the following expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We have |푅8푛−2| = 22푛−1 �2푛 푛 � + 2 � � 2푛 푛 + 1 2 푘푟푠 � � 2푛 − 1 푛 + 1 4 (푘푟2 + 푘푠2 + (−1) 푘−1 2 − 2) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5) |푆8푛| = 22푛 �2푛 푛 � + 2 � � 2푛 푛 + 1 2 푘푟푠 �� 2푛 푛 + 1 4 (푘푟2 − 푘푠2 + (−1) 푘+1 2 +푠) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='6) |푆8푛−2| = 22푛−1 �2푛 푛 � + 2 � � 2푛 푛 + 1 2 푘푟푠 � � 2푛 − 1 푛 + 1 4 (푘푟2 − 푘푠2 + (−1) 푘−1 2 +푠 − 2) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7) where in each case the sum extends over all tuples (푘, 푟, 푠) such that 푘 > 0 is odd and 푟 > 푠 > 0 are coprime and of opposite parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Auxiliary results In this section, we gather lemmas that we will use to evaluate the binomial sums derived in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' General.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In the next proposition, we record useful information about the asymptotics of binomial coefficients that are (almost) central, see [22, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We have �2푛 푛 � ∼ 4푛/√휋푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Furthermore, if 푘 is of order 표(푛2/3), then � 2푛 푛 + 푘 � ∼ �2푛 푛 � 푒− 푘2 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The following corollary of [17, Theorem 2] is a variant of Möbius inversion that we will use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Let 퐹 and 푓 be real-valued functions defined on R⩾1 and related through 퐹(푛) = � 1⩽푑⩽푛 푑 odd 푓 (푛/푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (More explicitly, the sum extends over all odd integers between 1 and 푛, not just the odd divisors of 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=') Then 푓 (푛) = � 1⩽푑⩽푛 푑 odd 휇(푑)퐹(푛/푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 5 Reciprocals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We define the circle sector 퐶휃 (푛) for 0 ⩽ 휃 ⩽ 휋/4 as 퐶휃 (푛) = {(푥, 푦) ∈ R2 >0 | 푥2 + 푦2 < 푛2, 푦 ⩽ 푥 tan(휃)}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' the part of the circle of radius 푛 centered at the origin in R2 that lies in the upper-right quadrant and is bounded by 푦 = 0 and 푦 = 푥 tan(휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Denote by 퐹휃 (푛) the number of integral, opposite-parity lattice points in the circle sector 퐶휃 (푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Let 푓휃 (푛) be the number of such that are also coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Then 푓휃 (푛) = � 1⩽푑⩽푛 푑 odd 휇(푑)퐹휃 (푛/푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' If (푥, 푦) is an integral, opposite-parity lattice point in the circle sector 퐶휃 (푛) with greatest common divisor 푑, then (푥/푑, 푦/푑) is a primitive, integral, opposite-parity lattice point in the circle sector 퐶휃 (푛/푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The opposite holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Noting that a pair of opposite-parity integers that are both at most 푛 must have odd greatest common divisor at most 푛, we find 퐹휃 (푛) = � 1⩽푑⩽푛 푑 odd 푓휃 (푛/푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2 gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ The next lemma shows that 퐹휃 (푛) and 푓휃 (푛) are linear in 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This is a key result that helps to overcome the main difficulty in evaluating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1), see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The following asymptotics for 퐹휃 and 푓휃 hold as 푛 goes to infinity: (a) 퐹휃 (푛) ∼ 휃푛2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (b) 푓휃 (푛) ∼ 2휃푛2/휋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Part (b) follows after combining part (a) with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3 and � 푑⩾1 푑 odd 휇(푑) 푑2 = � 푑⩾1 휇(푑) 푑2 − � 푑⩾1 푑 even 휇(푑) 푑2 = � 푑⩾1 휇(푑) 푑2 − � 푑⩾1 휇(2푑) 4푑2 = � 푑⩾1 휇(푑) 푑2 + 1 4 � 푑⩾1 푑 odd 휇(푑) 푑2 by multiplicativity of the Möbius function, so that � 푑⩾1 푑 odd 휇(푑) 푑2 = 4 3 � 푑⩾1 휇(푑) 푑2 = 8 휋2 , see [14, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For part (a), we start by distributing the lattice points in 퐶휃 (푛) over four subsets depending on the parity of each of the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Denote by 퐹00 the number of lattice points in 퐶휃 (푛) whose coordinates are both even, by 퐹01 the number of those whose 푥-coordinate is even and 푦-coordinate is odd, and similarly for 퐹10 and 퐹11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For each even number 푥0, the number of lattice points (푥0, 푦) with odd 푦 exceeds those with even 푦 by at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Since 푥0 lies between 1 and 푛, we find that 퐹00 + 푛/2 ⩾ 퐹01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Similarly, we deduce 퐹10 + (푛+1)/2 ⩾ 퐹11 and 퐹00 +푛 sin(휃)/2 ⩾ 퐹10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Therefore the difference between any two of the sets 퐹00, 퐹01, 퐹10, and 퐹11 is of order 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' On the other hand, the quantity 퐹00 equals the number of total lattice points in 퐶휃 (푛/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This is asymptotically equal to the area of 퐶휃 (푛/2), which is 휃푛2/8, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' [12, Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As 퐹00, 퐹01, 퐹10, and 퐹11 differ by a term of order 푛 at most, they are equal asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Therefore 퐹휃 (푛) = 퐹01 + 퐹10 ∼ 휃푛2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ Consider the set 퐵푛 = � (푘, 푟, 푠) ∈ Z3 ���� 푘 > 0 and odd, 푟 > 푠 > 0 coprime and of opposite parity, and 푘(푟2 + 푠2) ⩽ 푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Since the inequality 푘(푟2 + 푠2) ⩾ 2푘푟푠 holds for all positive integers 푘, 푟 and 푠, the set 퐵5푛 certainly contains all tuples (푘, 푟, 푠) over which the sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) extends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The next lemma concerns the asymptotic size of 퐵푛 and thus shows how the number of summands in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) depends on 푛 as 푛 tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 6 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The set 퐵푛 is of size asymptotically equal to 1 4휋 푛 log 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Take 휃 = 휋/4 in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Then |퐵푛| = � 1⩽푑⩽푛 푑 odd 푓휃 ��푛 푑 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) Calculating the sum up to 푑 = 푛+ := 푛/log log 푛, we find � 1⩽푑⩽푛+ 푑 odd 푓휃 �� 푛 푑 � ∼ 2휃푛 휋2 � 1⩽푑⩽푛+ 푑 odd 1 푑 ∼ 휃푛 log 푛 휋2 = 푛 log 푛 4휋 where the last asymptotic equality follows since log(푛/log log 푛) ∼ log 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This gives the claimed asymptotic size of 퐵푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The remaining terms of the sum in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1), where 푑 > 푛/log log 푛, are bounded by � 푛+<푑⩽푛 푑 odd 푓휃 ��푛 푑 � < 푛 푓휃 �� 푛 푛/log log 푛 � ∼ 푛 log log 푛 2휋 and thus do not contribute to the asymptotic size of 퐵푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ Let (푎, 푏, 푐) be a Pythagorean triple, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 푎2 + 푏2 = 푐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='8 implies that the number of Pythagorean triples with hypotenuse less than 푛 and of opposite parity (considering the triples (푎, 푏, 푐) and (푏, 푎, 푐) to be the same) is asymptotic to 1 4휋 푛 log 푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' an error term of the form 푐0푛 for an explicit constant 푐0 appears to have been given first by Sierpiński in [21, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (7)], with further improvements due to Stronina [23] and Nowak & Recknagel [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Skew-reciprocals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Here, we establish results analogous to those in §3, but for skew-reciprocal polyno- mials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The tuples (푘, 푟, 푠) over which the sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='6) extends are contained in 퐷5푛, where 퐷푛 = � (푘, 푟, 푠) ∈ Z3 ���� 푘 > 0 and odd, 푟 > 푠 > 0 coprime and of opposite parity, and 푘(푟2 − 푠2) ⩽ 푛 and 2푘푟푠 ⩽ 푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Neither of 푘(푟2 − 푠2) and 2푘푟푠 dominates the other for every choice of positive integers 푘, 푟 and 푠 with 푟 > 푠;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' indeed, the inequality 푘(푟2 − 푠2) > 2푘푟푠 holds if and only if ( √ 2 − 1)푟 > 푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Hence both of the inequalities (푟2 − 푠2) ⩽ 푛 and 2푘푟푠 ⩽ 푛 are required in the definition of 퐷푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Set 훼 = artanh( √ 2 − 1) = log � 1 + √ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' this is the inverse hyperbolic tangent of the angle between the 푟-axis and the line from the origin to the intersection point of the hyperbolas 푟2 − 푠2 = 푛 and 2푟푠 = 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Define the hyperbolic sectors 퐻휃 (푛) and 퐻∗ 휃 (푛) for 0 < 휃 ⩽ 훼 as 퐻휃 (푛) = {(푥1, 푦1) ∈ R2 >0 | 푥2 1 − 푦2 1 < 푛2, 푦1 ⩽ tanh(휃)푥1}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) 퐻∗ 휃 (푛) = {(푥2, 푦2) ∈ R2 >0 | 2푥2푦2 < 푛2, 푦2 < 푥2 ⩽ 푒2휃 푦2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) Note that 푒2휃 = (1 + tanh(휃))/(1 − tanh(휃)), and that both tanh(훼) and 푒2훼 are equal to √ 2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As in the reciprocal case, the reason to consider these sectors is that their areas scale linearly in 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The areas of 퐻휃 (푛) and of 퐻∗ 휃 (푛) are each equal to 휃푛2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The linear transformation sending 푥2 ↦→ (푥1 + 푦1)/ √ 2 and 푦2 ↦→ (푥1 − 푦1)/ √ 2 maps 퐻∗ 휃 (푛) to 퐻휃 (푛) and has determinant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Thus 퐻∗ 휃 (푛) and 퐻휃 (푛) have equal area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The area of 퐻휃 (푛) is 푛2 times as large as that of the region bounded by the hyperbola 푥2 1 − 푦2 1 = 1, the axis 푦1 = 0, and the ray through the origin and the point (cosh(휃), sinh(휃)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' But that is simply 휃/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ We summarise the analogues of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4 in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Denote by 퐺 휃 (푛) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 퐺∗ 휃 (푛)) the number of integral, opposite-parity lattice points in 퐻휃 (푛) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 퐻∗ 휃 (푛)), and by 푔휃 (푛) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 푔∗ 휃 (푛)) the number of such that are also coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Then the following hold: (a) 푔(푛) = � 휇(푑)퐺(푛/푑) where the sum extends over all odd 1 ⩽ 푑 ⩽ 푛, and similarly for 푔∗ 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 7 (b) 퐺 휃 (푛) ∼ 퐺∗ 휃 (푛) ∼ 휃푛2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (c) 푔휃 (푛) ∼ 푔∗ 휃 (푛) ∼ 2휃푛2/휋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' All proofs are analogous to those of the mentioned lemmas, where 퐻휃 (푛) (respectively 퐻∗ 휃 (푛)) plays the role of 퐶휃 (푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' That the asymptotic expressions for 푓휃, 푔휃, and 푔∗ 휃 are all equal comes from the fact that the circle sector 퐶휃 and the hyperbolic sectors 퐻휃 and 퐻∗ 휃 all have equal area, see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ The next result is the analogue of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The set 퐷푛 is of size asymptotically equal to 2훼 휋2 푛 log 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Note that |퐷푛| = � 1⩽푑⩽푛 푑 odd 푔훼 ��푛 푑 � + 푔∗ 훼 ��푛 푑 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4) Writing 푛+ = 푛/log log 푛 and reasoning as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5 that the terms in the sum with 푑 > 푛+ do not contribute, we find with help of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7(c) that |퐷푛| ∼ � 1⩽푑⩽푛+ 푑 odd 푔훼 ��푛 푑 � + 푔∗ 훼 ��푛 푑 � ∼ 4훼푛 휋2 � 1⩽푑⩽푛+ 푑 odd 1 푑 ∼ 2훼푛 log 푛 휋2 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ Let (푎, 푏, 푐) be a Pythagorean triple, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 푎2 + 푏2 = 푐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='8 implies that the number of Pythagorean triples with legs less than 푛 and of opposite parity (considering the triples (푎, 푏, 푐) and (푏, 푎, 푐) to be the same) is asymptotic to 2훼 휋2 푛 log 푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' see [4, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 2] for an improved error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The reciprocals In this section, we build up towards the proof of the part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 that concerns reciprocals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For the proof, we break up the sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) into several pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Fix a (large) integer 푁 and set 휖 = 푁−1 and 푚 = 5 � 푛 log 푛 (the number 5 is a convenient choice, but could be replaced by any real number greater than 2 √ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Write Σ1 = � (푘,푟,푠)∈퐵푁√푛 � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 1 4 (푘푟2 + 푘푠2 + (−1) 푘+1 2 ) � , Σ2 = � (푘,푟,푠)∈퐵푚\\퐵푁√푛 � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 1 4 (푘푟2 + 푘푠2 + (−1) 푘+1 2 ) � , and define Σ3 through 2Σ3 = |푅8푛| − 22푛�2푛 푛 � − 2Σ1 − 2Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Figure 1 shows how the domain consisting of lattice points over which the sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) extends is divided into parts associated with the sums Σ1, Σ2 and Σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The following subsections go into the asymptotics of each of these terms, showing that Σ1 is the dominant term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' To obtain an exact expression for the main term in the asymptotics of Σ1, precise control over both binomial coefficients in its summand is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In contrast, to show that Σ2 is negligible in comparison, we only need to control one binomial coefficient precisely, and for Σ3 it suffices to estimate both binomial coefficients appearing in the summand by the maximal value they can obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5 are key in this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We often use elementary estimates of sums by integrals without reference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' proofs for any such estimate may be found in [22, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 8 푠 푟 Σ3 � 5푛 푘 Σ2 � 푚 푘 Σ1 � 푁 √푛 푘 푟 = 푠 휋 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Slice of the domain containing 퐵5푛 at a fixed 푘, showing the subdomains related to the sums Σ푖 with 푖 = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The full domain (with 푘 varying) is part of the interior of an elliptic paraboloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The sums Σ2 and Σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In this subsection, we show that the term 22푛�2푛 푛 � and the sums Σ2 and Σ3 each have negligible contribution in comparison to 16푛 log 푛/√푛 when 휖 tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' First, recall that we have already seen in the introduction that 22푛�2푛 푛 � ≍ 16푛/√푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The sum Σ3 satisfies Σ3 ⩽ |퐵5푛| �2푛 푛 � � 2푛 푛 + � 푛 log 푛 � ≍ |퐵5푛| �2푛 푛 �2 푒− 푛 log 푛 푛 ≍ 16푛 log 푛 푛 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In conclusion, both 22푛�2푛 푛 � and Σ3 are of order 표(16푛 log 푛/√푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The sum Σ2 satisfies lim 휖 →0 lim 푛→∞ Σ2 16푛 log 푛/√푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Defining 퐶 = 퐶(푘,푟, 푠) = 4푘2푟2푠2 + (푘(푟2 + 푠2) + (−1) 푘+1 2 )2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 implies that Σ2 ∼ �2푛 푛 �2 � (푘,푟,푠)∈퐵푚\\퐵푁√푛 푒− 퐶 16푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Since 퐶 > 푘2(푟2 + 푠2)2 for all positive integers 푘, 푟, and 푠, the sum on the right-hand side is bounded from above by � 1⩽푟⩽√푚 2⩽푠⩽√푚 푘> 푁√푛 푟2+푠2 푒− 1 16푛 푘2(푟2+푠2)2 ⩽ ∫ √푚 1 ∫ √푚 0 � 1 + ∫ ∞ 푁√푛 푟2+푠2 푒− 1 16푛 푘2(푟2+푠2)2 d푘 � d푟 d푠 by applying elementary estimates for sums by integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Pulling out the 1 from the middle integral and evaluating the innermost integral yields � (푘,푟,푠)∈퐵푚\\퐵푁√푛 푒− 퐶 16푛 < 푚 + 2√휋푛(1 − erf(푁/4)) ∫ √푚 1 ∫ √푚 0 1 푟2 + 푠2 d푟 d푠, where erf(푥) = 2휋−1/2 ∫ 푥 0 푒−푡2 d푡 is the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Switching to polar coordinates with 푅2 = 푟2 + 푠2, the remaining double integral is bounded by ∫ √푚 1 ∫ √푚 0 1 푟2 + 푠2 d푟 d푠 < 휋 2 ∫ √ 2푚 1 1 푅 d푅 = 휋 4 log 2푚 ≍ log 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 9 Thus, as �2푛 푛 �2 ≍ 16푛/푛, the sum Σ2 is asymptotically at most (1 − erf(푁/4)) 16푛 log 푛 √푛 up to a multiplicative constant independent of 푛 and 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As erf(푥) goes to 1 as 푥 tends to ∞, this yields the claimed limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ The sum Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' To obtain a precise estimate of Σ1, we need to control both binomial coefficients in the summand of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This is achieved by dividing the domain over which the sum extends in boxes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Let 1 ⩽ 푖 ⩽ 푁 and 1 ⩽ 푗 ⩽ 푁2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Write 휃푖 = 푖휖휋/4 and consider the inequalities ( 푗 − 1)휖√푛 < 푘(푟2 + 푠2) ⩽ 푗휖√푛, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) tan(휃푖−1) < 푠/푟 ⩽ tan(휃푖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) this is a region enclosed between two circles and two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For fixed positive 푘, the inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) divide the right-upper quadrant of the disk 푟2 + 푠2 ⩽ 푁√푛/푘 in boxes of equal area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Define the set 푇푖 푗 as 푇푖 푗 = � (푘, 푟, 푠) ∈ Z3 ���� 푘 > 0 and odd, 푟 > 푠 > 0 coprime and of opposite parity, and (푘, 푟, 푠) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The lattice point sets 푇푖 푗 are asymptotically equal in size as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As 푛 tends to infinity, we have ��푇푖 푗 �� ∼ 휖2 8휋 √푛 log 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In particular, the size of 푇푖 푗 does not depend on 푖 and 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Write 푎 = � 푗휖√푛/푘 and 푏 = � ( 푗 − 1)휖√푛/푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For fixed positive 푘, the number of integral, coprime, opposite-parity lattice points (푟, 푠) in the box bounded by the inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) equals 푞(푖, 푗) := � 푓휃푖 (푎) − 푓휃푖−1 (푎)� − � 푓휃푖 (푏) − 푓휃푖−1 (푏)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4) For 푘 ⩽ 푛+ := √푛/log log 푛, the quantity 푞 satisfies the asymptotic equality 푞(푖, 푗) ∼ 2 휋2 (휃푖 − 휃푖−1)(푎2 − 푏2) = 휖2 2휋 √푛 푘 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4 (notice that this is just the area of the box multiplied by 4/휋2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' When 푘 > 푛+, the bound 푞(푖, 푗) ≪ log log 푛 holds as each of the four terms on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4) are at most of this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Therefore � 1⩽푘⩽푛+ 푘 odd 푞(푖, 푗) ∼ 휖2 2휋 √푛 � 1⩽푘⩽푛+ 푘 odd 1 푘 ∼ 휖2 8휋 √푛 log 푛 and � 푛+<푘⩽√푛 푘 odd 푞(푖, 푗) ≪ √푛 log log 푛, which implies ��푇푖 푗 �� = � 1⩽푘⩽√푛 푘 odd 푞(푖, 푗) ∼ 휖2 8휋 √푛 log 푛 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ Multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) through by 푟 and using both of the resulting inequalities, some rewriting of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) leads to 1 4푚(푖, 휖)( 푗 − 1)휖√푛 < 1 2푘푟푠 < 1 4 푀(푖, 휖) 푗휖√푛, where 푚(푖, 휖) = 2 tan(휃푖−1) cos2(휃푖) and 푀(푖, 휖) = 2 tan(휃푖) cos2(휃푖−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Note that 푚(푖, 휖) is increasing on the interval [−1, 푁] and 푀(푖, 휖) is increasing on [−1, 푁 + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 10 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The sum Σ1 satisfies lim 휖 →0 lim 푛→∞ Σ1 16푛 log 푛/√푛 = 1 4휋3/2 ∫ 1 0 1 � 1 + sin2(휋푡/2) d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We give an upper and a lower bound that converge to the same value as 휖 tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For the upper bound, note that Σ1 = � 1⩽푖⩽푁 1⩽ 푗⩽푁 2 � (푘,푟,푠)∈푇푖 푗 � 2푛 푛 + 1 2 푘푟푠 �� 2푛 푛 + 1 4 (푘푟2 + 푘푠2 + (−1)(푘+1)/2) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='6) ⩽ �2푛 푛 �2 � 1⩽푖⩽푁 |푇푖1| + � 1⩽푖⩽푁 2⩽ 푗⩽푁 2 ��푇푖 푗 �� � 2푛 푛 + 1 4푚(푖, 휖)( 푗 − 1)휖√푛 � � 2푛 푛 + 1 4 ( 푗 − 1)휖√푛 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The first sum in the last line, where 푗 = 1 is fixed, has negligible contribution as 휖 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In addition, the asymptotics of the last binomial coefficient is not altered by changing 푛 + 1 4 ( 푗 − 1)휖√푛 − 1 to 푛+ 1 4 ( 푗 −1)휖√푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Combined with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2, the sum Σ1 is therefore asymptotically no larger than 휖2 8휋2 16푛 log 푛 √푛 푁 � 푖=1 푁 2 � 푗=2 푒− 1 16 ( 푗−1)2 휖 2(1+푚(푖,휖 )2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7) The inner sum in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7) is smaller than ∫ ∞ 1 푒− 1 16 ( 푗−1)2 휖 2(1+푚(푖,휖 )2) d푗 = 2√휋 휖 � 1 + 푚(푖, 휖)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Plugging this into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7), moving out all constants from the sum but keeping all 휖’s in it shows that it remains to evaluate 푁 � 푖=1 휖 � 1 + 푚(푖, 휖)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Again, we employ an integral estimate (using that 푚(푖, 휖) is increasing on the interval [0, 푁]) to bound the last sum from above by ∫ 푁 0 휖 � 1 + 푚(푖, 휖)2 d푖 = ∫ 1 0 1 � 1 + 4 tan2 � (푥−휖 ) 휋 4 � cos4 � 푥 휋 4 � d푥 after the substitution 푥 = 푖휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As 휖 tends to 0, this becomes the integral shown in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Now we prove that the asymptotic lower bound is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Starting from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='6), notice that this can be bounded from below by � 1⩽푖⩽푁 1⩽ 푗⩽푁 2 ��푇푖 푗 �� � 2푛 푛 + 1 4 푀(푖, 휖) 푗휖√푛 � � 2푛 푛 + 1 4 푗휖√푛 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Again, dropping the +1 in the last binomial coefficient, this sum is asymptotically at least 휖2 8휋2 16푛 log 푛 √푛 푁 � 푖=1 푁 2 � 푗=1 푒− 1 16 푗2휖 2(1+푀 (푖,휖 )2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The inner sum is at least ∫ 푁 2 1 푒− 1 16 푗2휖 2(1+푀 (푖,휖 )2) d푗 = 2√휋 휖 � 1 + 푀(푖, 휖)2 � erf �� 1 + 푀(푖, 휖)2 4휖 � − erf � 휖 � (1 + 푀(푖, 휖)2) 4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 11 Since the error function is monotonously increasing, and 푀(푖, 휖) is monotonously increasing on [1, 푁] as well, the term involving the error functions is at least erf � 1 4휖 � − erf � 휖 √ 3 4 � which tends to 1 as 휖 tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We are left with the sum 푁 � 푖=1 휖 � 1 + 푀(푖, 휖)2 which is bounded from below by ∫ 푁 1 휖 � 1 + 푀(푖, 휖)2 d푖 = ∫ 1 휖 1 � 1 + 4 tan2( 푥 휋 4 ) cos4( (푥−휖 ) 휋 4 ) d푥 where 푥 = 푖휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In the limit 휖 → 0 this becomes the integral on the right-hand side in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The skew-reciprocals As in the reciprocal case, fix some (large) integer 푁, define 휖 = 푁−1 and 푚 = 5 � 푛 log 푛, and write |푆8푛| = 22푛 �2푛 푛 � + 2Σ′ 1 + 2Σ′ 2 + 2Σ′ 3 where Σ′ 1 = � (푘,푟,푠)∈퐷푁√푛 � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 1 4 (푘푟2 − 푘푠2 + (−1)푠+ 푘+1 2 ) � , Σ′ 2 = � (푘,푟,푠)∈퐷푚\\퐷푁√푛 � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 1 4 (푘푟2 − 푘푠2 + (−1)푠+ 푘+1 2 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' With methods very similar to the ones employed in the reciprocal case, in the double limit as first 푛 and then 푁 tends to infinity, each of 22푛�2푛 푛 � and the sums Σ′ 2 and Σ′ 3 are negligible compared to 16푛 log 푛/√푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Here, we focus on the evaluation of Σ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Let 1 ⩽ 푖 ⩽ 푁 and 1 ⩽ 푗 ⩽ 푁2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Recall that 훼 is the constant log � 1 + √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Write 휃푖 = 푖휖훼 and consider the inequalities ( 푗 − 1)휖√푛 < 푘(푟2 − 푠2) ⩽ 푗휖√푛, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) tanh(휃푖−1) < 푠/푟 ⩽ tanh(휃푖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) this is a region enclosed between two hyperbolas and two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The quantity tanh(휃) varies between 0 and tanh(훼) = √ 2 − 1 as 휃 varies between 0 and 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Therefore, fixing 푘, the regions described by the inequalities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) partition 퐻훼( � 푗휖√푛/푘) (with 퐻훼 as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Similarly, the inequalities ( 푗 − 1)휖√푛 < 2푘푟푠 ⩽ 푗휖√푛, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) 푒−2휃푖 < 푠/푟 ⩽ 푒−2휃푖−1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4) partition 퐻∗ 훼( � 푗휖√푛/푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Define the set 푇 ′ 푖 푗 = � (푘, 푟, 푠) ∈ Z3 ���� 푘 > 0 and odd, 푟 > 푠 > 0 coprime and of opposite parity, and (푘, 푟, 푠) satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) � , and let 푇 ′∗ 푖 푗 be the similar set of tuples that satisfy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As 푛 tends to infinity, we have |푇 ′ 푖 푗| ∼ |푇 ′∗ 푖 푗 | ∼ 훼휖2 2휋2 √푛 log 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In particular, the size of 푇 ′ 푖 푗 does not depend on 푖 and 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We argue as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Write 푎 = � 푗휖√푛/푘 and 푏 = � ( 푗 − 1)휖√푛/푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For fixed positive 푘, the number of integral, coprime, opposite-parity lattice points in the box bounded by the inequalities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) equals 푞(푖, 푗) := �푔휃푖 (푎) − 푔휃푖−1 (푎)� − �푔휃푖 (푏) − 푔휃푖−1 (푏)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5) For 푘 ⩽ 푛+ := √푛/log log 푛, we deduce the asymptotic equality 푞(푖, 푗) ∼ 2 휋2 (휃푖 − 휃푖−1)(푎2 − 푏2) = 2훼휖2 휋2 √푛 푘 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' When 푘 > 푛+, the bound 푞(푖, 푗) ≪ log log 푛 holds as each of the four terms on the right-hand side in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5) are at most of this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' By an argument entirely similar to the one in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2, we find |푇 ′ 푖 푗| = � 1⩽푘⩽√푛 푘 odd 푞(푖, 푗) ∼ � 1⩽푘⩽푛+ 푘 odd 푞(푖, 푗) ∼ 2훼휖2 휋2 √푛 � 1⩽푘⩽푛+ 푘 odd 1 푘 ∼ 훼휖2 2휋2 √푛 log 푛, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The same argument gives the result for 푇 ′∗ 푖 푗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ Write 푚′(푖, 휖) = sinh(2휃푖−1) and 푀 ′(푖, 휖) = sinh(2휃푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Manipulating the inequalities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2) leads to 1 4푚′(푖, 휖)( 푗 − 1)휖√푛 < 1 2푘푟푠 ⩽ 1 4 푀 ′(푖, 휖) 푗휖√푛, whereas the inequalities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='4) yield 1 4푚′(푖, 휖)( 푗 − 1)휖√푛 < 1 4푘(푟2 − 푠2) ⩽ 1 4 푀 ′(푖, 휖) 푗휖√푛 for the same functions 푚′ and 푀 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Write Σ = � 1⩽푖⩽푁 1⩽ 푗⩽푁 2 � (푘,푟,푠)∈푇 ′ 푖 푗 � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 1 4 (푘푟2 − 푘푠2 + (−1)푠+ 푘+1 2 ) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='6) and Σ∗ = � 1⩽푖⩽푁 1⩽ 푗⩽푁 2 � (푘,푟,푠)∈푇 ′∗ 푖 푗 � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 1 4 (푘푟2 − 푘푠2 + (−1)푠+ 푘+1 2 ) � , so that Σ′ 1 = Σ + Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Each of the sums Σ and Σ∗ can be asymptotically bounded from above by 훼휖2 2휋3 16푛 log 푛 √푛 푁 � 푖=1 푁 2 � 푗=1 푒− 1 16 ( 푗−1)2 휖 2(1+푚′(푖,휖 )2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7) and from below by 훼휖2 2휋3 16푛 log 푛 √푛 푁 � 푖=1 푁 2 � 푗=1 푒− 1 16 푗2 휖 2(1+푀′(푖,휖 )2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='8) In particular, Σ′ 1 is asymptotically equal to 2Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We give an upper and a lower bound that converge to the same value as 휖 tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For the upper bound, note that Σ ⩽ �2푛 푛 �2 � 1⩽푖⩽푁 |푇 ′ 푖1| + � 1⩽푖⩽푁 2⩽ 푗⩽푁 2 |푇 ′ 푖 푗| � 2푛 푛 + 1 4푚′(푖, 휖)( 푗 − 1)휖√푛 � � 2푛 푛 + 1 4 ( 푗 − 1)휖√푛 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The −1 appearing in the last binomial coefficient can simply be ignored, because it doesn’t affect the asymptotics in 푛 of that binomial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In addition, we see that the first term in the last line will 13 have negligible contribution as 휖 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The asymptotics for almost central binomial coefficients given in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 and for |푇 ′ 푖 푗| of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 show the last sum is asymptotically no larger than the sum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For the lower bound, we observe Σ ⩾ � 1⩽푖⩽푁 1⩽ 푗⩽푁 2 |푇 ′ 푖 푗| � 2푛 푛 + 1 4 푀 ′(푖, 휖) 푗휖√푛 � � 2푛 푛 + 1 4 푗휖√푛 + 1 � starting from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Again, dropping the +1 in the last binomial coefficient, this sum is asymptotically at least the sum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' After replacing 푇 ′ 푖 푗 by 푇 ′∗ 푖 푗 , the same argument holds for Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ We are now in the position to obtain our main result for Σ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' With 훼 = log � 1 + √ 2, the sum Σ′ 1 satisfies lim 휖 →0 lim 푛→∞ Σ′ 1 16푛 log 푛/√푛 = 2훼 휋5/2 ∫ 1 0 1 � 1 + sinh2(2훼푡) d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We show that the sums (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='8) are asymptotically equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This implies that 1 2Σ′ 1 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7) are asymptotically equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We start with the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The inner sum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7) is smaller than ∫ ∞ 1 푒− 1 16 ( 푗−1)2 휖 2(1+푚′(푖,휖 )2) d푗 = 2√휋 휖 � 1 + 푚′(푖, 휖)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Plugging this into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7), moving out all constants from the sum but keeping all 휖’s in it shows that it remains to evaluate 푁 � 푖=1 휖 � 1 + 푚′(푖, 휖)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Again, we employ an integral estimate (using that 푚′(푖, 휖) is increasing on the interval [0, 푁]) to bound the last sum from above by ∫ 푁 0 휖 � 1 + 푚′(푖, 휖)2 d푖 = ∫ 1 0 1 � 1 + sinh2(2(푡 − 휖)훼) d푡 after the substitution 푡 = 푖휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' As 휖 tends to 0, this becomes the integral shown in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' For the lower bound, the inner sum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='8) is at least ∫ 푁 2 1 푒− 1 16 푗2휖 2(1+푀 (푖,휖 )2) d푗 = 2√휋 휖 � 1 + 푀(푖, 휖)2 � erf �� 1 + 푀(푖, 휖)2 4휖 � − erf � 휖 � (1 + 푀(푖, 휖)2) 4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Since the error function is monotonously increasing, and 푀(푖, 휖) is monotonously increasing on [1, 푁] as well, the term involving the error functions is at least erf � 1 4휖 � − erf � 휖 2 √ 2 � which tends to 1 as 휖 tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We are left with the sum 푁 � 푖=1 휖 � 1 + 푀(푖, 휖)2 which is bounded from below by ∫ 푁 1 휖 � 1 + 푀(푖, 휖)2 d푖 = ∫ 1 휖 1 � 1 + sinh2(2훼푥) d푡 where 푡 = 푖휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In the limit 휖 → 0 this again becomes the integral on the right-hand side in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ 14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 We are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We first prove part (a) and then part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Whereas Σ1 and Σ2 depend on 휖, the total sum |푅8푛| does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In particular, lim 푛→∞ |푅8푛| 16푛 log 푛/√푛 = lim 휖 →0 lim 푛→∞ |푅8푛| 16푛 log 푛/√푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The last double limit can be split in several pieces using that |푅8푛| = 22푛�2푛 푛 � + 2Σ1 + 2Σ2 + 2Σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' In particular, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1), Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1, and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='3 show that lim 휖 →0 lim 푛→∞ |푅8푛| 16푛 log 푛/√푛 = lim 휖 →0 lim 푛→∞ 2Σ1 16푛 log 푛/√푛 = 1 2휋3/2 ∫ 1 0 1 � 1 + sin2(휋푡/2) d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' To evaluate the integral, substitute 푥 = sin4(휋푡/2) so that 2휋 d푡 = 푥−3/4(1 − √푥)−1/2 d푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Hence ∫ 1 0 1 � 1 + sin2(휋푡/2) d푡 = 1 2휋 ∫ 1 0 푥−3/4(1 − 푥)−1/2 d푥 = 1 2휋 퐵 �1 4, 1 2 � = Γ( 1 4)Γ( 1 2) 2휋Γ( 3 4) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) where 퐵 is the beta function, which satisfies 퐵(푚, 푛) = Γ(푚)Γ(푛)/Γ(푚 + 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Legendre’s duplica- tion formula for the gamma function yields Γ(1/2) = Γ(1/4)Γ(3/4)/ √ 2휋, showing that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1) equals Γ( 1 4)2/ √ 8휋3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The skew-reciprocal case is entirely similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' With the same steps, we deduce lim 푛→∞ |푆8푛| 16푛 log 푛/√푛 = lim 휖 →0 lim 푛→∞ 2Σ′ 1 16푛 log 푛/√푛 = 4훼 휋5/2 ∫ 1 0 1 � 1 + sinh2(2훼푡) d푡 where again 훼 = log � 1 + √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' To evaluate the integral, substituting 푥 = sinh(2훼푡) yields d푥 = 2훼 cosh(2훼푡) d푡 = 2훼 √ 푥2 + 1 d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Therefore ∫ 1 0 1 � 1 + sinh2(2훼푡) d푡 = 1 2훼 ∫ 1 0 1 푥2 + 1 d푥 = 1 2훼 (arctan(1) − arctan(0)) = 휋 8훼, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We prove the result for the reciprocal polynomials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' an analogous argument works for the skew-reciprocals as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Write 푛0 = 1 4 (푘푟2 + 푘푠2 + (−1) 푘+1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The second binomial coefficient in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='5) equals �2푛 − 1 푛 + 푛0 � = �1 2 − 푛0 2푛 � � 2푛 푛 + 푛0 � if 푘 ≡ 1 mod 4, � 2푛 − 1 푛 + 푛0 − 1 � = �1 2 + 푛0 2푛 � � 2푛 푛 + 푛0 � if 푘 ≡ 3 mod 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' these identities also hold when 푛0 = 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Therefore |푅8푛−2| = 1 2 |푅8푛| + Σ, where Σ = 1 2푛 � (푘,푟,푠)∈퐵5푛 1 4 (1 + (−1) 푘+1 2 푘(푟2 + 푠2)) � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 푛0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Write 푉푡 = 1 2푛 � (푘,푟,푠)∈퐵푡 1 4 (1 + 푘(푟2 + 푠2)) � 2푛 푛 + 1 2 푘푟푠 � � 2푛 푛 + 푛0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Then 푉5푛 is at least as big as Σ in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Estimating in each case the term 1 4 (1 + 푘(푟2 + 푠2)) 15 by the maximum value it can possibly attain, we see that 푉5푛 −푉푚 is asymptotically at most Σ3, whereas 푉푚 is asymptotically at most � log 푛/푛(Σ1 + Σ2) (both up to a multiplicative constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Both of these are negligible compared to |푅8푛|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Square discriminants in other degrees In this section, we discuss Littlewood polynomials with square discriminant in degree 푛 � 0, 6 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The following surprising result, attributed to Alexei Entin in [3, §4], shows that such polynomials do not even exist in even degree 푛 ≡ 2, 4 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='1 (Entin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Let 푛 ≡ 2, 4 mod 8 be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Then no Littlewood polynomial of degree 푛 has square discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Suppose that 푛 is even and 푓 ∈ F푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Set 푝푛(푋) = (푋푛+1 − 1)/(푋 − 1) and note that 푓 and 푝푛 coincide modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Since 푋푛+1 − 1 and its derivative are coprime modulo 2, the polynomial 푝푛 is separable over F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Thus 푝푛 is separable over the 2-adic field Q2 as well by Hensel’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The splitting field of 푝푛 over Q2, which is the cyclotomic extension Q2(휁)/Q2 where 휁 is a primitive 푛 + 1-th root of unity, is an unramified extension of Q2 because 2 and 푛 + 1 are coprime, see [15, Proposition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Writing 퐺( 푓 /퐾) for the Galois group of 푓 over a field 퐾, this implies that 퐺(푝푛/Q2) is isomorphic to 퐺(푝푛/F2) = 퐺( 푓 /F2) ⩽ 퐺( 푓 /Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' The discriminant of 푝푛 is a square in Z2 if and only if it is 1 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' A resultant calculation shows that Δ(푝푛) = (−1) 푛(푛−1) 2 (푛 + 1)푛−1, which is congruent to 5 mod 8 if 푛 ≡ 2, 4 mod 8 (and congruent to 1 mod 8 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Therefore 푓 cannot have square discriminant over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' □ In the case of odd-degree Littlewood polynomials, the situation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Call a degree-푛 polynomial 푓 nearly reciprocal if it is of the form 푓 (푋) = ±푋푛 푓 (푋−1), nearly skew-reciprocal if 푓 (푋) = ±푋푛 푓 (−푋−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' We give some examples: Littlewood polynomials with vanishing square discriminant exist in any odd degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Indeed, the nearly reciprocal polynomial given by (푋푛+1 − 1)(푋푛 + 푋푛−1 + · · · + 푋 + 1) = (푋 − 1)(푋푛 + 푋푛−1 + · · · + 푋 + 1)2 ∈ F2푛+1 has a multiple factor and thus its discriminant vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' An odd-degree Littlewood polynomial with vanishing square discriminant is not necessarily nearly (skew-)reciprocal, or the product of such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Indeed, the polynomial (푋 + 1)2(푋2 − 푋 + 1)(푋7 − 푋5 + 푋4 − 푋3 + 푋2 + 1) has vanishing discriminant, but the Galois group of its splitting field is 퐶2 × 푆7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' A computer experiment shows that all Littlewood polynomials of odd degree ⩽ 29 with nonva- nishing square discriminant have a cyclotomic factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' in fact, each such polynomial is divisible by 푋 + 1 or 푋 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' Does there exist an odd-degree Littlewood polynomial without cyclotomic factors that has square discriminant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' (If not, this would imply for example that no irreducible Littlewood polynomial of odd degree 푛 has Galois group contained in 퐴푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=') A related question, raised by Peled, Sen and Zeitouni [20, §7], is whether Littlewood polyno- mials with a repeated non-cyclotomic factor exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content=' This was recently answered in the affirmative in response to a question on MathOverflow [24]: Peter Taylor found the polynomial (푋18 + 푋16 + 2푋15 + 2푋13 + 푋12 + 2푋11 + 3푋10 + 3푋8 + 2푋7 + 푋6 + 2푋5 + 2푋3 + 1) × (푋2 + 1)(푋 − 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='hokken@uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} +page_content='nl 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfjg9a/content/2301.05656v1.pdf'} diff --git a/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf b/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4492cc519c188f0ab529c8e6e2a60e64c9d832f5 --- /dev/null +++ b/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9abfe7428317adb37e209780f8a60fa728dd4ea4886bf2e8c168b8ed2b943299 +size 162427 diff --git a/p9FIT4oBgHgl3EQfwita/content/tmp_files/2301.11352v1.pdf.txt b/p9FIT4oBgHgl3EQfwita/content/tmp_files/2301.11352v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ad68830cf984f6076c1ce8c55114989f576894b --- /dev/null +++ b/p9FIT4oBgHgl3EQfwita/content/tmp_files/2301.11352v1.pdf.txt @@ -0,0 +1,532 @@ +arXiv:2301.11352v1 [math.CA] 26 Jan 2023 +THE DISCRETE SPHERICAL MAXIMAL FUNCTION: +A NEW PROOF OF ℓ2-BOUNDEDNESS +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +Abstract. We provide a new direct proof of the ℓ2-boundedness of the Discrete Spherical Maximal Func- +tion that neither relies on abstract transference theorems (and hence Stein’s Spherical Maximal Function +Theorem) nor on delicate asymptotics for the Fourier transform of discrete spheres. +1. Introduction +The study of discrete analogues of central constructs of Euclidean harmonic analysis, initiated by Bourgain +[1], has grown into a vast, active area of research. +An important result in this development is the ℓp- +boundedness of the so-called discrete spherical maximal function [5]. +Beyond its own intrinsic interest, this operator, or more precisely certain “mollified variants”, play a +crucial role in studying certain geometric point configurations in positive density subsets of the integer +lattice, see [2]. +Let d ≥ 5, λ2 ∈ N, and Nλ := |{m ∈ Zd : |m| = λ}|. It is well-known, see for example [7], that +cdλd−2 ≤ Nλ ≤ Cdλd−2 +for some constants 0 < cd < Cd. For f : Zd → R define the discrete spherical averages +Aλf(n) = N −1 +λ +� +|m|=λ +f(n − m) +and the maximal operator +A∗f(n) = sup +λ +|Aλf(n)|. +The variables n, m in the two equations above, and throughout this short note, are always assumed to be +in Zd. Furthermore, the parameter λ will always be assumed to satisfy λ2 ∈ N. +In [5] it was shown that for p > d/(d − 2) one has the estimate +∥A∗f∥p ≤ Cp,d ∥f∥p +where ∥f∥p = (� +x |f(x)|p)1/p denotes the ℓp(Zd) norm of the function f. It was further noted in [5] that +the condition that d ≥ 5 and p > d/(d − 2) are both sharp. +The approach taken in [5] had three main steps. The first step was to approximate Aλ by an infinite sum +of simpler operators M a/q +λ +, each associated to a reduced fraction a/q, with 0 < a/q ≤ 1. A general abstract +transference theorem, which allows one to pass from certain convolution operators on Rd to analogous +operators on Zd, was then used to analyze each M a/q +λ +. In particular, this approach makes use of Stein’s +Spherical Maximal Function Theorem [6]. The final step of the argument is to show that the approximation +taken in the first step is adequate, this step uses the full asymptotic expansion for the Fourier transform of +(the indicator function of) the discrete sphere of radius λ in Zd. +In this note we provide a short direct proof of the ℓ2 case of the main result in [5]. Our direct proof relies +on the observation that one obtains gains in ℓ2 for maximal operators at a single dyadic scale, when applied +to functions whose Fourier transform is suitably localized away from rational points with suitably small +denominators, specifically Proposition 1 below. This combined with an almost orthogaonality argument +2010 Mathematics Subject Classification. 42B25. +The first and second authors were partially supported by grants NSF-DMS 1702411 and NSF-DMS 1600840, respectively. +1 + +2 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +quickly leads to the proof Theorem 1 below. Note that we do not need the full asymptotic expansion of the +underlying multipliers neither any transference arguments to utilise Stein’s spherical maximal theorem. +Our main result is the following, +Theorem 1. If d ≥ 5, then +∥A∗f∥2 ≤ Cd ∥f∥2. +2. Key estimates for maximal operators at a single dyadic scale +Recall that for f ∈ ℓ1(Zd) we define its Fourier transform �f : Td → C by +�f(α) = +� +n∈Zd +f(n)e−2πin·α. +Before stating Proposition 1 we need to introduce some additional notation. For any integer j ≥ 0 we let +qj = lcm{1, 2, . . ., 2j} and note that qj ≍ e2j. For any non-negative integers j and k that satisfy 2j ≤ k , we +let +(1) +Ωj,k := {α ∈ Td : α ∈ [−2j−k, 2j−k]d + (q−1 +j Z)d}. +Proposition 1. If d ≥ 5, k ∈ N, and 1 ≤ j ≤ log2(k) − 2, then one has the estimate +(2) +��� +sup +2k≤λ≤2k+1 |Aλf| +��� +2 ≪ 2−j/2j−1∥f∥2 +whenever supp �f ⊆ Ωc +j,k, where Ωc +j,k denotes the complement of Ωj,k. +In the Proposition above, and for the rest of this short note, we use the notation A ≪ B to denote that +A ≤ CB for some constant C that may depend on d, which we consider fixed and greater than or equal to 5. +The proof of Proposition 1 is presented in Section 4, while the reduction of Theorem 1 to Proposition 1 +is presented in Section 3 below. We conclude this section by noting that Proposition 1 immediately implies +the following “mollified variant” of Theorem 1 which is of independent interest. +Theorem 2. If d ≥ 5, η > 0, and L ≥ q4 +η, then one has the estimate +(3) +��� sup +λ≥η−2L +|Aλf| +��� +2 ≪ η ∥f∥2 +whenever supp �f ⊆ Ωc +η,L, with Ωη,L = {α ∈ Td : α ∈ [−L−1, L−1]d + (q−1 +η Z)d} and qη = lcm{1 ≤ q ≤ η−2}. +Indeed, note that in proving (3) one may restrict the sup to η−2L ≤ λ ≤ 2η−2L. Choosing k, j ∈ N such +that 2k ≤ η−2L ≤ 2k+1 and 2j ≥ η−2 we have that 2k−j ≤ L and hence Ωj,k ⊆ Ωη,L. Applying Proposition +1 with j and k chosen as above implies Theorem 2. +This provides a slight strengthening of Proposition 5 in [2], more importantly it provides a significantly +simpler direct proof. +3. Proof of Theorem 1 +3.1. A smooth sampling function supported on Ωj,k. Let ψ ∈ S(Rd) be a Schwartz function satisfying +1Q(ξ) ≤ �ψ(ξ) ≤ 12Q(ξ) +where Q = [−1/2, 1/2]d and +�ψ(ξ) := +� +Rd ψ(x)e−2πix·ξdx +denote the Fourier transform of ψ on Rd. For a given q ∈ N and L > q we define ψq,L : Zd → R as +ψq,L(m) = +�� q +L +�d ψ +� m +L +� +if m ∈ (qZ)d +0 +otherwise + +THE DISCRETE SPHERICAL MAXIMAL FUNCTION: +A NEW PROOF OF ℓ2-BOUNDEDNESS +3 +Writing m = qr + s with r ∈ Zd and s ∈ Zd/qZd, it follows from Poisson summation that +�ψq,L(α) = +� +m∈Zd +ψ(m)e−2πim·α +is a q−1-periodic function on Td that satisfies +�ψq,L(α) = +� +ℓ∈Zd +�ψ(L(α − ℓ/q)). +For a given k ∈ N and 0 ≤ j ≤ Jk := [log2(k)] − 2, we now define the sampling function +(4) +Ψj,k = ψqj,2k−j +and note that supp �Ψj,k ⊆ Ωj,k. +Finally we define ∆Ψj,k = Ψj+1,k −Ψj,k and note the important almost orthogonality property they enjoy. +Lemma 1. There exists a constant C = CΨ > 0 such that +� +k≥2j +|� +∆Ψk,j(α)|2 ≤ C +uniformly in j ∈ N and α ∈ Td. +Proof of Lemma 1. Note that Ωk+1,j ⊆ Ωk,j. Now fix j ∈ N. If α /∈ Ω2j,j, then � +∆Ψk,j(α) = 0. +If α ∈ Ω2j,j, then we define k1 = k1(j) := max{k ≥ 2j : α ∈ Ωk,j}. +Then there exists a unique +ℓ1 ∈ Zd such that |α − ℓ1/qj| ≤ 2j−k1. Clearly � +∆Ψk,j(α) = 0 if k > k1, while if 2j ≤ k ≤ k1 we have +�Ψk,j(α) = ˜Ψ(2k−j(α − ℓ1/qj)). It therefore follows, by writing ∆Ψk,j = (Ψk,j+1 − 1) + (1 − Ψk,j), that +|� +∆Ψk,j(α)| ≤ CΨ 2k−j|α − ℓ1/qj| ≤ CΨ 2k−k1 +and hence that +� +k≥2j +|� +∆Ψk,j(α)|2 ≤ CΨ +� +1≤k≤k1 +2−2(k1−k) ≤ 4 +3 CΨ. +□ +3.2. Proof that Proposition 1 implies Theorem 1. Let +(5) +Mkf := +sup +2k≤λ≤2k+1 |Aλf|. +Writing +f = f ∗ Ψk,0 + +Jk−1 +� +j=0 +f ∗ ∆Ψk,j + (f − f ∗ Ψk,Jk) +it follows by subadditivity that +(6) +Mkf ≤ Mk(f ∗ Ψk,0) + +Jk−1 +� +j=0 +Mk(f ∗ ∆Ψk,j) + Mk(f − f ∗ Ψk,Jk) +Theorem 1 will now follow from a few observations and applications of Proposition 1, in light of the fact +that +A∗f = sup +k +Mkf. +First we note that it is straightforward to verify that the first term on the right in (6) above satisfies +Mk(f ∗ Ψk,0) ≤ CΨHf +uniformly in k, where +Hf(n) = sup +ℓ>0 +1 +(2 · 2ℓ + 1)d +��� +� +m∈[−2ℓ,2ℓ]d∩Zd +f(n − m) +��� + +4 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +denotes the discrete Hardy-Littlewood maximal operator. Since, by the same arguments as in Euclidean +spaces, we have ∥Hf∥2 ≪ ∥f∥2, it follows that +sup +k +∥Mk(f ∗ Ψk,0)∥2 ≪ ∥f∥2. +For the middle terms in (6) we first note that +sup +k +Jk−1 +� +j=0 +Mk(f ∗ ∆Ψk,j) ≪ +� ∞ +� +k=0 +��� +Jk−1 +� +j=0 +Mk(f ∗ ∆Ψk,j) +��� +2�1/2 +Taking ℓ2 norms of both sides of the inequality above and applying Minkowski’s inequality, followed by +an application of Proposition 1, gives +���sup +k +� +0≤j≤Jk +Mk(f ∗ ∆Ψk,j) +��� +2 ≤ +� +j +� � +k≥2j +∥Mk(f ∗ ∆Ψk,j)∥2 +2 +�1/2 +≪ +� +j +2−j/2� � +k≥2j +∥f ∗ ∆Ψk,j∥2 +2 +�1/2 +≪ ∥f∥2 +where the last inequality above follows from Lemma 1. +One more application of Proposition 1 with j = [log2 k] − 2 to the last term in (6) gives +���sup +k +Mk(f − f ∗ Ψk,Jk) +��� +2 ≤ +� ∞ +� +k=1 +∥Mk(f − f ∗ Ψk,Jk)∥2 +2 +�1/2 +≪ +� ∞ +� +k=1 +k−1(log2 k)−2�1/2 +∥f∥2 ≪ ∥f∥2. +□ +4. Proof of Proposition 1 +Fix ε = 2−2k. We start by observing that +1{|m|=λ}(m) = +� 1 +0 +e2πi(|m|2−λ2)tdt = e2πελ2 � 1 +0 +e2πi|m|2(t+iε)e−2πiλ2t dt +where 1{|m|=λ} denotes the indicator function of the discrete sphere of radius λ in Zd. +Since Nλ ≍ λd−2 it therefore follows that +Mkf ≪ +sup +2k≤λ≤2k+1 +1 +λd−2 +� 1 +0 +|f ∗ st| dt +where st(m) = e2πi|m|2(t+iε), and hence that +∥Mkf∥2 ≪ ε(d−2)/2 +� 1 +0 +∥f ∗ st∥2 dt ≤ ε(d−2)/2 +�� 1 +0 +∥�st 1Ωc +j,k∥∞ dt +� +∥f∥2. +Thus, in order to prove Proposition 1 it suffices to show that +(7) +� 1 +0 +∥�st 1Ωc +j,k∥∞ dt ≪ ε−(d−2)/22−j/2j−1. +To do this we will employ the circle method and decompose the interval into Farey arcs, that is neigh- +borhoods Va,q of reduced rationals a/q which allows us to estimate �st(ξ) by using Poisson summation and +properties of Gaussian sums. Specifically, we decompose the interval [0, 1] into neighborhoods of rationals +whose denominator is smaller than 2k as follows: Let +H = {a/q : 1 ≤ q ≤ 2k, 0 < a ≤ q, (a, q) = 1} +and define +Va,q = +� +t ∈ [0, 1] : |t − a/q| = min +r∈H |t − r| +� +. +Note that, by Dirichlet’s principle, for every t ∈ [0, 1], there exists a/q ∈ H such that |t − a/q| ≤ 2−kq−1, +thus we have that |Va,q| ≤ 2−k+1q−1. Also, if a/q ̸= a′/q′ with (a, q) = (a′, q′) = 1 and 1 ≤ q, q′ ≤ 2k then + +THE DISCRETE SPHERICAL MAXIMAL FUNCTION: +A NEW PROOF OF ℓ2-BOUNDEDNESS +5 +|a/q − a′/q′| ≥ 1/(qq′) ≥ 2−kq−1, hence |Va,q| ≥ 2−kq−1. Thus the Farey arcs Va,q at level 2k provide a +partition (up to endpoints) of [0,1] into intervals of length |Va,q| ≈ 2−kq−1. +It follows from Poisson summation that for 1 ≤ a ≤ q, (a, q) = 1, 1 ≤ q ≤ 2k one has +(8) +|�st(α)| ≤ q−d/2(ε + |τ|)−d/2 � +ℓ∈Zd +e− π +2 |α−ℓ/q|2/(ε+ε−1|τ|2) +for each t ∈ Va,q with τ = t − a/q. The details of the calculation to derive estimate (8) are laid out more +carefully in [3], but they can be briefly summarize as follows: First write �st as a product of one dimensional +functions. An application of Poisson summation and a change of variables leaves a double sum that can +be recognized as a quadratic Gaussian sum, which can be bounded by q−1/2, and a sum of terms involving +�sτ(ℓ/q − α) which has a simple closed form. See formula (12) in [3]. +Since |τ| ≤ 2−kq−1, it follows that q2(ε + ε−1|τ|2) ≪ 1, and hence that +� +ℓ∈Zd +e− π +2 |α−ℓ/q|2/(ε+ε−1|τ|2) ≪ 1 +which in turn implies that if t ∈ Va,q with t = a/q + τ, then +(9) +∥�st∥∞ ≪ q−d/2(ε + |τ|)−d/2. +We write +(10) +� 1 +0 +∥�stχΩc +j,k ∥∞ dt = +� +q|qj +� +(a,q)=1 +� +Va,q +∥�st 1Ωc +j,k∥∞dt + +� +q∤qj +� +(a,q)=1 +� +Va,q +∥�st 1Ωc +j,k∥∞ dt. +In order to estimate the first double sum above we consider separately the case when |τ| ≥ 2j/2ε and +|τ| ≤ 2j/2ε. When |τ| ≥ 2j/2ε we use estimate (9) to bound it by +(11) +� +q|qj +� +(a,q)=1 +� +Va,q +q−d/2(ε + |τ|)−d/2dt ≪ +� +q|qj +q−d/2+1 +� ∞ +ε2j/2(ε + |τ|)−d/2dτ +≪ ε−(d−2)/22−j(d−2)/4 � +q|qj +q−d/2+1. +When |τ| ≤ 2j/2ε we note that because q|qj we have that +(12) +� +ℓ∈Zd +e− π +2 |α−ℓ/q|2/(ε+ε−1|τ|2) = e− π +2 |α−ℓ0/q|2/(ε+ε−1|τ|2) + +� +ℓ̸=ℓ0 +e− π +2 |α−ℓ/q|2/(ε+ε−1|τ|2) +where ℓ0 denotes the nearest integer to qα. For every α ∈ Ωc +j,k we have |α − ℓ0 +q | = |α − qjℓ0/q +qj +| ≥ 2j−k and +hence that +(13) +|e− π +2 |α−ℓ0/q|2/(ε+ε−1|τ|2)| ≪ e−c(2j−k)2/2j−2k ≪ e−c2j +since ε + ε−1|τ|2 ≤ 2 · 2jε ≪ 2j−2k. To estimate the sum where ℓ ̸= ℓ0 in (12) above we again use the fact +that ε + ε−1|τ|2 ≪ 2j−2k. Since |qα − ℓ| ≥ 1/2 for ℓ ̸= ℓ0 and q|qj with j ≤ log2 k − 2 it follows that +q2(ε + ε−1|τ|2) ≪ (22j)22j−2k ≤ 2k2−2k ≤ 2−k +and hence that +(14) +� +ℓ̸=ℓ0 +e− π +2 |α−ℓ/q|2/(ε+ε−1|τ|2) ≪ 2−c2k. +Combining estimates (13), and (14) it follows that +∥�st 1Ωc +j,k∥∞ ≪ q−d/2(ε + |τ|)−d/2 (e−c2j + 2−c2k) + +6 +NEIL LYALL +´AKOS MAGYAR +ALEX NEWMAN +PETER WOOLFITT +whenever |τ| ≤ 2j/2ε and q|qj. Further combining this with (11) we obtain that +� +q|qj +� +(a,q)=1 +� +Va,q +∥�st 1Ωc +j,k∥∞dt ≪ +� +q|qj +q−d/2+1ε−(d−2)/2 � +(e−c2j + 2−c2k) + 2−j(d−2)/4� +≪ ε−(d−2)/2 2−j(d−2)/4 � +q|qj +q−d/2+1 +≪ ε−(d−2)/2 2−j(d−2)/4 +as the sum over q converges for d ≥ 5. +In order to estimate the second double sum in (10) we need the following observation, whose proof we +delay until after completing the proof of Proposition 1. +Lemma 2. Given r > 1, k ∈ N, then +� +q∤Qk +q−r ≪r k−r+1(log k)−1 +where Qk = lcm{1 ≤ q ≤ k} and ≪r denotes less than a constant depending on r. +Using estimate (9) and Lemma 2 one can bound the second double sum in (10) above by +� +q∤qj +� +(a,q)=1 +� +Va,q +q−d/2(ε + |τ|)−d/2dt ≪ ε−(d−2)/2 � +q∤qj +q−d/2+1 ≪ ε−(d−2)/22−j(d−4)/2j−1 +whenever d ≥ 5 completing the proof of Proposition 1. +□ +Proof of Lemma 2. If q ∈ N such that q ∤ Qk, then either a large power of a small prime divides q, or a +large prime divides q. Explicitly, for prime p ≤ k, let ap = min{a ∈ N : k < pa}. If q ∤ Qk then one of the +following must occur: +(i) there exists a p > k such that q = pq1 +(ii) there exists a p < k such that q = papq1. +In the first case +� +(i) holds +q−r ≤ +� +p>k +� +q1∈N +p−rq−r +1 +≪r +� +p>k +p−r +≪r +� +m≥0 +� +p∈[2mk,2m+1k) +2−mrk−r +≪r +� +m≥0 +2−mrk−r +2mk +m log(k) ≪r k−r+1(log k)−1. +In the second case +� +(ii) holds +q−r ≤ +� +p≤k +p−rap � +q1∈N +q−r +1 +≪r +k +log k k−r. +□ +References +[1] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), no. 1, 39-72. +[2] N. Lyall and ´A. Magyar, Distances and trees in dense subsets of Zd, to appear in Israel J. Math. +[3] ´A. Magyar, Lp-bounds for spherical maximal operators on Zn, Rev. Mat. Iberoamericana 13 (1997), 307-313. +[4] ´A. Magyar, k-point configurations in sets of positive density of Zn, Duke Math. J., v 146/1, (2009) pp. 1-34. +[5] ´A. Magyar, E. M. Stein, and S. Wainger, Discrete analogues in harmonic analysis: Spherical averages, Ann. Math. (2) +155 (2002), no. 1, 189-208. +[6] E. M. Stein, Maximal functions I: Spherical means, Proc. Nat. Acad. Sci. 73 (1976), 2174-2175. +[7] R. C. Vaughan, The Hardy-Littlewood Method, Second ed., Cambridge University Press, Cambridge, 1997. + +THE DISCRETE SPHERICAL MAXIMAL FUNCTION: +A NEW PROOF OF ℓ2-BOUNDEDNESS +7 +Department of Mathematics, The University of Georgia, Athens, GA 30602, USA +Email address: lyall@math.uga.edu +Email address: magyar@math.uga.edu +Email address: alxjames@uga.edu +Email address: pwoolfitt@uga.edu + diff --git a/p9FIT4oBgHgl3EQfwita/content/tmp_files/load_file.txt b/p9FIT4oBgHgl3EQfwita/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a041811bd682990a421b31fed30ec13cf27f733 --- /dev/null +++ b/p9FIT4oBgHgl3EQfwita/content/tmp_files/load_file.txt @@ -0,0 +1,154 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf,len=153 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='11352v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='CA] 26 Jan 2023 THE DISCRETE SPHERICAL MAXIMAL FUNCTION: A NEW PROOF OF ℓ2-BOUNDEDNESS NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' We provide a new direct proof of the ℓ2-boundedness of the Discrete Spherical Maximal Func- tion that neither relies on abstract transference theorems (and hence Stein’s Spherical Maximal Function Theorem) nor on delicate asymptotics for the Fourier transform of discrete spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Introduction The study of discrete analogues of central constructs of Euclidean harmonic analysis, initiated by Bourgain [1], has grown into a vast, active area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' An important result in this development is the ℓp- boundedness of the so-called discrete spherical maximal function [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Beyond its own intrinsic interest, this operator, or more precisely certain “mollified variants”, play a crucial role in studying certain geometric point configurations in positive density subsets of the integer lattice, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Let d ≥ 5, λ2 ∈ N, and Nλ := |{m ∈ Zd : |m| = λ}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' It is well-known, see for example [7], that cdλd−2 ≤ Nλ ≤ Cdλd−2 for some constants 0 < cd < Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' For f : Zd → R define the discrete spherical averages Aλf(n) = N −1 λ � |m|=λ f(n − m) and the maximal operator A∗f(n) = sup λ |Aλf(n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' The variables n, m in the two equations above, and throughout this short note, are always assumed to be in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Furthermore, the parameter λ will always be assumed to satisfy λ2 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' In [5] it was shown that for p > d/(d − 2) one has the estimate ∥A∗f∥p ≤ Cp,d ∥f∥p where ∥f∥p = (� x |f(x)|p)1/p denotes the ℓp(Zd) norm of the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' It was further noted in [5] that the condition that d ≥ 5 and p > d/(d − 2) are both sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' The approach taken in [5] had three main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' The first step was to approximate Aλ by an infinite sum of simpler operators M a/q λ , each associated to a reduced fraction a/q, with 0 < a/q ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' A general abstract transference theorem, which allows one to pass from certain convolution operators on Rd to analogous operators on Zd, was then used to analyze each M a/q λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' In particular, this approach makes use of Stein’s Spherical Maximal Function Theorem [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' The final step of the argument is to show that the approximation taken in the first step is adequate, this step uses the full asymptotic expansion for the Fourier transform of (the indicator function of) the discrete sphere of radius λ in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' In this note we provide a short direct proof of the ℓ2 case of the main result in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Our direct proof relies on the observation that one obtains gains in ℓ2 for maximal operators at a single dyadic scale, when applied to functions whose Fourier transform is suitably localized away from rational points with suitably small denominators, specifically Proposition 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' This combined with an almost orthogaonality argument 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' 42B25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' The first and second authors were partially supported by grants NSF-DMS 1702411 and NSF-DMS 1600840, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' 1 2 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT quickly leads to the proof Theorem 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Note that we do not need the full asymptotic expansion of the underlying multipliers neither any transference arguments to utilise Stein’s spherical maximal theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Our main result is the following, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' If d ≥ 5, then ∥A∗f∥2 ≤ Cd ∥f∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Key estimates for maximal operators at a single dyadic scale Recall that for f ∈ ℓ1(Zd) we define its Fourier transform �f : Td → C by �f(α) = � n∈Zd f(n)e−2πin·α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Before stating Proposition 1 we need to introduce some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' For any integer j ≥ 0 we let qj = lcm{1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=', 2j} and note that qj ≍ e2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' For any non-negative integers j and k that satisfy 2j ≤ k , we let (1) Ωj,k := {α ∈ Td : α ∈ [−2j−k, 2j−k]d + (q−1 j Z)d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' If d ≥ 5, k ∈ N, and 1 ≤ j ≤ log2(k) − 2, then one has the estimate (2) ��� sup 2k≤λ≤2k+1 |Aλf| ��� 2 ≪ 2−j/2j−1∥f∥2 whenever supp �f ⊆ Ωc j,k, where Ωc j,k denotes the complement of Ωj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' In the Proposition above, and for the rest of this short note, we use the notation A ≪ B to denote that A ≤ CB for some constant C that may depend on d, which we consider fixed and greater than or equal to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' The proof of Proposition 1 is presented in Section 4, while the reduction of Theorem 1 to Proposition 1 is presented in Section 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' We conclude this section by noting that Proposition 1 immediately implies the following “mollified variant” of Theorem 1 which is of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' If d ≥ 5, η > 0, and L ≥ q4 η, then one has the estimate (3) ��� sup λ≥η−2L |Aλf| ��� 2 ≪ η ∥f∥2 whenever supp �f ⊆ Ωc η,L, with Ωη,L = {α ∈ Td : α ∈ [−L−1, L−1]d + (q−1 η Z)d} and qη = lcm{1 ≤ q ≤ η−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Indeed, note that in proving (3) one may restrict the sup to η−2L ≤ λ ≤ 2η−2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Choosing k, j ∈ N such that 2k ≤ η−2L ≤ 2k+1 and 2j ≥ η−2 we have that 2k−j ≤ L and hence Ωj,k ⊆ Ωη,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Applying Proposition 1 with j and k chosen as above implies Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' This provides a slight strengthening of Proposition 5 in [2], more importantly it provides a significantly simpler direct proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Proof of Theorem 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' A smooth sampling function supported on Ωj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Let ψ ∈ S(Rd) be a Schwartz function satisfying 1Q(ξ) ≤ �ψ(ξ) ≤ 12Q(ξ) where Q = [−1/2, 1/2]d and �ψ(ξ) := � Rd ψ(x)e−2πix·ξdx denote the Fourier transform of ψ on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' For a given q ∈ N and L > q we define ψq,L : Zd → R as ψq,L(m) = �� q L �d ψ � m L � if m ∈ (qZ)d 0 otherwise THE DISCRETE SPHERICAL MAXIMAL FUNCTION: A NEW PROOF OF ℓ2-BOUNDEDNESS 3 Writing m = qr + s with r ∈ Zd and s ∈ Zd/qZd, it follows from Poisson summation that �ψq,L(α) = � m∈Zd ψ(m)e−2πim·α is a q−1-periodic function on Td that satisfies �ψq,L(α) = � ℓ∈Zd �ψ(L(α − ℓ/q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' For a given k ∈ N and 0 ≤ j ≤ Jk := [log2(k)] − 2, we now define the sampling function (4) Ψj,k = ψqj,2k−j and note that supp �Ψj,k ⊆ Ωj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Finally we define ∆Ψj,k = Ψj+1,k −Ψj,k and note the important almost orthogonality property they enjoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' There exists a constant C = CΨ > 0 such that � k≥2j |� ∆Ψk,j(α)|2 ≤ C uniformly in j ∈ N and α ∈ Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Note that Ωk+1,j ⊆ Ωk,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Now fix j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' If α /∈ Ω2j,j, then � ∆Ψk,j(α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' If α ∈ Ω2j,j, then we define k1 = k1(j) := max{k ≥ 2j : α ∈ Ωk,j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Then there exists a unique ℓ1 ∈ Zd such that |α − ℓ1/qj| ≤ 2j−k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Clearly � ∆Ψk,j(α) = 0 if k > k1, while if 2j ≤ k ≤ k1 we have �Ψk,j(α) = ˜Ψ(2k−j(α − ℓ1/qj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' It therefore follows, by writing ∆Ψk,j = (Ψk,j+1 − 1) + (1 − Ψk,j), that |� ∆Ψk,j(α)| ≤ CΨ 2k−j|α − ℓ1/qj| ≤ CΨ 2k−k1 and hence that � k≥2j |� ∆Ψk,j(α)|2 ≤ CΨ � 1≤k≤k1 2−2(k1−k) ≤ 4 3 CΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Proof that Proposition 1 implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Let (5) Mkf := sup 2k≤λ≤2k+1 |Aλf|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Writing f = f ∗ Ψk,0 + Jk−1 � j=0 f ∗ ∆Ψk,j + (f − f ∗ Ψk,Jk) it follows by subadditivity that (6) Mkf ≤ Mk(f ∗ Ψk,0) + Jk−1 � j=0 Mk(f ∗ ∆Ψk,j) + Mk(f − f ∗ Ψk,Jk) Theorem 1 will now follow from a few observations and applications of Proposition 1, in light of the fact that A∗f = sup k Mkf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' First we note that it is straightforward to verify that the first term on the right in (6) above satisfies Mk(f ∗ Ψk,0) ≤ CΨHf uniformly in k, where Hf(n) = sup ℓ>0 1 (2 · 2ℓ + 1)d ��� � m∈[−2ℓ,2ℓ]d∩Zd f(n − m) ��� 4 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT denotes the discrete Hardy-Littlewood maximal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Since, by the same arguments as in Euclidean spaces, we have ∥Hf∥2 ≪ ∥f∥2, it follows that sup k ∥Mk(f ∗ Ψk,0)∥2 ≪ ∥f∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' For the middle terms in (6) we first note that sup k Jk−1 � j=0 Mk(f ∗ ∆Ψk,j) ≪ � ∞ � k=0 ��� Jk−1 � j=0 Mk(f ∗ ∆Ψk,j) ��� 2�1/2 Taking ℓ2 norms of both sides of the inequality above and applying Minkowski’s inequality, followed by an application of Proposition 1, gives ���sup k � 0≤j≤Jk Mk(f ∗ ∆Ψk,j) ��� 2 ≤ � j � � k≥2j ∥Mk(f ∗ ∆Ψk,j)∥2 2 �1/2 ≪ � j 2−j/2� � k≥2j ∥f ∗ ∆Ψk,j∥2 2 �1/2 ≪ ∥f∥2 where the last inequality above follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' One more application of Proposition 1 with j = [log2 k] − 2 to the last term in (6) gives ���sup k Mk(f − f ∗ Ψk,Jk) ��� 2 ≤ � ∞ � k=1 ∥Mk(f − f ∗ Ψk,Jk)∥2 2 �1/2 ≪ � ∞ � k=1 k−1(log2 k)−2�1/2 ∥f∥2 ≪ ∥f∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Proof of Proposition 1 Fix ε = 2−2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' We start by observing that 1{|m|=λ}(m) = � 1 0 e2πi(|m|2−λ2)tdt = e2πελ2 � 1 0 e2πi|m|2(t+iε)e−2πiλ2t dt where 1{|m|=λ} denotes the indicator function of the discrete sphere of radius λ in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Since Nλ ≍ λd−2 it therefore follows that Mkf ≪ sup 2k≤λ≤2k+1 1 λd−2 � 1 0 |f ∗ st| dt where st(m) = e2πi|m|2(t+iε), and hence that ∥Mkf∥2 ≪ ε(d−2)/2 � 1 0 ∥f ∗ st∥2 dt ≤ ε(d−2)/2 �� 1 0 ∥�st 1Ωc j,k∥∞ dt � ∥f∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Thus, in order to prove Proposition 1 it suffices to show that (7) � 1 0 ∥�st 1Ωc j,k∥∞ dt ≪ ε−(d−2)/22−j/2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' To do this we will employ the circle method and decompose the interval into Farey arcs, that is neigh- borhoods Va,q of reduced rationals a/q which allows us to estimate �st(ξ) by using Poisson summation and properties of Gaussian sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Specifically, we decompose the interval [0, 1] into neighborhoods of rationals whose denominator is smaller than 2k as follows: Let H = {a/q : 1 ≤ q ≤ 2k, 0 < a ≤ q, (a, q) = 1} and define Va,q = � t ∈ [0, 1] : |t − a/q| = min r∈H |t − r| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Note that, by Dirichlet’s principle, for every t ∈ [0, 1], there exists a/q ∈ H such that |t − a/q| ≤ 2−kq−1, thus we have that |Va,q| ≤ 2−k+1q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Also, if a/q ̸= a′/q′ with (a, q) = (a′, q′) = 1 and 1 ≤ q, q′ ≤ 2k then THE DISCRETE SPHERICAL MAXIMAL FUNCTION: A NEW PROOF OF ℓ2-BOUNDEDNESS 5 |a/q − a′/q′| ≥ 1/(qq′) ≥ 2−kq−1, hence |Va,q| ≥ 2−kq−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Thus the Farey arcs Va,q at level 2k provide a partition (up to endpoints) of [0,1] into intervals of length |Va,q| ≈ 2−kq−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' It follows from Poisson summation that for 1 ≤ a ≤ q, (a, q) = 1, 1 ≤ q ≤ 2k one has (8) |�st(α)| ≤ q−d/2(ε + |τ|)−d/2 � ℓ∈Zd e− π 2 |α−ℓ/q|2/(ε+ε−1|τ|2) for each t ∈ Va,q with τ = t − a/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' The details of the calculation to derive estimate (8) are laid out more carefully in [3], but they can be briefly summarize as follows: First write �st as a product of one dimensional functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' An application of Poisson summation and a change of variables leaves a double sum that can be recognized as a quadratic Gaussian sum, which can be bounded by q−1/2, and a sum of terms involving �sτ(ℓ/q − α) which has a simple closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' See formula (12) in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Since |τ| ≤ 2−kq−1, it follows that q2(ε + ε−1|τ|2) ≪ 1, and hence that � ℓ∈Zd e− π 2 |α−ℓ/q|2/(ε+ε−1|τ|2) ≪ 1 which in turn implies that if t ∈ Va,q with t = a/q + τ, then (9) ∥�st∥∞ ≪ q−d/2(ε + |τ|)−d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' We write (10) � 1 0 ∥�stχΩc j,k ∥∞ dt = � q|qj � (a,q)=1 � Va,q ∥�st 1Ωc j,k∥∞dt + � q∤qj � (a,q)=1 � Va,q ∥�st 1Ωc j,k∥∞ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' In order to estimate the first double sum above we consider separately the case when |τ| ≥ 2j/2ε and |τ| ≤ 2j/2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' When |τ| ≥ 2j/2ε we use estimate (9) to bound it by (11) � q|qj � (a,q)=1 � Va,q q−d/2(ε + |τ|)−d/2dt ≪ � q|qj q−d/2+1 � ∞ ε2j/2(ε + |τ|)−d/2dτ ≪ ε−(d−2)/22−j(d−2)/4 � q|qj q−d/2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' When |τ| ≤ 2j/2ε we note that because q|qj we have that (12) � ℓ∈Zd e− π 2 |α−ℓ/q|2/(ε+ε−1|τ|2) = e− π 2 |α−ℓ0/q|2/(ε+ε−1|τ|2) + � ℓ̸=ℓ0 e− π 2 |α−ℓ/q|2/(ε+ε−1|τ|2) where ℓ0 denotes the nearest integer to qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' For every α ∈ Ωc j,k we have |α − ℓ0 q | = |α − qjℓ0/q qj | ≥ 2j−k and hence that (13) |e− π 2 |α−ℓ0/q|2/(ε+ε−1|τ|2)| ≪ e−c(2j−k)2/2j−2k ≪ e−c2j since ε + ε−1|τ|2 ≤ 2 · 2jε ≪ 2j−2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' To estimate the sum where ℓ ̸= ℓ0 in (12) above we again use the fact that ε + ε−1|τ|2 ≪ 2j−2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Since |qα − ℓ| ≥ 1/2 for ℓ ̸= ℓ0 and q|qj with j ≤ log2 k − 2 it follows that q2(ε + ε−1|τ|2) ≪ (22j)22j−2k ≤ 2k2−2k ≤ 2−k and hence that (14) � ℓ̸=ℓ0 e− π 2 |α−ℓ/q|2/(ε+ε−1|τ|2) ≪ 2−c2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Combining estimates (13), and (14) it follows that ∥�st 1Ωc j,k∥∞ ≪ q−d/2(ε + |τ|)−d/2 (e−c2j + 2−c2k) 6 NEIL LYALL ´AKOS MAGYAR ALEX NEWMAN PETER WOOLFITT whenever |τ| ≤ 2j/2ε and q|qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Further combining this with (11) we obtain that � q|qj � (a,q)=1 � Va,q ∥�st 1Ωc j,k∥∞dt ≪ � q|qj q−d/2+1ε−(d−2)/2 � (e−c2j + 2−c2k) + 2−j(d−2)/4� ≪ ε−(d−2)/2 2−j(d−2)/4 � q|qj q−d/2+1 ≪ ε−(d−2)/2 2−j(d−2)/4 as the sum over q converges for d ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' In order to estimate the second double sum in (10) we need the following observation, whose proof we delay until after completing the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Given r > 1, k ∈ N, then � q∤Qk q−r ≪r k−r+1(log k)−1 where Qk = lcm{1 ≤ q ≤ k} and ≪r denotes less than a constant depending on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Using estimate (9) and Lemma 2 one can bound the second double sum in (10) above by � q∤qj � (a,q)=1 � Va,q q−d/2(ε + |τ|)−d/2dt ≪ ε−(d−2)/2 � q∤qj q−d/2+1 ≪ ε−(d−2)/22−j(d−4)/2j−1 whenever d ≥ 5 completing the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' □ Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' If q ∈ N such that q ∤ Qk, then either a large power of a small prime divides q, or a large prime divides q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' Explicitly, for prime p ≤ k, let ap = min{a ∈ N : k < pa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' If q ∤ Qk then one of the following must occur: (i) there exists a p > k such that q = pq1 (ii) there exists a p < k such that q = papq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' In the first case � (i) holds q−r ≤ � p>k � q1∈N p−rq−r 1 ≪r � p>k p−r ≪r � m≥0 � p∈[2mk,2m+1k) 2−mrk−r ≪r � m≥0 2−mrk−r 2mk m log(k) ≪r k−r+1(log k)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' In the second case � (ii) holds q−r ≤ � p≤k p−rap � q1∈N q−r 1 ≪r k log k k−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content=' □ References [1] J.' metadata={'source': 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ℓ2-BOUNDEDNESS 7 Department of Mathematics, The University of Georgia, Athens, GA 30602, USA Email address: lyall@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='edu Email address: magyar@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='edu Email address: alxjames@uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='edu Email address: pwoolfitt@uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf'} diff --git a/qdE0T4oBgHgl3EQfaQC1/content/tmp_files/2301.02333v1.pdf.txt b/qdE0T4oBgHgl3EQfaQC1/content/tmp_files/2301.02333v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9766c97262d192b4a8cec37379b0369b1c3554d --- /dev/null +++ b/qdE0T4oBgHgl3EQfaQC1/content/tmp_files/2301.02333v1.pdf.txt @@ -0,0 +1,2399 @@ +MHVG2MTS: Multilayer Horizontal Visibility Graphs for +Multivariate Time Series Analysis +Vanessa Freitas Silva1, Maria Eduarda Silva2, Pedro Ribeiro1, and Fernando Silva1 +1CRACS-INESC TEC, Faculdade de Ciˆencias, Universidade do Porto +2LIAAD-INESC TEC, Faculdade de Economia, Universidade do Porto +Abstract +Understanding the properties of time-indexed multivariate data has been a predominant topic mainly to address open issues in multi- +variate time series analysis, such as finding appropriate measures to analyze temporal dependence and cross-dimension dependencies, +as well as visualizing multidimensional data. Usually, the methodologies used to analyze multivariate time series are based on adapting +approaches for univariate settings or on assumptions and parameters for specific problems. A different strategy uses complex network +to obtain an additional and reduced representation of temporal and causal properties of the time series data. Recent strategies involve +mapping multivariate time series into high-level network structures, specifically into multiplex networks representing interconnections +between contemporary timestamps of different time series components. +In this work, we propose a new mapping method that takes advantage of the entire structure of multilayer networks. We introduce +the multilayer horizontal visibility graph that is based on the new concept of cross-horizontal visibility between lagged timestamps +of different components, which allows describing the cross-dimension dependencies via inter-layer edges. We use a set of existing +topological measures of multilayer networks as well as a novel measure to evaluate and validate our approach, which is parameter-free, +does not require data pre-processing and is applicable to any kind of multivariate time series data. +We provide an extensive experimental evaluation, where we explore the proposed topological measures, showing that the inter-layer +edges based on cross-horizontal visibility preserve more information about the time series data after the mappings, information that +would inevitably be lost using mapping methods that result in single-layer and multiplex structures. We also verify that the information +mapped by the inter-layer edges is not enough on its own, but that it complements the data information captured by the commonly +used intra-layer edges. Furthermore, we complement our analysis by performing a multivariate time series clustering task based on the +proposed measure set of the proposed mapping method, demonstrating its validity. +Keywords: multivariate time series mappings, multilayer horizontal visibility graphs, multivariate time series features +1 +Introduction +Recent technological developments led to the wide availability of large amounts of high dimensional time- +indexed data for which appropriate methodological and computational tools are required. These multidimen- +sional time-indexed data, usually designated as multivariate time series, have become ubiquitous in all domains +from climate studies or health monitoring to financial data analysis, and are characterized by serial correlation +as well as cross-sectional dependencies, in what is often designated as the curse of dimensionality. +To overcome the lack of appropriate methodological and computational tools to characterize high-dimensional +time-indexed data, feature-based approaches for time series analysis have been proposed in the literature. Time +series features are traditionally based on statistics and models for time series analysis and often rely on pre- +processing and/or assumptions that are not usually satisfied. A particular methodology that is free of such +requirements consists in mapping the time series into a complex network and then extracting topological fea- +tures of the network for time series mining tasks and forecasting. In fact, network science, the research area +that studies how to extract information from complex networks (Albert and Barab´asi, 2002; Barab´asi, 2016), +1 +arXiv:2301.02333v1 [cs.SI] 5 Jan 2023 + +provides a vast set of topological graph measurements (Costa et al., 2007), a well-defined set of problems such +as community detection (Fortunato, 2010) or link prediction (L¨u and Zhou, 2011), and a large track record +of successful application of complex network methodologies to different fields (Vespignani, 2018), including +graph classification (Peach et al., 2021). +Univariate time series are mapped into single-layer networks based on the concepts of visibility, transition +probability, and proximity (Zou et al., 2019; Silva et al., 2021). Multivariate time series may be mapped into +single or multiple-layer networks. In the former, the nodes represent the component time series and the edges +represent the relationships between the nodes (component time series) computed using statistical methods or +models. These methods imply that all important information on the dynamics of each time series component, +such as serial correlation, is lost in the mapping. Mapping methods that represent multivariate time series as +multiplex networks were proposed with the objective of preserving both the dynamical (over time) and the +cross-sectional information contained in the multivariate data (Lacasa et al., 2015; Eroglu et al., 2018; Silva +et al., 2021). In these multiplex networks, each component univariate time series is mapped into a layer (using +a univariate time series mapping in which each timestamp is represented by a node) and different layers are +connected via the common nodes (time stamps). Inevitably, lagged cross-correlations, which sometimes are the +most important information, are lost in the mapping process. +To overcome this limitation, we propose a new mapping to represent a multivariate time series as a multiple- +layer complex network in this work. Multiple-layer networks, or multilayer networks, are complex structures +capable of establishing internal connections (within the same layer) and external connections (between different +layers) which allow the creation of very complete and flexible data structures (Kivel¨a et al., 2014). From a high- +level view, multilayer networks have a structure compatible with that of multivariate time series. The proposed +mapping is based on a new horizontal visibility concept, the cross-horizontal visibility, developed to capture the +cross-dependencies between pairs of component time series. Thus, the multiplex visibility graphs (Lacasa et al., +2015) are extended with the incorporation of inter-layer edges established according to the cross-horizontal vis- +ibility between different nodes (time stamps). These new edges/connections can capture dependencies between +different timestamps of different variables. The resulting networks are denoted as multilayer horizontal vis- +ibility graphs. Furthermore, we propose a set of global topological multilayer network features as a novel +set of features for multivariate time series comprising: intra-layer topological features, inter-layer topological +features, all-layer topological features, and relational features which are topological features that relate com- +ponents of the network (such as layers or edges). Within relational features, we propose a new topological +feature aimed at relating intra and inter-layer connections. These different subsets of topological features allow +to analyze and compare the underlying properties of intra-layer and inter-layer edges, and to assess the contri- +bution of the proposed mapping method in relation to the multiplex methods in the literature. This proposed +methodology is represented in Figure 1. +We use synthetic multivariate time series generated from a selected set of multivariate time series models to test +and evaluate the framework proposed in this work. +2 + +Figure 1: Schematic diagram of the network-based features approach to time series mining tasks. +1.1 +Contributions +The main contributions of this work are as follows: +• We introduce a new method for mapping multivariate time series into a complete multilayer network +structure. This mapping is based on the concept of horizontal visibility and on a multiplex visibility +graph to take better advantage of the structural capacity of multilayer networks. As far as we know, +the incorporation of inter-layer edges between different entities of multilayer networks has not been +previously used in the literature to analyze multivariate time series data. +• We propose a new topological feature for multilayer networks and present a different set of multilayer +network topological features selected for the analysis of the proposed mapping method and to reduce the +dimensionality of the multivariate time series data. As far as we know, no other work presents different +topological features of multilayer networks, based on their intra- and inter-layer edges, in the context of +time series analysis. +• We also perform a detailed exploratory and empirical analysis of the different sets of features used in this +work, showcasing its validity and usefulness. +1.2 +Organization +We have organized this document as follows. Section 2 introduces the basic concepts of multivariate time series +and multilayer networks, setting the notation for the remainder of the paper, and also presents the background on +mapping methods useful for understanding the proposed method. Next, Section 3 presents the new concept of +visibility between time series components and the new mapping multivariate time series proposed. In Section 4 +we present the set of topological features that we extend to multilayer networks as well as the new proposed +topological feature to multilayer networks. Section 5 presents a study of this mapping method via analysis of +the corresponding feature set, in order to characterize the properties of the multivariate time series, also presents +a multivariate time series clustering task as a validation of the proposed method. Finally, Section 6 presents the +conclusions, some comments, and possible future work. +3 + +Mapping +Multivariate Time +Network Feature +Feature Analysis +(Multiplex/Multilayer) +Extraction +Series Set +Layers +Similarity +JSDintr +y3 +Multiplex Network Features +All-layer +Relational +Intra-layer +.Inter-layer +JSD1? +y3 +Y4 +： +Multilayer Network Fea2 +Background +This Section introduces the main concepts and notation necessary for the remainder of the paper. +2.1 +Multivariate Time Series +An Univariate Time Series (UTS) is a sequence of (scalar) observations indexed by time t, usually denoted +by {Yt}T +t=1. Unlike a random sample, such observations are ordered in time and usually present serial cor- +relation that must be accounted for in the analysis. If at each time t we obtain a vector of m observations, +Y t = [Y1,t, Y2,t, . . . , Ym,t]′, where ′ represents the transpose, then the data set Y = {Y t}T +t=1 is called a +Multivariate Time Series (MTS). Henceforward, the UTS components of the MTS Y are denoted by Y α = +[Yα,1, Yα,2, . . . , Yα,T ], α = 1, . . . , m and thus we can denote the MTS by its components, Y = {Y α}m +α=1. +MTS data present not only serial correlation within each component, Y α, but also a correlation between the +different UTS’s, Y α and Y β with α ̸= β, both contemporaneous and lagged correlation. Thus, analyzing MTS +depends on key dependence measures such as the autocorrelation function (ACF), which measures the linear +predictability of a UTS, +ρ(s, t) = corr(Yt, Ys) = +cov(Ys, Yt) +� +var(Ys)var(Yt) +, +(1) +and the cross-correlation function (CCF), which measures the correlation between any two components of the +MTS, α and β, say, at times s and t, +ρα,β(s, t) = corr(Yα,s, Yβ,t). +(2) +Time series analysis refers to the collection of procedures developed to systematically solve the statistical +problems posed by the serial correlation. There is a plethora of (linear and non-linear) statistical models in the +literature adequate to describe the behavior of UTS (Shumway and Stoffer, 2017). And although the theory +of UTS extends naturally to the multivariate case, such as the mean, covariance, ACF, and CCF functions, +new concepts arise. MTS analysis requires tools, methods, and models for mining information from multiple +measurements which present both temporal and cross-sectional correlations. +2.2 +Multilayer Networks +A graph (or network), G, is a mathematical structure defined by a pair (V, E), where V represents the set of +nodes and E the set of edges (connections) between pairs of nodes. Two nodes vi and vj are called neighbors +if they are connected, (vi, vj) ∈ E. If there is no direction from a source node to a target node the edges are +undirected: (vi, vj) ∈ E implies that (vj, vi) ∈ E. A graph can be represented by an adjacency matrix, A, and +Ai,j is 1 when (vi, vj) ∈ E and is 0 otherwise. +A Multilayer Network (MNet) is a complete and general structure suitable for modeling multiple complex +systems through their interactions, intra- and inter-systems. A MNet is generally defined as a quadruplet M = +(VM, EM, V, L) where V and L represent the set of entities and the set of layers of M, respectively, and VM and +EM represent the global sets of nodes and edges, respectively. The VM ⊆ V × L1 × . . . × Lm, where Lα ∈ L +is an elementary layer, is a set of node-layer combinations in which a node is present in the corresponding +layer Lα. The EM ⊆ VM × VM is the set of edges that contain the pairs of possible combinations of nodes +and elementary layers (Kivel¨a et al., 2014). We denominate as intra-layer edges, the connections between +nodes of the same layer, (vα +i , vα +j ), and inter-layer edges the connections between nodes of different layers, +(vα +i , vβ +j ) with α ̸= β. Two particular cases of multilayer networks are the monoplex network when m = 1 +and M reduces to a (single-layer) network, G, and the multiplex network, when M is a sequence of m graphs, +{Gα}m +α=1 = {(V α, Eα)}m +α=1, usually with a node set common to all elementary layers, and inter-layer edges +connecting only the counterpart nodes across the layers, that is connecting (vα +i , vβ +i ), α ̸= β (Boccaletti et al., +2014). Figure 2 exemplifies the representation of simple multilayer and multiplex networks. +4 + +(a) Multilayer Network +(b) Multiplex Network +Figure 2: An illustrative example of two toy multilayer networks with five entities, V = {1, 2, 3, 4, 5}, and +two layers L = {L1, L2}. (a) represents a toy multilayer network and (b) a toy multiplex network. Solid lines +represent the intra-layer edges and dashed lines represent the inter-layer edges. +Source: Modified from Silva et al. (2021). +A node-aligned1 MNet has an associated adjacency tensor of order 4, AAA, where the tensor element Ai,j,α,β is 1 +when (vα +i , vβ +j ) ∈ EM and is 0 otherwise (Kivel¨a et al., 2014). If the MNet is not node-aligned, we can consider +empty nodes to complete the tensor structure. Another representation is obtained by flattening AAA into a supra- +adjacency matrix, A, where intra-layer edges are associated with diagonal element blocks and inter-layer edges +with off-diagonal element blocks. Figure 3 represents the supra-adjacency matrices of the networks illustrated +in Figure 2. From these element blocks we can infer three types of subgraphs: +• intra-layer graphs, Gα, represented by the square matrices of order |V α| formed by the diagonal element +blocks (intra-layer edges, Aα +i,j), ie., +� Aα 0 +0 +0 +� +, +• inter-layer graphs, Gα,β, represented by the square matrices of order |V α| + |V β| constructed from +off-diagonal element blocks (inter-layer edges, Aα,β +i,j and Aβ,α +j,i , and no intra-layer edges, Aα +i,j = 0 and +Aβ +i,j = 0) 2, ie., +� +0 +Aα,β +Aβ,α +0 +� +, and +• all-layer graphs, Gα,β +all , represented by the square matrices of size |V α| + |V β| constructed by both on +and off-diagonal element blocks (intra-layer edges, Aα +i,j and Aβ +i,j, and inter-layer edges, Aα,β +i,j and Aβ,α +j,i ), +ie., +� +Aα +Aα,β +Aβ,α +Aβ +� +. +Network science has been a very useful tool to answer the most diverse problems in several scientific fields (Vespig- +nani, 2018). A large number of topological, statistical, spectral, and combinatorial properties metrics that +extract information from networks are available in the literature (Albert and Barab´asi, 2002; Barab´asi, 2016; +Costa et al., 2007; Peach et al., 2021; Silva et al., 2022). We can group these metrics into global, local, and +”intermediate” features. The first group quantifies properties involving all network elements, the second prop- +erties over a given node or edge, and the last properties that involve subsets of the network, such as subgraphs. +These metrics used in monoplex contexts can be extended to MNets. Most of the common topological metrics +can be extended straightforwardly to intra-layer metrics by just computing them over the intra-layer edges. +These metrics can also be extended to the whole MNet, computing them over both intra-layer and inter-layer +edges (Kivel¨a et al., 2014; Huang et al., 2021). Other approaches rely on measurements and properties in the +tensor analysis literature (Kivel¨a et al., 2014). +1A multilayer network is node-aligned if all layers contain all entities, that is, VM = V × L1 × . . . × Lm. +2Note also that the inter-layer graphs have the characteristics of a bipartite graph. Where a bipartite graph is a graph Gα,β whose +node set V α,β can be divided into two disjoint and independent sets V α and V β (V α,β = V α ∪ V β and V α ∩ V β = ∅) and every +edge connects a node in V α to a node in V β. +5 + +4 +3 +- +2 +- +I +1 +- +1 +- +1 +4 +3 +I +2 +54 +3 +2 +5 +4 +3 +2 +5(a) Multilayer Network +(b) Multiplex Network +Figure 3: An illustrative example of two supra-adjacency matrices. (a) a supra-adjacency matrix of a toy +multilayer network and (b) a supra-adjacency matrix of a toy multiplex network. Colored blocks represent the +intra-layer graphs and gray blocks represent the inter-layer graphs. +2.3 +Mapping Time Series into Complex Networks +In the last decade, several network-based time series analysis approaches have been proposed. These ap- +proaches are based on the mappings of univariate and multivariate time series into the network domain, either +into single-layer or multiple-layer networks (Silva et al., 2021). The mappings proposed in the literature are +essentially based on concepts of visibility, transition probability, proximity, time series models, and statis- +tics (Silva et al., 2021; Zou et al., 2019). In this section, we review the concepts of visibility graphs that are +required for the next section. +Visibility Graphs (VG) establish connections (edges) between the timestamps (nodes) using visibility lines +between the observations, where nodes are associated with the natural ordering of observations. There are two +native variants of this method, the Natural Visibility Graph (NVG) (Lacasa et al., 2008) and the Horizontal +Visibility Graph (HVG) (Luque et al., 2009). The idea of these methods is that each UTS observation, Yt, is +seen as a vertical bar with a height equal to its numerical value and that these bars are laid in a landscape where +(the top of) a bar is visible from (the tops of) other bars. Each timestamp, t, is mapped into a node, vt, and +the corresponding edges (vi, vj), for i, j = 1 . . . T, i ̸= j, are established if there is a visibility line between +the corresponding data bars that is not intercepted. Formally, in the NVG and HVG, two nodes vi and vj are +connected if for all tk, ti < tk < tj, (tk, Yk) satisfies +Yk < Yj + (Yi − Yj)(tj − tk) +(tj − ti) +NVG +(3) +Yk < Yi +∧ +Yk < Yj +HVG. +(4) +We give a simple illustration of these methods in Figure 4. +VGs are always connected, each node vi sees at least its nearest neighbors, vi−1 and vi+1, are always undi- +rected unless we consider the direction of the time axis, and are invariant under affine transformations of the +data (Lacasa et al., 2008), each transformation Xt = aYt + b, for a ∈ R, b ∈ R, and t = 1, . . . , T, leads to the +same VG (Silva et al., 2021). +6 + +Figure 4: An illustrative example of the two visibility graph algorithms. (a) toy time series and corresponding +visibility lines between data bars (observations). Solid pink lines represent the natural visibility lines corre- +sponding to the NVG method, and dotted blue lines represent the horizontal visibility lines corresponding to +the HVG method. (b) network generated by the corresponding mappings. The NVG is the graph with all edges, +including the dashed pink edges, and the HVG is the subgraph that does not include these edges. +Source: Adapted from Silva et al. (2022). +Based on the definition of MNet, Lacasa and co-authors (Lacasa et al., 2015) proposed an extension of the +visibility mapping for MTS analysis, the Multiplex Visibility Graphs (MVG). Formally, a MVG of m layers, +M, is constructed so that layer set, {Lα}m +α=1, corresponds to the NVGs (or HVGs), {Gα}m +α=1, associated with +the time series components, {Y α}m +α=1. M is represented by the adjacency matrix vector, AM, whose elements +are the adjacency matrices of each layer, AM = {A1, A2, . . . , Am} with Aα +i,j = 1 if the nodes vα +i and vα +j are +connected in layer Lα and is 0 otherwise. Figure 5 illustrates the method. +Figure 5: An illustrative example of the multiplex natural visibility graph algorithm. (a) displays a toy multi- +variate time series, Y = {Y 1, Y 2, Y 3}, (b) the corresponding multiplex network, with three layers generated +by the multiplex natural visibility graph algorithm. +7 + +3 +MHVG: a New Multivariate Time Series Mapping +Visibility methods have shown to be very promising in capturing time series characteristics reflecting local and +global properties of the data, and not requiring pre-processing. +This section presents a new visibility algorithm to map an MTS into a multilayer horizontal visibility graph. +This algorithm is based on a new visibility concept, cross-horizontal visibility which is an extension of the +traditional horizontal visibility. +3.1 +Cross-Horizontal Visibility +Consider two time series Zα = (Zα,1, . . . , Zα,T ) and Zβ = (Zβ,1, . . . , Zβ,T ) on the same scale. Two arbitrary +data values (ti, Zα,ti) and (tj, Zβ,tj) are said to have cross-horizontal visibility, Cross-HV if +Zα,ti ∧ Zβ,tj > max (Zα,t, Zβ,t), +for all t, ti < t < tj, i, j = 1, . . . , T, i ̸= j. +(5) +This definition implies that all data values have Cross-HV to its neighbours and that the visibility is reciprocal, +meaning that if (t, Zα,t) has Cross-HV to (s, Zβ,s), then (s, Zβ,s), has Cross-HV to (t, Zα,t). The concept of +cross-horizontal visibility, Cross-HV, is illustrated in Figure 6 with two toy time series and for the first four data +points, with the bi-coloured lines indicating (the reciprocal) visibility between the corresponding time series. +Figure 6: Schematic diagram of the cross horizontal visibility concept. (a) Illustrates a toy bivariate time series +Zyellow, Zblue, (same scale) and the corresponding maximum time series; (b) represents the cross-horizontal +visibility, Cross-HVG, by solid bi-color lines (yellow and blue) connecting the data bars of the time series +components Zyellow and Zblue, for the first four timestamps. +3.2 +Multilayer Horizontal Visibility Graph +A Multilayer Horizontal Visibility Graph (MHVG) is obtained by mapping a MTS, Y = {Y α}m +α=1, into a +MNet structure, M = (VM, EM, V, L), using the concepts of HV and Cross-HV, as follows. Each unique +timestamp, t, is mapped into an unique entity in VM and each component time series, Y α is mapped into a +layer, Lα ∈ L, α = 1, . . . , m, using the HVG method described in Section 2.3, thus establishing the intra-layer +edges, (vα +i , vα +j ) ∈ EM, i, j = 1, . . . , T, i ̸= j. Then inter-layer edges (vα +i , vβ +j ) ∈ EM, between any two layers +Lα and Lβ, α, β = 1, . . . , m, α ̸= β and i, j = 1, . . . , T, i ̸= j are established using the Cross-HV described +above in Section 3.1. Note that to establish Cross-HV all the time series Y α, α = 1, . . . , m must be in the same +scale which may require a pre-processing step of the data set Y , comprising the Min-Max scaling of each time +series. The mapping is illustrated in Figure 7, with toy bivariate time series, for the sake of simplicity. +8 + +blue +4 +2 +3Figure 7: Schematic diagram of the multilayer horizontal visibility graph algorithm: (a) original time series, (b) +Min-Max re-scaled time series and maximum time series, (c) illustration of cross-HV with the edges between +adjacent timestamps omitted for simplicity (detail for the first four timestamps), (d) cross-horizontal visibility +graph: solid black lines represent the intra-layer edges (the HVGs), dashed lines the inter-layer edges (the Cross- +HVGs) and the red lines highlight inter-layer edges between nodes corresponding to non-adjacent timestamps. +From the generated MHVG, we can identify the intra-layer graphs, {Gα}m +α=1 and the inter-layer graphs, Gα,β, +for α, β = 1, . . . , m and α ̸= β. {Gα}m +α=1 correspond to the HVG of each individual time series component and +it is represented by the adjacency matrix Aα with Aα +i,j = 1 if (vα +i , vα +j ) ∈ EM and 0 otherwise. Gα,β corresponds +to the cross-horizontal visibility graph (Cross-HVG) of each pair of different time series components and it is +represented by the adjacency matrix Bα,β = +� +0 +Aα,β +Aβ,α +0 +� +with Aα,β +i,j = 1 and Aβ,α +j,i = 1 if (vα +i , vβ +j ) ∈ EM, and +0 otherwise. +Algorithm 1 describes the concept of Cross-HV and Algorithm 2 describes mapping a multivariate time series +into a Multilayer Horizontal Visibility Graph. In the Appendix, we describe the auxiliary functions to support +the implementation of the method, Algorithm 3 describes the function that creates an HVG, and Algorithm 4 +describes the function that creates the inter-layer edges. +9 + +Algorithm 1: Cross-Horizontal Visibility Graph +Input: Two rescaled time series, Za and Zb, (tsA, tsB), the corresponding layers, La and Lb, (layerA, layerB), and the +maximum time series, max +� +Za, Zb� +, (tsMax) +Procedure CHVG(tsA, tsB, tsMax, layerA, layerB) +1 +T ← tsMax.size() +▷ The time series lengths +for node i in layerA.get Nodes() do +for node j from i + 1 to T in layerB.get Nodes() do +if node i can ’see’ j then +2 +mnet.add Edge(i, j, layerA, layerB) +▷ Add inter-layer edge (va +i , vb +j) +end +end +for node j from i − 1 to 0 in layerB.get Nodes() do +if node i can ’see’ j then +3 +mnet.add Edge(i, j, layerA, layerB) +▷ Add inter-layer edge (va +i , vb +j) +end +end +end +4 +return +Algorithm 2: Multilayer Horizontal Visibility Graph +Input: A set of time series components, {Y a}m +a=1, (mts) +Output: A multilayer network, M, (mnet) +Procedure MHVG(mts) +1 +m ← mts.size() +▷ The number of time series components +2 +mnet ← {} +▷ The empty MNet M +3 +n mts ← {} +▷ List to store the rescaled time +series components +for a ← 1 to m do +4 +mnet.layers[a] ← {} +▷ The empty layer La +5 +HVG(mts[a], mnet.layers[a]) +▷ Map the time series Y a on the HVG +La (Eq. 4) +6 +n mts[a] ← MinMax(mts[a]) +▷ Rescale time series Za +end +for a ← 1 to m − 1 do +for b ← a + 1 to m do +7 +tsMax ← MaxTS(n mts[a], n mts[b]) +▷ Get the maximum rescaled time series +8 +CrossHVG(n mts[a], n mts[b], tsMax, mnet.layers[a], mnet.layers[b]) +▷ Map the pairwise time +series Za and Zb +on Cross-HVG (Eq. 5) +end +end +9 +return mnet +10 + +4 +A Novel Set of Multivariate Time Series Features +Mapping time series into complex networks and then using network topological features as features in univariate +time series mining tasks has become a popular approach due to its dimensionality reduction capabilities (Silva +et al., 2022; Fulcher, 2018; Wang et al., 2006). +In this work, we propose a set of MHVG topological features to analyze MTS data which includes: i) common +topological features extended to MNets and ii) a new feature constructed for MNets. +4.1 +Topological features extended to MNets +Common network topological features such as node centrality, graph distances, clustering, and community can +be naturally extended to a MNet structure and all the subgraphs mentioned in Section 2.2. To illustrate, consider +a local centrality measure such as the degree ki of a node vi, which represents the number of its adjacent edges. +In a MNet, we can compute three variants of ki, for each layer α = 1, . . . , m where we use the symbol ≺ (and +⪯) to express the inter-layer edges from a ”source” layer, α (and including intra-layer edges of the ”source” +layer) to a ”destination” layer, β, +• intra-layer degree: kα +i = � +j Aα +ij +• inter-layer degree: kα≺β +i += � +j Aα,β +ij +• all-layer degree: kα⪯β +i += kα +i + kα≺β +i +with β ̸= α. Note that local inter and all-layer topological measurements are asymmetric measures, that is, +kα≺β +i +̸= kβ≺α +i +and kα⪯β +i +̸= kβ⪯α +i +, since the measure is relative to node-layer vα +i or node-layer vβ +i . +In general, any common (local) topological feature Fi can be easily extended to intra-layer features, F α +i , just +computing them over individual layers, to inter-layer features, F α≺β +i +, computing over inter-layer edges, and to +all-layer features, F α⪯β +i +, which compute over both intra-layer and inter-layer edges. +An important feature associated with the degree is the degree distribution P(k) that measures the fraction of +nodes in a graph with degree k. In this work, we analyze the three variants of degree distributions, P(kα), P(kα≺β) +and P(kα⪯β), in layer Lα, α = 1, . . . , m, associated with its intra-layer degree, inter-layer degree and all-layer +degree, respectively. +To measure the similarity between pairs of layers in an MNet, we also use the Jensen–Shannon divergence +(JSD) which measures the distance between two distributions. As an example, the JSD between intra-layer +degree distributions P(kα) and P(kβ), (JSDα,β +intra) is defined as follows: +JSD(P(kα)||P(kβ)) = 1 +2KLD(P(kα)||Q(k)) + 1 +2KLD(P(kβ)||Q(k)) +where Q(k) = 1 +2(P(kα) + P(kβ)) and KLD is the Kullback–Leibler divergence: +KLD(P(kα)||Q(k)) = +� +k +P(kα) log2 +�P(kα) +Q(k) +� +. +Similarly, we define JSD for the inter-layer degree distributions (JSDα,β +inter) and the all-layer degree distribu- +tions (JSDα,β +all ). Note that JSD is a symmetrical version of the asymmetrical measure KLD. In the remainder +of this work, we will refer to similarity measures, such as JSD, as relational features. +11 + +In addition, we also extend global topological features to MNets. These features involve all (sub)graph elements +and therefore are symmetric. As an example, consider the average degree ¯k which calculates the arithmetic +mean of the degree ki of all nodes in the graph. As before, we can compute three variants of ¯k in a MHVG, +• average intra-degree: ¯kα = +1 +|Vα| +� +i kα +i +• average inter-degree: ¯kα,β = +1 +|Vα|+|Vβ| +�� +i kα≺β +i ++ � +j kβ≺α +j +� +• average all-degree: ¯kα,β +all = +1 +|Vα|+|Vβ| +�� +i kα⪯β +i ++ � +j kβ⪯α +j +� +In short, we can compute a (global) topological feature F in the subgraphs of the MNet: intra (F α), inter +(F α,β), and all-layer graphs (F α,β +all ). +Motivated by the set of features proposed in Silva et al. (2022), namely based on the concepts of node centrality, +graph distances, clustering, and community and the three types of MNet measurements defined above, we +propose intra-layer, inter-layer, and all-layer, for each pair of layers, features as follows: +• Average Degree: the average intra-degree ¯kα, average inter-degree ¯kα,β and average all-degree ¯kα,β +all , as +formulated above. +• Average path length: geodesic distances di,j, i ̸= j between node vi and vj corresponding to the length +of the shortest paths between them, where the path length is the number of edges in the path. The average +(intra-/inter-/all-)path length ( ¯dα, ¯dα,β and ¯dα,β +all ) is the arithmetic mean of the shortest paths among all +pairs of nodes in (intra, inter, and all-layer) graph. +• Number of communities: The number of (intra-/inter-/all-)communities, (Sα, Sα,β and Sα,β +all ), is the +amount of groups/communities of nodes that are densely connected on the subgraph. These communities +are found by performing random walks on the subgraph (intra, inter, and all-layer graph), so that short +random walks tend to stay in the same community until the modularity value (defined below) cannot be +increased anymore. +• Modularity: (Intra-/Inter-/All-)modularity, (Qα, Qα,β and Qα,β +all ), measures how good a specific division +of the corresponding subgraph G is into (intra-/inter-/all-)communities. +Table 1 provides the formulation details. +4.2 +Ratio Degree: a new MNet topological feature +The ratio degree is a new topological feature for multilayer graphs, introduced here to relate intra-layer and +inter-layer visibility. +The ratio degree of node vα +i of layer Lα is defined as +rα⪯β +i += kα≺β +i +kα +i +(6) +for any layer Lβ. The average ratio degree, ¯rα⪯β, is the arithmetic mean of the ratio degree of the nodes of +layer Lα. Note that the ratio degree is asymmetric, and thus it is not necessarily true that ¯rα⪯β = ¯rβ⪯α. +4.3 +The MHVG Features Set +The set of features defined above and summarized in Table 1 forms a set of features extracted from MHVG, as +depicted in Figure 8, that we propose to characterize a MTS. +12 + +Table 1: Topological multilayer network features 3. +Feature +Formulation +Note +Average Degree +¯kα = +1 +Nα +� +i +kα +i +¯kα,β = +1 +Nα,β +�� +i +kα≺β +i ++ +� +j +kβ≺α +j +� +¯kα,β +all = +1 +Nα,β +�� +i +kα⪯β +i ++ +� +j +kβ⪯α +j +� +Degree Distribution +P(kα) = nkα +Nα +n: number of nodes vα +i +with the corresponding +degree k +P(kα≺β) = nkα≺β +Nα +P(kα⪯β) = nkα⪯β +Nα +Average Path +Length +¯dα = +1 +Nα(Nα − 1) +� +i̸=j +dα +i,j +di,j: length of the shortest +paths between vi and vj in +the corresponding subgraph +¯dα,β = +1 +Nα,β(Nα,β − 1) +� +i̸=j +dα,β +i,j +¯dα,β +all = +1 +Nα,β(Nα,β − 1) +� +i̸=j +dα,β +i,j,all +Number of +Communities +Sα = |Cα| +C: set of communities in +corresponding subgraph +Sα,β = |Cα,β| +Sα,β +all = |Cα,β +all | +Modularity +Qα = +1 +2|Eα| +� +i,j +� +Bi,j − kikj +2|Eα| +� +δ (ci, cj) +B = Aα +Qα,β = +1 +2|Eα,β| +� +i,j +� +Bi,j − +kikj +2|Eα,β| +� +δ (ci, cj) +B = +� +0 +Aα,β +Aβ,α +0 +� +Qα,β +all = +1 +2|Eα,β +all | +� +i,j +� +Bi,j − +kikj +2|Eα,β +all | +� +δ (ci, cj) +B = +� +Aα +Aαβ +Aβα +Aβ +� +Jensen–Shannon +Divergence +JSDα,β +intra = JSD(P(kα)||P(kβ)) +JSDα,β +inter = JSD(P(kα≺β)||P(kβ≺α)) +JSDα,β +all = JSD(P(kα⪯β)||P(kβ⪯α)) +Average Ratio +Degree +¯rα⪯β = +1 +|Nα| +� +i +rα⪯β +i +2Remember that V α is the set of nodes in layer Lα, we define Nα = |V α| and Nα,β = |V α| + |V β| the number of nodes in the +corresponding layer(s). +3Remember that V α is the set of nodes in layer Lα, we define Nα = |V α| and Nα,β = |V α| + |V β| the number of nodes in the +corresponding layer(s). +13 + +Figure 8: Schematic diagram of the multilayer network features set extraction process. A multivariate time +series Y is mapped into a multilayer horizontal visibility graph. And for each of its subgraphs (intra-layer, inter- +layer, and all-layer graphs) are computer the global topological features (¯k, ¯d, S, Q and P(k)) and relational +features (¯r and JSD). +5 +Empirical evaluation of MHVG +In this section, we investigate whether the mapping method and the features set introduced above are useful +for characterizing MTS data and evaluate the performance of the methodology for MTS mining tasks. We +use synthetic bivariate time series, generated from bivariate time series models to control for MTS correlation +(serial and cross) properties. First, we make some considerations about the implementation of the methodology. +5.1 +Implementation Details +The mapping and topological features proposed in this work do not, conceptually, depend on the implementation +details. However, to show the practicality of the proposed method and to provide the reader with the ability to +reproduce it, we describe in more detail how we computed our methodology, illustrated in Figure 8. Note that +for illustrative purposes, we used m = 2, but the method is extensible to any value of m. +To map a multivariate time series Y into an MHVG, we follow Algorithm 2. The intra-layer HVG’s, {Gα}m +α=1, +for each time series component, {Yα,t}, α = 1, . . . , m, t = 1, . . . , T, are created using the implemented Algo- +rithm 3 proposed in (Luque et al., 2009). And the inter-layer edges are added following the mapping method +based on cross-horizontal visibility criteria proposed in Section 3.1, and using the implemented Algorithm 4. +Subgraphs corresponding to intra-, inter-, and all-layers, are fixed via corresponding adjacency submatrices of +the MHVG (see Sections 2.2). And the corresponding topological features described above are computed using +the methodologies and algorithms described below. +14 + +Time Series +Multivariate +Feature Extraction +8 +l dl s1:Ql: P(kl) +10 +3 +8 +K² : d² : s2 : Q²: P(k2) +Multilayer Horizontal +6 +10 +Visibility Graph +5 +T? di I st? IQf P(kt) +1±2 : #2±1 iJSD1.2 +IJSD12 +all +intra +④ +5 +10 +7 +8 +K1,2 +P(k1,2) +3 +4 +9The average degree (¯k) and average ratio degree (¯r) are calculated by the arithmetic mean of the degrees ki +and ratio degrees ri, respectively, of all node vi in the respective subgraph. In this work, the average path +length ( ¯d) follows an algorithm that computes the average shortest path length between all pairs of nodes (of +respective subgraphs) using a breadth-first search algorithm. To calculate the number of communities (S), +the function used makes use of the known ”Louvain” algorithm that finds community structures by multi- +level optimization of modularity (Q) feature (see Blondel et al. (2008) for more details). And the degree +distributions (P(k)) and Jensen–Shannon divergence (JSD) are implemented as described above section. +We used C++ and its needed set of libraries (such as igraph and standard libraries) to implement the data +structure to store an MNet and compute the functions to extract the topological features. For reproducibil- +ity purposes, the datasets and results are made available in https://github.com/vanessa-silva/ +MHVG2MTS. +5.2 +Synthetics Datasets +We consider a set of six bivariate time series models (m = 2), denoted by Data Generating Processes (DGPs), +summarized in Table 2. These MTS models present a set of particular characteristics in terms of serial and +cross-correlation (see Section2.1), namely: white noise (WN) processes simulate noise effects , one process +does not present any kind of correlation, and the other presents strong contemporaneous correlation; vector au- +toregression (VAR) processes simulate smooth linear data, presenting both serial and cross-correlation; vector +generalized autoregressive conditional heteroskedasticity (VGARCH) processes simulate nonlinear data with +persistent periods of high or low volatility. The parameters of each DGP are chosen so that the data exhibits a +range of serial and cross-correlation properties as described in Table 2. A detailed description of the DGP and +their properties as well as computational details are presented in Appendix B. +For each DGP in Table 2, we generated 100 instances of length T = 10000. As illustrated in Figure 8, we +map each bivariate time series into an MHVG, highlight the intra-, inter-, and all-layer graphs, and extract the +corresponding topological features. +Table 2: Synthetic data generating processes (bivariate time series models). Parameters, the main characteristic +of the data sets, and notation are also included. See Appendix B for more details. +DGP +Parameters +Characteristics +Notation +Independent White Noise +ϵt ∼ N(0, 1) +Noise effect +No correlation +iBWN +Correlated White Noise +� ϵ1,t +ϵ2,t +� +∼ N +� +0, +� 1.00 0.86 +0.86 1.50 +�� +Noise effect +No serial correlation +Cross-correlation +cBWN +Weak VAR(1) +ϕ = +� 2.50 +0.50 +� +, φ = +� 0.20 0.10 +0.02 0.10 +� +Weak correlation +(serial and cross) +wVAR +ϵt ∼ +� 1.00 0.10 +0.10 1.50 +� +Strong VAR(1) +ϕ = +� 0 +0 +� +, φ = +� 0.70 0.02 +0.30 0.80 +� +Strong correlation +(serial and cross, lagged +and contemporaneous) +sVAR +ϵt ∼ +� 1.00 0.86 +0.86 1.50 +� +Weak VGARCH(1, 1) +ω = +� 0.05 +0.02 +� +, α = +� 0.10 0.00 +0.00 0.05 +� +No serial correlation +Weak cross-correlation +wGARCH +β = +� 0.85 0.00 +0.00 0.88 +� +, ϵt ∼ +� 1.00 0.10 +0.10 1.50 +� +Strong VGARCH(1, 1) +ω = +� 0.05 +0.02 +� +, α = +� 0.10 0.00 +0.00 0.05 +� +Strong contemporaneous +cross-correlation +sGARCH +β = +� 0.85 0.00 +0.00 0.88 +� +, ϵt ∼ +� 1.00 0.86 +0.86 1.50 +� +15 + +To illustrate the procedure, we represent in Figure 9 one instance with 300 observations of each DGP and the +corresponding cross-correlation (CCF) plot (first two columns of the plot), the intra-, inter-, and all-layers de- +gree distributions on a semi-logarithmic scale (last three columns of the plot). These degree distributions are +computed as the arithmetic mean of the degree distributions of the 100 simulated instances. The plots clearly +show that the degree distributions are different across the DGPs. In fact, Luque et al. (2009) has shown that +the intra-layer degree distribution for white noise (uncorrelated data) follows a power law +� +P(k) = 1 +3 +� 2 +3 +�k−2� +and our results indicate that strong serial correlation leads to intra-layer degree distributions that are positively +skewed: as illustrated in Appendix B, the sVAR is the only DGP that produces data with strong serial cor- +relation. The degree distribution for the inter-layer subgraphs does not have an algebraic close form even in +the simplest case of two uncorrelated white noises. However, extensive simulations indicate that it does not +follow the power law P(k) = 1 +3 +� 2 +3 +�k−2, as illustrated in the first line, the third column of Figure 9. The plots +also indicate that inter-layer degree distribution depends both on the correlation between the two time series +(CCF represented in the second column of the plot in Figure 9) and the serial correlation within each time +series. Moreover, we note that inter-layer degree distributions for sVAR are positively skewed, for GARCH +models, wGARCH and sGARCH, are exponentially shaped while the remaining are approximately linear. Once +again, a slower decay of the lagged correlation leads to a longer tail in the degree distribution. Also, the de- +gree distribution curves corresponding to the GARCH models stand out from the others, especially the inter- +and all-layer degree distributions. The exponential shape of the inter-layer degree distributions is induced by +the heteroscedasticity and volatility clusters in the data which limit cross-horizontal visibility to the nearest +neighbours. +5.3 +MTS features via MHVG +The results for all the 21 features introduced in Section 4 and all DGPs, organized by subgraph structure, are +summarized, mean (standard deviation), in Tables 3 and 4. The values have been Min-Max normalized (across +models) for comparison purposes since the range of the different features varies across the different DGPs. The +cells in the tables are coloured with a gradient based on the mean values: cells with a maximum value of 1 are +coloured red, cells with a minimum value of 0 are coloured white, and the remainder with a hue of red colour +proportional to its value. +Table 3: Mean values (standard deviation) of the 100 instances of each DGP for each topological global feature +from intra-layer graphs, G1 and G2, and inter-layer graph, G1,2, resulting from the corresponding MHVGs. +DGP +Average +Average +Number of +Modularity +Degree +Path Length +Communities +¯k1 +¯k2 +¯k1,2 +¯d1 +¯d2 +¯d1,2 +S1 +S2 +S1,2 +Q1 +Q2 +Q1,2 +iBWN +0.805 +0.756 +0.615 +0.044 +0.049 +0.012 +0.265 +0.312 +0.206 +0.150 +0.207 +0.319 +(0.081) +(0.077) +(0.033) +(0.019) +(0.023) +(0.024) +(0.083) +(0.083) +(0.064) +(0.055) +(0.056) +(0.093) +cBWN +0.802 +0.752 +0.940 +0.045 +0.050 +0.007 +0.260 +0.310 +0.126 +0.150 +0.196 +0.181 +(0.083) +(0.080) +(0.078) +(0.022) +(0.022) +(0.015) +(0.084) +(0.090) +(0.080) +(0.051) +(0.062) +(0.141) +wVAR +0.790 +0.759 +0.615 +0.058 +0.056 +0.009 +0.342 +0.338 +0.211 +0.287 +0.277 +0.308 +(0.079) +(0.090) +(0.033) +(0.022) +(0.025) +(0.016) +(0.093) +(0.100) +(0.062) +(0.058) +(0.062) +(0.099) +sVAR +0.561 +0.601 +0.683 +0.449 +0.328 +0.011 +0.791 +0.700 +0.195 +0.893 +0.857 +0.263 +(0.121) +(0.108) +(0.092) +(0.050) +(0.039) +(0.025) +(0.104) +(0.121) +(0.069) +(0.046) +(0.064) +(0.103) +wVGARCH +0.540 +0.554 +0.102 +0.380 +0.325 +0.252 +0.239 +0.282 +0.696 +0.185 +0.207 +0.669 +(0.159) +(0.140) +(0.135) +(0.136) +(0.097) +(0.188) +(0.104) +(0.083) +(0.212) +(0.057) +(0.081) +(0.210) +sVGARCH +0.542 +0.505 +0.146 +0.390 +0.385 +0.232 +0.234 +0.300 +0.670 +0.179 +0.212 +0.626 +(0.184) +(0.186) +(0.174) +(0.138) +(0.149) +(0.217) +(0.103) +(0.105) +(0.220) +(0.065) +(0.065) +(0.240) +16 + +Table 4: Mean values (standard deviation) for each topological global and relational features from all-layer +graphs, G1,2 +all , resulting from MHVGs of DGP, computed over the 100 instances of each DGP. +DGP +Average +Average +Num. of +Modular. +Average +Jensen–Shannon +Degree +Path L. +Comm. +Ratio Deg. +Divergence +¯k1,2 +all +¯d1,2 +all +S1,2 +all +Q1,2 +all +¯r1⪯2 +¯r2⪯1 +JSDα,β +intra +JSDα,β +inter +JSDα,β +all +iBWN +0.617 +0.042 +0.237 +0.338 +0.586 +0.582 +0.170 +0.034 +0.090 +(0.033) +(0.018) +(0.107) +(0.052) +(0.028) +(0.037) +(0.071) +(0.0355) +(0.047) +cBWN +0.940 +0.051 +0.382 +0.507 +0.942 +0.931 +0.166 +0.061 +0.244 +(0.078) +(0.018) +(0.087) +(0.064) +(0.071) +(0.084) +(0.079) +(0.076) +(0.236) +wVAR +0.616 +0.054 +0.305 +0.413 +0.565 +0.571 +0.207 +0.034 +0.096 +(0.033) +(0.022) +(0.107) +(0.049) +(0.032) +(0.034) +(0.074) +(0.041) +(0.051) +sVAR +0.682 +0.457 +0.457 +0.842 +0.574 +0.579 +0.682 +0.120 +0.314 +(0.092) +(0.04955) +(0.127) +(0.05673) +(0.106) +(0.091) +(0.128) +(0.148) +(0.225) +wVGARCH +0.104 +0.297 +0.310 +0.206 +0.103 +0.097 +0.154 +0.085 +0.075 +(0.134) +(0.065) +(0.108) +(0.076) +(0.136) +(0.127) +(0.065) +(0.110) +(0.065) +sVGARCH +0.147 +0.388 +0.431 +0.3954 +0.149 +0.145 +0.149 +0.114 +0.137 +(0.173) +(0.120) +(0.118) +(0.088) +(0.179) +(0.170) +(0.077) +(0.161) +(0.172) +The results indicate that each set of features - intra-layer (first two columns of each feature in Table 3), inter- +layer (third column of each feature in Table 3), all-layer (first four columns of Table 4) and relational (last five +columns of Table 4) - distinguishes two groups of MTS depending on properties pertaining to correlation (serial +and cross) and volatility clustering. +To understand which MNet topological features capture the specific properties of the MTS models, we perform +PCA on the feature space. Figure 10 represents a bi-plot obtained using the intra-, inter-, all-layer, and relational +features, with the two principal components (PC) explaining 83.8% of the variance. The bi-plots resulting from +PCA in restricted feature sets are represented in Figure 14 (Appendix B). Overall, we can say that the average +degree and average ratio degree, ¯k and ¯r, are positively and negatively correlated, respectively, with the average +path length, ¯d. The community-related features of the intra- and all-layer graphs are positively correlated but +less correlated with the community-related features of the inter-layer graphs. The features that most contribute +to the first two PCs are the ¯k1,2, S1,2 and Q1,2 of the inter-layer graphs, the ¯k1,2 +all of the all-layer graphs, the ¯r1⪯2 +and ¯r2⪯1 of the relational layers, and Q1 and Q2 of the intra-layer graphs (see Figure 15 of Appendix B). +Figure 10 clearly shows four groups of models, GARCH, sVAR, cBWN and a group constituted by the wVAR +and the iBWN and identifies the topological features that characterize them. +The strong ACF and CCF of the sVAR are represented by high values for the number of communities and +modularity in its intra- and all-layer graphs. Inter-layer graphs present higher values of community-related +features for GARCH models. The average path length represents the GARCH models, in particular, the average +path length of the all-layer graphs tries to distinguish both wGARCH and sGARCH. The strong contemporaneous +CCF of the cBWN is represented by high values of average ratio degree features, such as the average degree +values of its inter and all-layer graphs. The iBWN and wVAR, are represented by high values of intra-layer +average degree. +The above results indicate that the topological features extracted from MHVG are adequate as a set of MTS +features. To further explore this idea, we proceed with clustering analysis of the DGPs via MNet topological +features. +17 + +5.4 +Mining Time Series with MNetF +In this section, we illustrate the usefulness of MNet features in MTS data mining tasks with a case study +regarding DGP clustering via a feature-based approach (Maharaj et al., 2019). Given a set of MTS, we compute +the MNet feature vectors, which are then Min-Max rescaled into the [0, 1] range and organized in a feature +data matrix. The PCs are computed, with no need for z-score normalization in PCA computation, and finally, a +clustering algorithm, the k-means, is applied to all PCs. We opt for k-means since it is fast and widely used in +the literature, but it requires the pre-introduction of the number of clusters, which for the purpose of this work +is not a problem. We use the appropriate evaluation metrics: Average Silhouette (AS), Adjusted Rand Index +(ARI), and Normalized Mutual Information (NMI). The AS does not need the ground truth, while the ARI and +NMI do. The range of values for NMI is [0, 1] and for ARI and AS is [−1, 1]. The results are summarized in +Figure 11. +As illustrated in Figure 11, the different metrics indicate different number of clusters for the data set: ARI +indicates k = 5 followed by k = 6 which is the ground truth value, while the NMI indicates either k = 6 +or k = 7. Metric AS, using the silhouette method to assess the quality of the clusters indicates k = 3. It is +interesting to note that the elements of the three clusters are: the sVAR models in one cluster, the GARCH +models in another, and the WN and wVAR models in the third cluster, indicating that the DGPs were clustered +according to correlation (serial and cross) and volatility properties. +Figure 12 represents the results from clustering the 100 instances of the 6 DGPs, using the 21 MNet features and +the k-means algorithm with k = 6. It is noticeable the perfect attribution of cBWN and sVAR samples across +two different clusters (clusters 1 and 3), the attribution of iBWN and wVAR samples across the same cluster +(cluster 2), and a homogeneous attribution of GARCH samples (wGARCH and sGARCH) across two clusters (4 +and 5), and some samples of cBWN and sGARCH across cluster 6 since k = 6 and the GARCH samples are the +most disparate in the feature space. +The clustering exercise was performed considering subsets of the MNet feature set. The results summarized in +Table 5 indicate that inter-layer edges contain, in fact, information about the MTS, leading to better clustering +results. +Subgraphs with both intra-layer edges and inter-layer edges add information that leads to improvements in the +clustering results (compare the last three rows of the Table with the first two). The results show that cross- +HVG of inter-layer edges capture different properties from MTS data, that, as we expected, translate into better +clustering results. Also note that the results of ARI and NMI from the set of intra-layer features are good results, +because the DGP under analysis involves the same statistical process for the two time series components whose +properties inherent to each process are also captured by the HVG mapping methods. +Table 5: Clustering evaluation metrics for the different clustering analyses resulting from different MNet feature +vectors. The values reflect the mean of 10 repetitions of the proposed method for different feature vectors and +for the ground truth (k = 6). The highest values are highlighted. +Feature Set +ARI +NMI +AS +[−1, 1] +[0, 1] +[−1, 1] +Intra-layer +0.522 +0.614 +0.289 +Inter-layer +0.294 +0.418 +0.508 +All-layer +0.667 +0.725 +0.505 +Relational +0.575 +0.646 +0.622 +MNet +0.629 +0.713 +0.452 +18 + +Figure 9: Analysis plots of DGP. The first column shows a subset of timestamps of each DGP, the second plots +the cross-correlation function between their corresponding time series components, and the last three show +the degree distribution of the MHVGs. The plots corresponding to the degree distributions are on a semi- +logarithmic scale. Lines of different colors refer to each DGP model (Y ), where the darkest colors refer to their +first components (Y 1) and the lighter ones to the second (Y 2). +19 + +MTS +CCF +P(k): Intra-Layer +P(k): Inter-Layer +P(k): All-Layer +Independent N +3 +1.00 - +1e+00 +1e+00 +1e+00 +2 +0.75 +1 +1e-01 +1e-01 +1e-01 - +0.50 +0 +1e-02 +1e-02 +1e-02 +-1 +0.25 - +0.00 +1e-03 +1e-03 +1e-03 +-0.25 +1e-04 +1e-04 +1e-04 - +1000 +1100 +1200 +1300 +-15-10 +2 +10 +18 +22 +2 +9 +10 +14 +16 20 24 28 32 +1.00 - +1e+00 +1e+00 - +1e+00 - +2 +0.75 +1e-01 +1e-01 +1e-01 +0 +0.50 +1e-02 +1e-02 +1e-02 +-2- +0.25 +0.00 +1e-03 +1e-03 +1e-03 +-0.25 +1e-04 +1e-04 +1e-04 - +1000 +1100 +1200 +1300 +-15 -10 +2 +6 +10 +18 +22 +2 +6 +10 +14 +6 +1.00 +1e+00 +1e+00 +1e+00 +Weak VAR +4- +0.75 +1e-01 +1e-01 +1e-01 +2 +0S'0 +1e-02 +1e-02 +1e-02 +0 +0.25 +iL +1e-03 +1e-03 +1e-03 +2 +00'0 +-0.25 +1e-04 +1e-04 +1e-04 +1000 +1100 +1200 +1300 +-15-10 +2 +10 +18 +22 +2 +9 +10 +14 +1620242832 +-00 +1e+00 +1e+00 +1e+00 - +VAR +Wa +0.75 +1e-01 +1e-01 +1e-01 - +0.50 +1e-02 +1e-02 +1e-02 +0.25 - +0.00 - +1e-03 +1e-03 +1e-03 +ii +1e-04 +1e-04 +-0.25 +1e-04 +1000 +1100 +1200 +1300 +-15 -10 +10 +15 +2 +10 +14 +18 +22 +2 +9 +10 +14 +16 20 24 28 32 +50 +1.00 - +1e+00 +1e+00 +1e+00 - +Weak GARCH +25 - +0.75 +1e-01 +1e-01 +1e-01 - +0.50 +1e-02 +1e-02 +1e-02 +-25 +0.25 - +1e-03 +1e-03 +1e-03 +50 - +-0.25 +1e-04 +1e-04 +1e-04 - +1000 +1100 +1200 +1300 +-15-10 +0 +15 +2 +10 +14 +18 +22 +2 +6 +10 +14 +12 16 20 24 28 32 +1.00 +1e+00 +1e+00 - +1e+00 +GARCH +2 +0.75 +1e-01 +1e-01 +1e-01 - +0 +09'0 +1e-02 - +1e-02 +1e-02 - +0.25 +1e-03 +1e-03 +1e-03 +6 +-0.25 +1e-04 +1e-04 +1e-04 - +1000 +1100 +1200 +1300 +-15-10 +-5 +0 +1015 +2 +10 +14 +18 +22 +2 +6 +10 +14 +12 16 20 24 2832 +Time +Lag +k +k +kFigure 10: Bi-plot of the first two principal components (PC) of principal component analysis for the Data Gen- +erating Process (DGP). Each DGP is represented by a different color and the arrows represent the contributions +of the MNet features to the PCs, the larger the size, sharpness, and closer to the red the greater the contribu- +tion of the feature. Features set together are positively correlated and those placed on opposite quadrants are +negatively correlated. +20 + +PCA: MNet Features +1.5 +Models +1.0 +cBWN +iBWN +SGARCH +Dim2 (29.4%) +sVAR +WGARCH +0.5 +IS +!intra +WVAR +Contrib +10.0 +7.5 +0.0 +5.0 +2.5 +0.5 +Dim1 (54.4%)(a) ARI +(b) NMI +(c) AS +Figure 11: Results of DGP’s clustering evaluations using the three evaluation metrics, ARI, NMI, and AS, for +the different number of clusters, k, given as input in the k-means algorithm. The results refer to 10 repetitions +of the clustering analysis using the MNet features set. +Figure 12: Attribution of the samples corresponding to instances of MTS models to the different clusters, ac- +cording to intra, inter, all-layer, and relational feature set. The different models are represented on the horizontal +axis and by a unique color. The bivariate time series samples are represented by the colored points according to +its model process. The vertical axis represents the cluster number to which a bivariate time series is assigned. +21 + +MNet Features +0.6 +0.5 +0.4 +Adj. +0.3 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Number of clusters (k)MNet Features +0.7 +Information +0.6 +0.5 +0.4 +0.3 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Number of Clusters (k)MNet Features +0.5 +0.4 +0.3 +0.2 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Number of Clusters (k)Cluster Analysis +6 +4 +cluster +N +cBWN +iBWN +SGARCH +SVAR +WGARCH +WVAR +Model6 +Conclusions +In this paper, we propose a new mapping method to represent multivariate time series as multilayer networks. +Our procedure relies on two main steps: first each time series component is mapped into a layer in a multilayer +networks structure, following the horizontal visibility concept available in the literature, and for each pair of +time series components in the multivariate time series, inter-layer edges are established based on a new concept +of cross-horizontal visibility proposed in this work. We are interested in analyzing the concept of horizontal +visibility as the base concept, given the promising results achieved in the previous works (Silva, 2018; Silva +et al., 2022), but the idea of the proposed mapping method, the Cross-(H)VG algorithm, naturally extends to +the concept of natural visibility and other versions of VG’s. +To assess the proposed method, we analyzed a specific set of topological features of multilayer networks. These +features are based on concepts of node centrality, graph distances, clustering, communities, and similarity +measures. Each feature extracted from all the subgraphs of the resulting multilayer network structure: intra- +layer graphs (only intra-layer edges), inter-layer graphs (only inter-layer edges) and all-graphs (with both intra +and inter-layer edges). +We perform an empirical evaluation on a set of 600 synthetic bivariate time series, grouped in 6 different +and specific statistical models, that result in a data set of 600 MHVGs. To understand the potential of our +proposed mapping method, we first analyze the degree distributions of the intra, inter, and all-layer subgraphs +of MHVGs. We were able to identify the specific properties of multivariate time series models, namely, we +were able to relate weak and strong cross-correlation with shapes of the inter-layer degree distribution curves +and weak and strong autocorrelation with shapes of the intra-layer degree distribution curves. In particular, we +were also able to relate the persistence of strong correlations to distributions (that result in positively skewed +shape) that have a longer right tail. Adding to the correlation properties (both auto and cross, contemporaneous +and lagged), the properties of the statistical models, such as heteroscedasticity and smoothness, results in inter +and all-layer degree distributions with different shape curves. +We also investigated the global features of the subgraphs (intra, inter, and all-layer). Community-related fea- +tures from intra and all-layer graph highlight the strongly VAR models, with high and persistent autocorrelation +and cross-correlation, as well as with smoothness, and from inter-layer graphs highlight the heteroscedasticity +models, both weak and strong VGARCH models. However, the values of average path length from all-layer +graphs seem to distinguish the properties of weakly and strongly correlated. The average intra-degree has higher +values for independent white noise and weak VAR models, but not distinguishing them. +The new relational feature proposed in this work, average ratio degree, seems to differentiate well the highly +correlated contemporary white noise models, which leads to a similarity in its inter-layer degree and intra- +layer degree features. In the context of this work, based on the synthetic models chosen for analysis, the +Jensen–Shannon divergence measure is only useful to characterize strong VAR models that present very strong +and very persistent correlations, unlike the other models. However, this feature can be quite useful in real +contexts, where the different variables of a multivariate time series can follow different dynamic models that will +be captured by this feature. We intend to investigate this and other characteristics, as well as other topological +features of multilayer networks in our future works. +To further showcase the validity of the proposed mapping method and feature set, we perform a clustering +analysis of the synthetic bivariate time series. The results allow concluding that features that use the inter- +layer edges of the MHVGs add valuable information to intra-layer features thus allowing to distinguish of the +different statistical processes underlying the multivariate data models. +22 + +To conclude, this work proposes a procedure to map multivariate time series into multilayer networks as a mean +to obtain a set of multivariate time series features that can be used for mining tasks. Open issues for future work +are new topological features of multilayer networks and a more extensive empirical study with benchmark +multivariate time series. +Availability of data and materials +The raw data are available at https://github.com/vanessa-silva/MHVG2MTS. +Acknowledgments +This work is financed by National Funds through the Portuguese funding agency, FCT, within project UIDB/50014/2020. +Vanessa Silva is also supported by an FCT PhD research grant (PD/BD/139630/2018). +References +Albert, R. and Barab´asi, A.-L. (2002). Statistical mechanics of complex networks. Reviews of modern physics, +74(1):47. +Barab´asi, A.-L. (2016). Network Science. Cambridge University Press, Cambridge, United Kingdom. +Barbosa, S. M. (2012). mAr: Multivariate AutoRegressive analysis. R package version 1.1-2. +Blondel, V. D., Guillaume, J.-L., Lambiotte, R., and Lefebvre, E. (2008). Fast unfolding of communities in +large networks. Journal of statistical mechanics: theory and experiment, 2008(10):P10008. +Boccaletti, S., Bianconi, G., Criado, R., Del Genio, C. I., G´omez-Gardenes, J., Romance, M., Sendina-Nadal, +I., Wang, Z., and Zanin, M. (2014). The structure and dynamics of multilayer networks. Physics Reports, +544(1):1–122. +Cipra, T. (2020). Time series in economics and finance. Springer, Wiesbaden, Deutschland. +Costa, L. d. F., Rodrigues, F. A., Travieso, G., and Villas Boas, P. R. (2007). Characterization of complex +networks: A survey of measurements. Advances in physics, 56(1):167–242. +Eroglu, D., Marwan, N., Stebich, M., and Kurths, J. (2018). Multiplex recurrence networks. Physical Review +E, 97(1):012312. +Fortunato, S. (2010). Community detection in graphs. Physics reports, 486(3-5):75–174. +Fulcher, B. D. (2018). Feature-based time-series analysis. In Feature engineering for machine learning and +data analytics, pages 87–116. CRC Press, Boca Raton, Florida. +Huang, X., Chen, D., Ren, T., and Wang, D. (2021). A survey of community detection methods in multilayer +networks. Data Mining and Knowledge Discovery, 35(1):1–45. +Kivel¨a, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y., and Porter, M. A. (2014). Multilayer +networks. Journal of Complex Networks, 2(3):203–271. +Lacasa, L., Luque, B., Ballesteros, F., Luque, J., and Nuno, J. C. (2008). From time series to complex networks: +The visibility graph. Proceedings of the National Academy of Sciences, 105(13):4972–4975. +23 + +Lacasa, L., Nicosia, V., and Latora, V. (2015). Network structure of multivariate time series. Scientific Reports, +5(1):15508. +L¨u, L. and Zhou, T. (2011). Link prediction in complex networks: A survey. Physica A: statistical mechanics +and its applications, 390(6):1150–1170. +Luque, B., Lacasa, L., Ballesteros, F., and Luque, J. (2009). Horizontal visibility graphs: Exact results for +random time series. Physical Review E, 80(4):046103. +Maharaj, E. A., D’Urso, P., and Caiado, J. (2019). Time series clustering and classification. CRC Press, Boca +Raton, Florida. +Nakatani, T. (2014). ccgarch: An R Package for Modelling Multivariate GARCH Models with Conditional +Correlations. R package version 0.2.3. +Peach, R. L., Arnaudon, A., Schmidt, J. A., Palasciano, H. A., Bernier, N. R., Jelfs, K. E., Yaliraki, S. N., +and Barahona, M. (2021). HCGA: Highly comparative graph analysis for network phenotyping. Patterns, +2(4):100227. +Shumway, R. H. and Stoffer, D. S. (2017). Time Series Analysis and its Applications: with R examples. 1431- +875X. Springer, New York, United States, 4 edition. +Silva, V. F. (2018). Time series analysis based on complex networks. Msc thesis, University of Porto. +Silva, V. F., Silva, M. E., Ribeiro, P., and Silva, F. (2021). Time series analysis via network science: Concepts +and algorithms. WIREs Data Mining and Knowledge Discovery, 11(3):e1404. +Silva, V. F., Silva, M. E., Ribeiro, P., and Silva, F. (2022). Novel features for time series analysis: a complex +networks approach. Data Mining and Knowledge Discovery, 36:1062—-1101. +Sucarrat, G. (2015). lgarch: Simulation and Estimation of Log-GARCH Models. R package version 0.6-2. +Tsay, R. S. (2013). Multivariate time series analysis: with R and financial applications. John Wiley & Sons, +Hoboken, New Jersey. +Vespignani, A. (2018). Twenty years of network science. +Wang, X., Smith, K., and Hyndman, R. J. (2006). Characteristic-based clustering for time series data. 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Physics Reports, 787:1–97. +24 + +A +MHVG: Algorithms +Algorithm 3: Horizontal Visibility Graph +Input: A time series, Y a, (ts), and a layer, La, (layer) +Procedure HVG(ts, layer) +1 +T ← ts.size() +▷ The time series length +for i ← 1 to T do +2 +mnet.add Node(i, layer) +▷ Add node-layer va +i +corresponding +to timestamp ti +end +for i ← 1 to T − 1 do +3 +mnet.add Edge(i, i+1, layer) +▷ Add intra-layer edge (va +i , va +i+1) +4 +k ← 1 +for j ← 2 to T − i do +5 +condition ← ts[i + k] +6 +p ← ts[i + j] +/* Test HVG condition: +Eq. 4 +*/ +if condition ≥ ts[i] then +7 +break +else if condition < p then +8 +mnet.add Edge(i, i+j, layer) +▷ Add intra-layer edge (va +i , va +i+j) +9 +k ← j +end +end +10 +return +25 + +Algorithm 4: Cross-Horizontal Visibility Graph +Input: Two rescaled time series, Za and Zb, (tsA, tsB), the corresponding layers, La and Lb, (layerA, layerB), and the +maximun time series, max +� +Za, Zb� +,(tsMax) +Procedure CHVG(tsA, tsB, tsMax, layerA, layerB) +1 +T ← tsMax.size() +▷ The time series lengths +/* Test cross-horizontal visibility to the right of Ya,i +*/ +for i ← 1 to T − 1 do +2 +k ← i + 1 +for j ← i + 1 to T do +if j = i + 1 then +3 +mnet.add Edge(i, i+1, layerA, layerB) +▷ Add inter-layer edge (va +i , vb +i+1) +else +4 +condition ← tsMax[k] +5 +p ← tsB[j] +/* Test Cross-HVG condition: +Eq. 5 +*/ +if condition ≥ tsA[i] then +6 +break +end +if condition < p then +7 +mnet.add Edge(i, j, layerA, layerB) +▷ Add inter-layer edge (va +i , vb +j) +8 +k ← j +else if condition < tsMax[j] then +9 +k ← j +▷ Update index k when tsMax[j] > tsMax[k] +end +end +end +end +/* Test cross-horizontal visibility to the left of Ya,i +*/ +for i ← 2 to T do +10 +k ← i − 1 +for j ← i − 1 to 1 do +if j = i − 1 then +11 +mnet.add Edge(i, i-1, layerA, layerB) +▷ Add inter-layer edge (va +i , vb +i−1) +else +12 +condition ← tsMax[k] +13 +p ← tsB[j] +/* Test Cross-HVG condition: +Eq. 5 +*/ +if condition ≥ tsA[i] then +14 +break +end +if condition < p then +15 +mnet.add Edge(i, j, layerA, layerB) +▷ Add inter-layer edge (va +i , vb +j) +16 +k ← j +else if condition < tsMax[j] then +17 +k ← j +▷ Update index k when tsMax[j] > tsMax[k] +end +end +end +end +26 + +B +Multivariate Time Series Models +Main references Cipra (2020); Wei (2019); Shumway and Stoffer (2017); Tsay (2013). +Linear Models +BWN The vector white noise process, ϵt, is a vector of sequences of i.i.d. random variables with mean vector +0 and and covariance matrix function Σ, where Σ is an m × m symmetric positive definite matrix. +The components of the white noise process are serially uncorrelated corr(ϵi,t, ϵi,s) = 0, for t ̸= s, but +may be contemporaneously correlated, corr(ϵi,t, ϵj,t) ̸= 0. It is the simplest multivariate time series +process that reflects information that is neither directly observable nor predictable. We generate white +noise processes, ϵt ∼ N(0, 1), that are not correlated, that is, are independent, and we refer to theses +processes as iBWN, and white noise processes contemporaneously correlated that we refer to them as +cBWN, +� ϵ1,t +ϵ2,t +� +∼ N +� +0, +� 1.00 0.86 +0.86 1.50 +�� +. +VAR(1) The vector autoregression process is a natural extension of the univariate autoregressive (AR) process +that the variable values depends linearly on its own previous values and on a stochastic term. We defined +a VAR(1) process as a vector AR process of order 1 if it satisfies the following equation: +Y t = ϕ + φY t−1 + ϵt, +(7) +where ϵt is the vector white noise, φ is the vector of autoregressive constants and ϕ is the vector of +intercepts. We generate a VAR(1) of 2 dimensions with the following vector of parameters: +� Y1,t +Y2,t +� += +� ϕ1,1 +ϕ2,1 +� ++ +� φ1,1 φ1,2 +φ2,1 φ2,2 +�� Y1,t−1 +Y2,t−1 +� ++ +� ϵ1,t +ϵ2,t +� +, +(8) +where ϕ = +� 2.50 +0.50 +� +, φ = +� 0.20 0.10 +0.02 0.10 +� +and ϵt ∼ +� 1.00 0.10 +0.10 1.50 +� +to generate weakly correlated VAR(1) processes, +and ϕ = +� 0 +0 +� +, φ = +� 0.70 0.02 +0.30 0.80 +� +and ϵt ∼ +� 1.00 0.86 +0.86 1.50 +� +to generate strongly correlated VAR(1) processes. +We refer to the two models generated as wVAR and sVAR, respectively. +Non Linear Models +VGARCH(1, 1) Also generalized autoregressive conditional heteroskedasticity (GARCH) models can be gen- +eralized to multidimensional settings, extending the principle of univariate conditional heteroscedasticity +to mutual volatility. We generate a bivariate GARCH(1, 1) model according to the following volatility +equation: +σt = ω + αϵ2t−1 + βσt−1, +(9) +where σt denotes the volatility in the variables Y t. We generate a VGARCH(1, 1) of 2 dimensions with +the following vector of parameters: +� σ11,t +σ22,t +� += +� ω1,1 +ω2,1 +� ++ +� α1,1 α1,2 +α2,1 α2,2 +�� ϵ2 +1,t−1 +ϵ2 +2,t−1 +� ++ +� β1,1 β1,2 +β2,1 β2,2 +�� σ11,t +σ22,t +� +, +(10) +where ϵt ∼ +� 1.00 0.10 +0.10 1.50 +� +to generate weakly correlated VGARCH(1, 1) processes, and ϵt ∼ +� 1.00 0.86 +0.86 1.50 +� +to generate strongly correlated VGARCH(1, 1) processes. +To both processes we use ω = +� 0.05 +0.02 +� +, +α = +� 0.10 0.00 +0.00 0.05 +� +and β = +� 0.85 0.00 +0.00 0.88 +� +. We refer to the two models generated as wGARCH and sGARCH, +respectively. +Bivariate time series are generated from the above DGP using the R packages: lgarch (Sucarrat, 2015), +mAr (Barbosa, 2012) and ccgarch (Nakatani, 2014). +27 + +B.1 +Autocorrelation Function Plots +See Figure 13. +Figure 13: Plot of the autocorrelation function of an instance of each of the DGP models. The first column +refers to the ACF’s of the first time series component (Y 1) of each model, while the second column refers to +the second component (Y 2). +28 + +Independent WN +1.00 +0.75 - +0.75 +0.50 +0.50 +0.25 +0.25 +0.00 +0.25 +Correlated WN +1.00 +1.00 +0.75 - +0.75 +0.50 +0.50 +0.25 +0.25 +0.00 +0.010 +0.25 +Weak VAR +1.00 +1.00 +0.75 +0.75 +0.50 +0.50 +0.25 +0.25 +0.00 +0.00 +0.25 +Strong VAR +1.00 +0.75 +0.75 +0.50 +0.50 +0.25 +0.25 +0.00 +0.25 +Weak GARCH +1.00 +1.00 +0.75 +0.75 +0.50 +0.50 +0.25 +0.25 +0.00 +0.010 +0.25 +Strong GARCH +1.00 +1.00 +0.75 +0.75 +0.50 +0.50 +0.25 +0.25 +0.00 +0.00 +0.25 +Lag +LagB.2 +Principal Component Analysis Results +See Figure 14 and Figure 15. +(a) Intra-layer features +(b) Inter-layer features +(c) All-layer features +(d) Relational features +Figure 14: Bi-plot of the first two principal components (PC) of principal component analysis for the Data +Generating Process (DGP) using the different feature vectors. Each DGP is represented by a different color and +the arrows represent the contributions of the set of features to the PC’s, the larger the size, sharpness, and closer +to the red the greater the contribution of the feature. +Figure 15: Bar plot with contributions of MNet features to the total of all 20 principal components formed by +the PCA. The red dashed line on the plot indicates the expected average contribution. +29 + +PCA: Intra-layer Features +0.8 +Contrib +20 +15 +Dim2 (23.3%) +10 +.4 +Models +CBWN +iBWN +SGARCH +0'0 +sVAR +WGARCH +Q 1 +WVAR +-0.4 - +-0.5 +0.0 +0.5 +1.0 +Dim1 (64.1%)PCA: Inter-layer Features +0.50 - +* +: +Models +6 +0.25 - +CBWN +iBWN +SGARCH +Dim2 (5.2%) +: +sVAR +. +！ +2 +WGARCH +2 +0.00 +WVAR +.8 +.3 +Contrib +30 +-0.25 - +20 +10 +-0.50 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +Dim1 (91.5%)PCA: All-layer Features +0.50 +0.25 - +Models +CBWN +iBWN +SGARCH +Dim2 (36.2%) +0.00 +sVAR +WGARCH +WVAR +Contrib +.0.25 +50 +40 +30 +-0.50 - +20 +-0.75- +-0.5 +0.0 +0.5 +Dim1 (57.7%)PCA: Relational-layer Features +0.5 +Models +CBWN +iBWN +0.0 +SGARCH +Dim2 (18.7%) +sVAR +WGARCH +WVAR +Contrib +.0.5 +30 +20 +10 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Dim1 (69.2%)Contribution of Variables to A Dimensions +10.0 +7.5 +(%) +Contributions +5.0 +2.5 +0.0 +k_1,2 +k +r_1,2 +r_2,1 +Q_1 +s_1,2 +Q 2 +Q_1,2 +s_1 +Js_intra +MNet Features \ No newline at end of file diff --git a/qdE0T4oBgHgl3EQfaQC1/content/tmp_files/load_file.txt b/qdE0T4oBgHgl3EQfaQC1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd0589923712304c1349af73331a0e94b2f167f2 --- /dev/null +++ b/qdE0T4oBgHgl3EQfaQC1/content/tmp_files/load_file.txt @@ -0,0 +1,1297 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf,len=1296 +page_content='MHVG2MTS: Multilayer Horizontal Visibility Graphs for Multivariate Time Series Analysis Vanessa Freitas Silva1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Maria Eduarda Silva2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Pedro Ribeiro1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' and Fernando Silva1 1CRACS-INESC TEC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Faculdade de Ciˆencias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Universidade do Porto 2LIAAD-INESC TEC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Faculdade de Economia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Universidade do Porto Abstract Understanding the properties of time-indexed multivariate data has been a predominant topic mainly to address open issues in multi- variate time series analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' such as finding appropriate measures to analyze temporal dependence and cross-dimension dependencies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' as well as visualizing multidimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Usually, the methodologies used to analyze multivariate time series are based on adapting approaches for univariate settings or on assumptions and parameters for specific problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' A different strategy uses complex network to obtain an additional and reduced representation of temporal and causal properties of the time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Recent strategies involve mapping multivariate time series into high-level network structures, specifically into multiplex networks representing interconnections between contemporary timestamps of different time series components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In this work, we propose a new mapping method that takes advantage of the entire structure of multilayer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We introduce the multilayer horizontal visibility graph that is based on the new concept of cross-horizontal visibility between lagged timestamps of different components, which allows describing the cross-dimension dependencies via inter-layer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We use a set of existing topological measures of multilayer networks as well as a novel measure to evaluate and validate our approach, which is parameter-free, does not require data pre-processing and is applicable to any kind of multivariate time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We provide an extensive experimental evaluation, where we explore the proposed topological measures, showing that the inter-layer edges based on cross-horizontal visibility preserve more information about the time series data after the mappings, information that would inevitably be lost using mapping methods that result in single-layer and multiplex structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We also verify that the information mapped by the inter-layer edges is not enough on its own, but that it complements the data information captured by the commonly used intra-layer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Furthermore, we complement our analysis by performing a multivariate time series clustering task based on the proposed measure set of the proposed mapping method, demonstrating its validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Keywords: multivariate time series mappings, multilayer horizontal visibility graphs, multivariate time series features 1 Introduction Recent technological developments led to the wide availability of large amounts of high dimensional time- indexed data for which appropriate methodological and computational tools are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These multidimen- sional time-indexed data, usually designated as multivariate time series, have become ubiquitous in all domains from climate studies or health monitoring to financial data analysis, and are characterized by serial correlation as well as cross-sectional dependencies, in what is often designated as the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To overcome the lack of appropriate methodological and computational tools to characterize high-dimensional time-indexed data, feature-based approaches for time series analysis have been proposed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Time series features are traditionally based on statistics and models for time series analysis and often rely on pre- processing and/or assumptions that are not usually satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' A particular methodology that is free of such requirements consists in mapping the time series into a complex network and then extracting topological fea- tures of the network for time series mining tasks and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In fact, network science, the research area that studies how to extract information from complex networks (Albert and Barab´asi, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Barab´asi, 2016), 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='02333v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='SI] 5 Jan 2023 provides a vast set of topological graph measurements (Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2007), a well-defined set of problems such as community detection (Fortunato, 2010) or link prediction (L¨u and Zhou, 2011), and a large track record of successful application of complex network methodologies to different fields (Vespignani, 2018), including graph classification (Peach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Univariate time series are mapped into single-layer networks based on the concepts of visibility, transition probability, and proximity (Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Multivariate time series may be mapped into single or multiple-layer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In the former, the nodes represent the component time series and the edges represent the relationships between the nodes (component time series) computed using statistical methods or models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These methods imply that all important information on the dynamics of each time series component, such as serial correlation, is lost in the mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Mapping methods that represent multivariate time series as multiplex networks were proposed with the objective of preserving both the dynamical (over time) and the cross-sectional information contained in the multivariate data (Lacasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Eroglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In these multiplex networks, each component univariate time series is mapped into a layer (using a univariate time series mapping in which each timestamp is represented by a node) and different layers are connected via the common nodes (time stamps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Inevitably, lagged cross-correlations, which sometimes are the most important information, are lost in the mapping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To overcome this limitation, we propose a new mapping to represent a multivariate time series as a multiple- layer complex network in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Multiple-layer networks, or multilayer networks, are complex structures capable of establishing internal connections (within the same layer) and external connections (between different layers) which allow the creation of very complete and flexible data structures (Kivel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' From a high- level view, multilayer networks have a structure compatible with that of multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The proposed mapping is based on a new horizontal visibility concept, the cross-horizontal visibility, developed to capture the cross-dependencies between pairs of component time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Thus, the multiplex visibility graphs (Lacasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2015) are extended with the incorporation of inter-layer edges established according to the cross-horizontal vis- ibility between different nodes (time stamps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These new edges/connections can capture dependencies between different timestamps of different variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The resulting networks are denoted as multilayer horizontal vis- ibility graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Furthermore, we propose a set of global topological multilayer network features as a novel set of features for multivariate time series comprising: intra-layer topological features, inter-layer topological features, all-layer topological features, and relational features which are topological features that relate com- ponents of the network (such as layers or edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Within relational features, we propose a new topological feature aimed at relating intra and inter-layer connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These different subsets of topological features allow to analyze and compare the underlying properties of intra-layer and inter-layer edges, and to assess the contri- bution of the proposed mapping method in relation to the multiplex methods in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' This proposed methodology is represented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We use synthetic multivariate time series generated from a selected set of multivariate time series models to test and evaluate the framework proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 2 Figure 1: Schematic diagram of the network-based features approach to time series mining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1 Contributions The main contributions of this work are as follows: We introduce a new method for mapping multivariate time series into a complete multilayer network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' This mapping is based on the concept of horizontal visibility and on a multiplex visibility graph to take better advantage of the structural capacity of multilayer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' As far as we know, the incorporation of inter-layer edges between different entities of multilayer networks has not been previously used in the literature to analyze multivariate time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We propose a new topological feature for multilayer networks and present a different set of multilayer network topological features selected for the analysis of the proposed mapping method and to reduce the dimensionality of the multivariate time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' As far as we know, no other work presents different topological features of multilayer networks, based on their intra- and inter-layer edges, in the context of time series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We also perform a detailed exploratory and empirical analysis of the different sets of features used in this work, showcasing its validity and usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2 Organization We have organized this document as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Section 2 introduces the basic concepts of multivariate time series and multilayer networks, setting the notation for the remainder of the paper, and also presents the background on mapping methods useful for understanding the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Next, Section 3 presents the new concept of visibility between time series components and the new mapping multivariate time series proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In Section 4 we present the set of topological features that we extend to multilayer networks as well as the new proposed topological feature to multilayer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Section 5 presents a study of this mapping method via analysis of the corresponding feature set, in order to characterize the properties of the multivariate time series, also presents a multivariate time series clustering task as a validation of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Finally, Section 6 presents the conclusions, some comments, and possible future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 3 Mapping Multivariate Time Network Feature Feature Analysis (Multiplex/Multilayer) Extraction Series Set Layers Similarity JSDintr y3 Multiplex Network Features All-layer Relational Intra-layer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='Inter-layer JSD1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' y3 Y4 ： Multilayer Network Fea2 Background This Section introduces the main concepts and notation necessary for the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1 Multivariate Time Series An Univariate Time Series (UTS) is a sequence of (scalar) observations indexed by time t, usually denoted by {Yt}T t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Unlike a random sample, such observations are ordered in time and usually present serial cor- relation that must be accounted for in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' If at each time t we obtain a vector of m observations, Y t = [Y1,t, Y2,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , Ym,t]′, where ′ represents the transpose, then the data set Y = {Y t}T t=1 is called a Multivariate Time Series (MTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Henceforward, the UTS components of the MTS Y are denoted by Y α = [Yα,1, Yα,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , Yα,T ], α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , m and thus we can denote the MTS by its components, Y = {Y α}m α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' MTS data present not only serial correlation within each component, Y α, but also a correlation between the different UTS’s, Y α and Y β with α ̸= β, both contemporaneous and lagged correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Thus, analyzing MTS depends on key dependence measures such as the autocorrelation function (ACF), which measures the linear predictability of a UTS, ρ(s, t) = corr(Yt, Ys) = cov(Ys, Yt) � var(Ys)var(Yt) , (1) and the cross-correlation function (CCF), which measures the correlation between any two components of the MTS, α and β, say, at times s and t, ρα,β(s, t) = corr(Yα,s, Yβ,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (2) Time series analysis refers to the collection of procedures developed to systematically solve the statistical problems posed by the serial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' There is a plethora of (linear and non-linear) statistical models in the literature adequate to describe the behavior of UTS (Shumway and Stoffer, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' And although the theory of UTS extends naturally to the multivariate case, such as the mean, covariance, ACF, and CCF functions, new concepts arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' MTS analysis requires tools, methods, and models for mining information from multiple measurements which present both temporal and cross-sectional correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2 Multilayer Networks A graph (or network), G, is a mathematical structure defined by a pair (V, E), where V represents the set of nodes and E the set of edges (connections) between pairs of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Two nodes vi and vj are called neighbors if they are connected, (vi, vj) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' If there is no direction from a source node to a target node the edges are undirected: (vi, vj) ∈ E implies that (vj, vi) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' A graph can be represented by an adjacency matrix, A, and Ai,j is 1 when (vi, vj) ∈ E and is 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' A Multilayer Network (MNet) is a complete and general structure suitable for modeling multiple complex systems through their interactions, intra- and inter-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' A MNet is generally defined as a quadruplet M = (VM, EM, V, L) where V and L represent the set of entities and the set of layers of M, respectively, and VM and EM represent the global sets of nodes and edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The VM ⊆ V × L1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' × Lm, where Lα ∈ L is an elementary layer, is a set of node-layer combinations in which a node is present in the corresponding layer Lα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The EM ⊆ VM × VM is the set of edges that contain the pairs of possible combinations of nodes and elementary layers (Kivel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We denominate as intra-layer edges, the connections between nodes of the same layer, (vα i , vα j ), and inter-layer edges the connections between nodes of different layers, (vα i , vβ j ) with α ̸= β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Two particular cases of multilayer networks are the monoplex network when m = 1 and M reduces to a (single-layer) network, G, and the multiplex network, when M is a sequence of m graphs, {Gα}m α=1 = {(V α, Eα)}m α=1, usually with a node set common to all elementary layers, and inter-layer edges connecting only the counterpart nodes across the layers, that is connecting (vα i , vβ i ), α ̸= β (Boccaletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 2 exemplifies the representation of simple multilayer and multiplex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 4 (a) Multilayer Network (b) Multiplex Network Figure 2: An illustrative example of two toy multilayer networks with five entities, V = {1, 2, 3, 4, 5}, and two layers L = {L1, L2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (a) represents a toy multilayer network and (b) a toy multiplex network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Solid lines represent the intra-layer edges and dashed lines represent the inter-layer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Source: Modified from Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' A node-aligned1 MNet has an associated adjacency tensor of order 4, AAA, where the tensor element Ai,j,α,β is 1 when (vα i , vβ j ) ∈ EM and is 0 otherwise (Kivel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' If the MNet is not node-aligned, we can consider empty nodes to complete the tensor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Another representation is obtained by flattening AAA into a supra- adjacency matrix, A, where intra-layer edges are associated with diagonal element blocks and inter-layer edges with off-diagonal element blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 3 represents the supra-adjacency matrices of the networks illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' From these element blocks we can infer three types of subgraphs: intra-layer graphs, Gα, represented by the square matrices of order |V α| formed by the diagonal element blocks (intra-layer edges, Aα i,j), ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', � Aα 0 0 0 � , inter-layer graphs, Gα,β, represented by the square matrices of order |V α| + |V β| constructed from off-diagonal element blocks (inter-layer edges, Aα,β i,j and Aβ,α j,i , and no intra-layer edges, Aα i,j = 0 and Aβ i,j = 0) 2, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', � 0 Aα,β Aβ,α 0 � , and all-layer graphs, Gα,β all , represented by the square matrices of size |V α| + |V β| constructed by both on and off-diagonal element blocks (intra-layer edges, Aα i,j and Aβ i,j, and inter-layer edges, Aα,β i,j and Aβ,α j,i ), ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', � Aα Aα,β Aβ,α Aβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Network science has been a very useful tool to answer the most diverse problems in several scientific fields (Vespig- nani, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' A large number of topological, statistical, spectral, and combinatorial properties metrics that extract information from networks are available in the literature (Albert and Barab´asi, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Barab´asi, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Peach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We can group these metrics into global, local, and ”intermediate” features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The first group quantifies properties involving all network elements, the second prop- erties over a given node or edge, and the last properties that involve subsets of the network, such as subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These metrics used in monoplex contexts can be extended to MNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Most of the common topological metrics can be extended straightforwardly to intra-layer metrics by just computing them over the intra-layer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These metrics can also be extended to the whole MNet, computing them over both intra-layer and inter-layer edges (Kivel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Other approaches rely on measurements and properties in the tensor analysis literature (Kivel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 1A multilayer network is node-aligned if all layers contain all entities, that is, VM = V × L1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' × Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 2Note also that the inter-layer graphs have the characteristics of a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Where a bipartite graph is a graph Gα,β whose node set V α,β can be divided into two disjoint and independent sets V α and V β (V α,β = V α ∪ V β and V α ∩ V β = ∅) and every edge connects a node in V α to a node in V β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 5 4 3 2 I 1 1 1 4 3 I 2 54 3 2 5 4 3 2 5(a) Multilayer Network (b) Multiplex Network Figure 3: An illustrative example of two supra-adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (a) a supra-adjacency matrix of a toy multilayer network and (b) a supra-adjacency matrix of a toy multiplex network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Colored blocks represent the intra-layer graphs and gray blocks represent the inter-layer graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='3 Mapping Time Series into Complex Networks In the last decade, several network-based time series analysis approaches have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These ap- proaches are based on the mappings of univariate and multivariate time series into the network domain, either into single-layer or multiple-layer networks (Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The mappings proposed in the literature are essentially based on concepts of visibility, transition probability, proximity, time series models, and statis- tics (Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In this section, we review the concepts of visibility graphs that are required for the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Visibility Graphs (VG) establish connections (edges) between the timestamps (nodes) using visibility lines between the observations, where nodes are associated with the natural ordering of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' There are two native variants of this method, the Natural Visibility Graph (NVG) (Lacasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2008) and the Horizontal Visibility Graph (HVG) (Luque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The idea of these methods is that each UTS observation, Yt, is seen as a vertical bar with a height equal to its numerical value and that these bars are laid in a landscape where (the top of) a bar is visible from (the tops of) other bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Each timestamp, t, is mapped into a node, vt, and the corresponding edges (vi, vj), for i, j = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' T, i ̸= j, are established if there is a visibility line between the corresponding data bars that is not intercepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Formally, in the NVG and HVG, two nodes vi and vj are connected if for all tk, ti < tk < tj, (tk, Yk) satisfies Yk < Yj + (Yi − Yj)(tj − tk) (tj − ti) NVG (3) Yk < Yi ∧ Yk < Yj HVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (4) We give a simple illustration of these methods in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' VGs are always connected, each node vi sees at least its nearest neighbors, vi−1 and vi+1, are always undi- rected unless we consider the direction of the time axis, and are invariant under affine transformations of the data (Lacasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2008), each transformation Xt = aYt + b, for a ∈ R, b ∈ R, and t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , T, leads to the same VG (Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 6 Figure 4: An illustrative example of the two visibility graph algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (a) toy time series and corresponding visibility lines between data bars (observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Solid pink lines represent the natural visibility lines corre- sponding to the NVG method, and dotted blue lines represent the horizontal visibility lines corresponding to the HVG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (b) network generated by the corresponding mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The NVG is the graph with all edges, including the dashed pink edges, and the HVG is the subgraph that does not include these edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Source: Adapted from Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Based on the definition of MNet, Lacasa and co-authors (Lacasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2015) proposed an extension of the visibility mapping for MTS analysis, the Multiplex Visibility Graphs (MVG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Formally, a MVG of m layers, M, is constructed so that layer set, {Lα}m α=1, corresponds to the NVGs (or HVGs), {Gα}m α=1, associated with the time series components, {Y α}m α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' M is represented by the adjacency matrix vector, AM, whose elements are the adjacency matrices of each layer, AM = {A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , Am} with Aα i,j = 1 if the nodes vα i and vα j are connected in layer Lα and is 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 5 illustrates the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 5: An illustrative example of the multiplex natural visibility graph algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (a) displays a toy multi- variate time series, Y = {Y 1, Y 2, Y 3}, (b) the corresponding multiplex network, with three layers generated by the multiplex natural visibility graph algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 7 3 MHVG: a New Multivariate Time Series Mapping Visibility methods have shown to be very promising in capturing time series characteristics reflecting local and global properties of the data, and not requiring pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' This section presents a new visibility algorithm to map an MTS into a multilayer horizontal visibility graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' This algorithm is based on a new visibility concept, cross-horizontal visibility which is an extension of the traditional horizontal visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1 Cross-Horizontal Visibility Consider two time series Zα = (Zα,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , Zα,T ) and Zβ = (Zβ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , Zβ,T ) on the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Two arbitrary data values (ti, Zα,ti) and (tj, Zβ,tj) are said to have cross-horizontal visibility, Cross-HV if Zα,ti ∧ Zβ,tj > max (Zα,t, Zβ,t), for all t, ti < t < tj, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , T, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (5) This definition implies that all data values have Cross-HV to its neighbours and that the visibility is reciprocal, meaning that if (t, Zα,t) has Cross-HV to (s, Zβ,s), then (s, Zβ,s), has Cross-HV to (t, Zα,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The concept of cross-horizontal visibility, Cross-HV, is illustrated in Figure 6 with two toy time series and for the first four data points, with the bi-coloured lines indicating (the reciprocal) visibility between the corresponding time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 6: Schematic diagram of the cross horizontal visibility concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (a) Illustrates a toy bivariate time series Zyellow, Zblue, (same scale) and the corresponding maximum time series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (b) represents the cross-horizontal visibility, Cross-HVG, by solid bi-color lines (yellow and blue) connecting the data bars of the time series components Zyellow and Zblue, for the first four timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2 Multilayer Horizontal Visibility Graph A Multilayer Horizontal Visibility Graph (MHVG) is obtained by mapping a MTS, Y = {Y α}m α=1, into a MNet structure, M = (VM, EM, V, L), using the concepts of HV and Cross-HV, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Each unique timestamp, t, is mapped into an unique entity in VM and each component time series, Y α is mapped into a layer, Lα ∈ L, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , m, using the HVG method described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='3, thus establishing the intra-layer edges, (vα i , vα j ) ∈ EM, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , T, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Then inter-layer edges (vα i , vβ j ) ∈ EM, between any two layers Lα and Lβ, α, β = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , m, α ̸= β and i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , T, i ̸= j are established using the Cross-HV described above in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Note that to establish Cross-HV all the time series Y α, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , m must be in the same scale which may require a pre-processing step of the data set Y , comprising the Min-Max scaling of each time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The mapping is illustrated in Figure 7, with toy bivariate time series, for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 8 blue 4 2 3Figure 7: Schematic diagram of the multilayer horizontal visibility graph algorithm: (a) original time series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (b) Min-Max re-scaled time series and maximum time series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (c) illustration of cross-HV with the edges between adjacent timestamps omitted for simplicity (detail for the first four timestamps),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (d) cross-horizontal visibility graph: solid black lines represent the intra-layer edges (the HVGs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' dashed lines the inter-layer edges (the Cross- HVGs) and the red lines highlight inter-layer edges between nodes corresponding to non-adjacent timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' From the generated MHVG, we can identify the intra-layer graphs, {Gα}m α=1 and the inter-layer graphs, Gα,β, for α, β = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , m and α ̸= β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' {Gα}m α=1 correspond to the HVG of each individual time series component and it is represented by the adjacency matrix Aα with Aα i,j = 1 if (vα i , vα j ) ∈ EM and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Gα,β corresponds to the cross-horizontal visibility graph (Cross-HVG) of each pair of different time series components and it is represented by the adjacency matrix Bα,β = � 0 Aα,β Aβ,α 0 � with Aα,β i,j = 1 and Aβ,α j,i = 1 if (vα i , vβ j ) ∈ EM, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Algorithm 1 describes the concept of Cross-HV and Algorithm 2 describes mapping a multivariate time series into a Multilayer Horizontal Visibility Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In the Appendix, we describe the auxiliary functions to support the implementation of the method, Algorithm 3 describes the function that creates an HVG, and Algorithm 4 describes the function that creates the inter-layer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 9 Algorithm 1: Cross-Horizontal Visibility Graph Input: Two rescaled time series, Za and Zb, (tsA, tsB), the corresponding layers, La and Lb, (layerA, layerB), and the maximum time series, max � Za, Zb� , (tsMax) Procedure CHVG(tsA, tsB, tsMax, layerA, layerB) 1 T ← tsMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='size() ▷ The time series lengths for node i in layerA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='get Nodes() do for node j from i + 1 to T in layerB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='get Nodes() do if node i can ’see’ j then 2 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='add Edge(i, j, layerA, layerB) ▷ Add inter-layer edge (va i , vb j) end end for node j from i − 1 to 0 in layerB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='get Nodes() do if node i can ’see’ j then 3 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='add Edge(i, j, layerA, layerB) ▷ Add inter-layer edge (va i , vb j) end end end 4 return Algorithm 2: Multilayer Horizontal Visibility Graph Input: A set of time series components, {Y a}m a=1, (mts) Output: A multilayer network, M, (mnet) Procedure MHVG(mts) 1 m ← mts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='size() ▷ The number of time series components 2 mnet ← {} ▷ The empty MNet M 3 n mts ← {} ▷ List to store the rescaled time series components for a ← 1 to m do 4 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='layers[a] ← {} ▷ The empty layer La 5 HVG(mts[a], mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='layers[a]) ▷ Map the time series Y a on the HVG La (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 4) 6 n mts[a] ← MinMax(mts[a]) ▷ Rescale time series Za end for a ← 1 to m − 1 do for b ← a + 1 to m do 7 tsMax ← MaxTS(n mts[a], n mts[b]) ▷ Get the maximum rescaled time series 8 CrossHVG(n mts[a], n mts[b], tsMax, mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='layers[a], mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='layers[b]) ▷ Map the pairwise time series Za and Zb on Cross-HVG (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 5) end end 9 return mnet 10 4 A Novel Set of Multivariate Time Series Features Mapping time series into complex networks and then using network topological features as features in univariate time series mining tasks has become a popular approach due to its dimensionality reduction capabilities (Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Fulcher, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In this work, we propose a set of MHVG topological features to analyze MTS data which includes: i) common topological features extended to MNets and ii) a new feature constructed for MNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1 Topological features extended to MNets Common network topological features such as node centrality, graph distances, clustering, and community can be naturally extended to a MNet structure and all the subgraphs mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To illustrate, consider a local centrality measure such as the degree ki of a node vi, which represents the number of its adjacent edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In a MNet, we can compute three variants of ki, for each layer α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , m where we use the symbol ≺ (and ⪯) to express the inter-layer edges from a ”source” layer, α (and including intra-layer edges of the ”source” layer) to a ”destination” layer, β, intra-layer degree: kα i = � j Aα ij inter-layer degree: kα≺β i = � j Aα,β ij all-layer degree: kα⪯β i = kα i + kα≺β i with β ̸= α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Note that local inter and all-layer topological measurements are asymmetric measures, that is, kα≺β i ̸= kβ≺α i and kα⪯β i ̸= kβ⪯α i , since the measure is relative to node-layer vα i or node-layer vβ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In general, any common (local) topological feature Fi can be easily extended to intra-layer features, F α i , just computing them over individual layers, to inter-layer features, F α≺β i , computing over inter-layer edges, and to all-layer features, F α⪯β i , which compute over both intra-layer and inter-layer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' An important feature associated with the degree is the degree distribution P(k) that measures the fraction of nodes in a graph with degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In this work, we analyze the three variants of degree distributions, P(kα), P(kα≺β) and P(kα⪯β), in layer Lα, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , m, associated with its intra-layer degree, inter-layer degree and all-layer degree, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To measure the similarity between pairs of layers in an MNet, we also use the Jensen–Shannon divergence (JSD) which measures the distance between two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' As an example, the JSD between intra-layer degree distributions P(kα) and P(kβ), (JSDα,β intra) is defined as follows: JSD(P(kα)||P(kβ)) = 1 2KLD(P(kα)||Q(k)) + 1 2KLD(P(kβ)||Q(k)) where Q(k) = 1 2(P(kα) + P(kβ)) and KLD is the Kullback–Leibler divergence: KLD(P(kα)||Q(k)) = � k P(kα) log2 �P(kα) Q(k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Similarly, we define JSD for the inter-layer degree distributions (JSDα,β inter) and the all-layer degree distribu- tions (JSDα,β all ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Note that JSD is a symmetrical version of the asymmetrical measure KLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In the remainder of this work, we will refer to similarity measures, such as JSD, as relational features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 11 In addition, we also extend global topological features to MNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These features involve all (sub)graph elements and therefore are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' As an example, consider the average degree ¯k which calculates the arithmetic mean of the degree ki of all nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' As before, we can compute three variants of ¯k in a MHVG, average intra-degree: ¯kα = 1 |Vα| � i kα i average inter-degree: ¯kα,β = 1 |Vα|+|Vβ| �� i kα≺β i + � j kβ≺α j � average all-degree: ¯kα,β all = 1 |Vα|+|Vβ| �� i kα⪯β i + � j kβ⪯α j � In short, we can compute a (global) topological feature F in the subgraphs of the MNet: intra (F α), inter (F α,β), and all-layer graphs (F α,β all ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Motivated by the set of features proposed in Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (2022), namely based on the concepts of node centrality, graph distances, clustering, and community and the three types of MNet measurements defined above, we propose intra-layer, inter-layer, and all-layer, for each pair of layers, features as follows: Average Degree: the average intra-degree ¯kα, average inter-degree ¯kα,β and average all-degree ¯kα,β all , as formulated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Average path length: geodesic distances di,j, i ̸= j between node vi and vj corresponding to the length of the shortest paths between them, where the path length is the number of edges in the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The average (intra-/inter-/all-)path length ( ¯dα, ¯dα,β and ¯dα,β all ) is the arithmetic mean of the shortest paths among all pairs of nodes in (intra, inter, and all-layer) graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Number of communities: The number of (intra-/inter-/all-)communities, (Sα, Sα,β and Sα,β all ), is the amount of groups/communities of nodes that are densely connected on the subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These communities are found by performing random walks on the subgraph (intra, inter, and all-layer graph), so that short random walks tend to stay in the same community until the modularity value (defined below) cannot be increased anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Modularity: (Intra-/Inter-/All-)modularity, (Qα, Qα,β and Qα,β all ), measures how good a specific division of the corresponding subgraph G is into (intra-/inter-/all-)communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Table 1 provides the formulation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2 Ratio Degree: a new MNet topological feature The ratio degree is a new topological feature for multilayer graphs, introduced here to relate intra-layer and inter-layer visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The ratio degree of node vα i of layer Lα is defined as rα⪯β i = kα≺β i kα i (6) for any layer Lβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The average ratio degree, ¯rα⪯β, is the arithmetic mean of the ratio degree of the nodes of layer Lα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Note that the ratio degree is asymmetric, and thus it is not necessarily true that ¯rα⪯β = ¯rβ⪯α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='3 The MHVG Features Set The set of features defined above and summarized in Table 1 forms a set of features extracted from MHVG, as depicted in Figure 8, that we propose to characterize a MTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 12 Table 1: Topological multilayer network features 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Feature Formulation Note Average Degree ¯kα = 1 Nα � i kα i ¯kα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β = 1 Nα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β �� i kα≺β i + � j kβ≺α j � ¯kα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β all = 1 Nα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β �� i kα⪯β i + � j kβ⪯α j � Degree Distribution P(kα) = nkα Nα n: number of nodes vα i with the corresponding degree k P(kα≺β) = nkα≺β Nα P(kα⪯β) = nkα⪯β Nα Average Path Length ¯dα = 1 Nα(Nα − 1) � i̸=j dα i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j: length of the shortest paths between vi and vj in the corresponding subgraph ¯dα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β = 1 Nα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β(Nα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β − 1) � i̸=j dα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j ¯dα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β all = 1 Nα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β(Nα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β − 1) � i̸=j dα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='all Number of Communities Sα = |Cα| C: set of communities in corresponding subgraph Sα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β = |Cα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β| Sα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β all = |Cα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β all | Modularity Qα = 1 2|Eα| � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j � Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j − kikj 2|Eα| � δ (ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' cj) B = Aα Qα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β = 1 2|Eα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β| � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j � Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j − kikj 2|Eα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β| � δ (ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' cj) B = � 0 Aα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β Aβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='α 0 � Qα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β all = 1 2|Eα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β all | � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j � Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='j − kikj 2|Eα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β all | � δ (ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' cj) B = � Aα Aαβ Aβα Aβ � Jensen–Shannon Divergence JSDα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β intra = JSD(P(kα)||P(kβ)) JSDα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β inter = JSD(P(kα≺β)||P(kβ≺α)) JSDα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β all = JSD(P(kα⪯β)||P(kβ⪯α)) Average Ratio Degree ¯rα⪯β = 1 |Nα| � i rα⪯β i 2Remember that V α is the set of nodes in layer Lα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' we define Nα = |V α| and Nα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='β = |V α| + |V β| the number of nodes in the corresponding layer(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 3Remember that V α is the set of nodes in layer Lα, we define Nα = |V α| and Nα,β = |V α| + |V β| the number of nodes in the corresponding layer(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 13 Figure 8: Schematic diagram of the multilayer network features set extraction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' A multivariate time series Y is mapped into a multilayer horizontal visibility graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' And for each of its subgraphs (intra-layer, inter- layer, and all-layer graphs) are computer the global topological features (¯k, ¯d, S, Q and P(k)) and relational features (¯r and JSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 5 Empirical evaluation of MHVG In this section, we investigate whether the mapping method and the features set introduced above are useful for characterizing MTS data and evaluate the performance of the methodology for MTS mining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We use synthetic bivariate time series, generated from bivariate time series models to control for MTS correlation (serial and cross) properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' First, we make some considerations about the implementation of the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1 Implementation Details The mapping and topological features proposed in this work do not, conceptually, depend on the implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' However, to show the practicality of the proposed method and to provide the reader with the ability to reproduce it, we describe in more detail how we computed our methodology, illustrated in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Note that for illustrative purposes, we used m = 2, but the method is extensible to any value of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To map a multivariate time series Y into an MHVG, we follow Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The intra-layer HVG’s, {Gα}m α=1, for each time series component, {Yα,t}, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , m, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' , T, are created using the implemented Algo- rithm 3 proposed in (Luque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' And the inter-layer edges are added following the mapping method based on cross-horizontal visibility criteria proposed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1, and using the implemented Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Subgraphs corresponding to intra-, inter-, and all-layers, are fixed via corresponding adjacency submatrices of the MHVG (see Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' And the corresponding topological features described above are computed using the methodologies and algorithms described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 14 Time Series Multivariate Feature Extraction 8 l dl s1:Ql: P(kl) 10 3 8 K² : d² : s2 : Q²: P(k2) Multilayer Horizontal 6 10 Visibility Graph 5 T?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' di I st?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' IQf P(kt) 1±2 : #2±1 iJSD1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2 IJSD12 all intra ④ 5 10 7 8 K1,2 P(k1,2) 3 4 9The average degree (¯k) and average ratio degree (¯r) are calculated by the arithmetic mean of the degrees ki and ratio degrees ri, respectively, of all node vi in the respective subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In this work, the average path length ( ¯d) follows an algorithm that computes the average shortest path length between all pairs of nodes (of respective subgraphs) using a breadth-first search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To calculate the number of communities (S), the function used makes use of the known ”Louvain” algorithm that finds community structures by multi- level optimization of modularity (Q) feature (see Blondel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (2008) for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' And the degree distributions (P(k)) and Jensen–Shannon divergence (JSD) are implemented as described above section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We used C++ and its needed set of libraries (such as igraph and standard libraries) to implement the data structure to store an MNet and compute the functions to extract the topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' For reproducibil- ity purposes, the datasets and results are made available in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='com/vanessa-silva/ MHVG2MTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2 Synthetics Datasets We consider a set of six bivariate time series models (m = 2), denoted by Data Generating Processes (DGPs), summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These MTS models present a set of particular characteristics in terms of serial and cross-correlation (see Section2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1), namely: white noise (WN) processes simulate noise effects , one process does not present any kind of correlation, and the other presents strong contemporaneous correlation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' vector au- toregression (VAR) processes simulate smooth linear data, presenting both serial and cross-correlation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' vector generalized autoregressive conditional heteroskedasticity (VGARCH) processes simulate nonlinear data with persistent periods of high or low volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The parameters of each DGP are chosen so that the data exhibits a range of serial and cross-correlation properties as described in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' A detailed description of the DGP and their properties as well as computational details are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' For each DGP in Table 2, we generated 100 instances of length T = 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' As illustrated in Figure 8, we map each bivariate time series into an MHVG, highlight the intra-, inter-, and all-layer graphs, and extract the corresponding topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Table 2: Synthetic data generating processes (bivariate time series models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Parameters, the main characteristic of the data sets, and notation are also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' See Appendix B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' DGP Parameters Characteristics Notation Independent White Noise ϵt ∼ N(0, 1) Noise effect No correlation iBWN Correlated White Noise � ϵ1,t ϵ2,t � ∼ N � 0, � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 �� Noise effect No serial correlation Cross-correlation cBWN Weak VAR(1) ϕ = � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � , φ = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 � Weak correlation (serial and cross) wVAR ϵt ∼ � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � Strong VAR(1) ϕ = � 0 0 � , φ = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='80 � Strong correlation (serial and cross, lagged and contemporaneous) sVAR ϵt ∼ � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � Weak VGARCH(1, 1) ω = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='02 � , α = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='05 � No serial correlation Weak cross-correlation wGARCH β = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='88 � , ϵt ∼ � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � Strong VGARCH(1, 1) ω = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='02 � , α = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='05 � Strong contemporaneous cross-correlation sGARCH β = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='88 � , ϵt ∼ � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � 15 To illustrate the procedure, we represent in Figure 9 one instance with 300 observations of each DGP and the corresponding cross-correlation (CCF) plot (first two columns of the plot), the intra-, inter-, and all-layers de- gree distributions on a semi-logarithmic scale (last three columns of the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These degree distributions are computed as the arithmetic mean of the degree distributions of the 100 simulated instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The plots clearly show that the degree distributions are different across the DGPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In fact, Luque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (2009) has shown that the intra-layer degree distribution for white noise (uncorrelated data) follows a power law � P(k) = 1 3 � 2 3 �k−2� and our results indicate that strong serial correlation leads to intra-layer degree distributions that are positively skewed: as illustrated in Appendix B, the sVAR is the only DGP that produces data with strong serial cor- relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The degree distribution for the inter-layer subgraphs does not have an algebraic close form even in the simplest case of two uncorrelated white noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' However, extensive simulations indicate that it does not follow the power law P(k) = 1 3 � 2 3 �k−2, as illustrated in the first line, the third column of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The plots also indicate that inter-layer degree distribution depends both on the correlation between the two time series (CCF represented in the second column of the plot in Figure 9) and the serial correlation within each time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Moreover, we note that inter-layer degree distributions for sVAR are positively skewed, for GARCH models, wGARCH and sGARCH, are exponentially shaped while the remaining are approximately linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Once again, a slower decay of the lagged correlation leads to a longer tail in the degree distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Also, the de- gree distribution curves corresponding to the GARCH models stand out from the others, especially the inter- and all-layer degree distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The exponential shape of the inter-layer degree distributions is induced by the heteroscedasticity and volatility clusters in the data which limit cross-horizontal visibility to the nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='3 MTS features via MHVG The results for all the 21 features introduced in Section 4 and all DGPs, organized by subgraph structure, are summarized, mean (standard deviation), in Tables 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The values have been Min-Max normalized (across models) for comparison purposes since the range of the different features varies across the different DGPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The cells in the tables are coloured with a gradient based on the mean values: cells with a maximum value of 1 are coloured red, cells with a minimum value of 0 are coloured white, and the remainder with a hue of red colour proportional to its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Table 3: Mean values (standard deviation) of the 100 instances of each DGP for each topological global feature from intra-layer graphs, G1 and G2, and inter-layer graph, G1,2, resulting from the corresponding MHVGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' DGP Average Average Number of Modularity Degree Path Length Communities ¯k1 ¯k2 ¯k1,2 ¯d1 ¯d2 ¯d1,2 S1 S2 S1,2 Q1 Q2 Q1,2 iBWN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='756 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='044 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='179) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='170) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='077) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='161) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='172) The results indicate that each set of features - intra-layer (first two columns of each feature in Table 3), inter- layer (third column of each feature in Table 3), all-layer (first four columns of Table 4) and relational (last five columns of Table 4) - distinguishes two groups of MTS depending on properties pertaining to correlation (serial and cross) and volatility clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To understand which MNet topological features capture the specific properties of the MTS models, we perform PCA on the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 10 represents a bi-plot obtained using the intra-, inter-, all-layer, and relational features, with the two principal components (PC) explaining 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='8% of the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The bi-plots resulting from PCA in restricted feature sets are represented in Figure 14 (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Overall, we can say that the average degree and average ratio degree, ¯k and ¯r, are positively and negatively correlated, respectively, with the average path length, ¯d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The community-related features of the intra- and all-layer graphs are positively correlated but less correlated with the community-related features of the inter-layer graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The features that most contribute to the first two PCs are the ¯k1,2, S1,2 and Q1,2 of the inter-layer graphs, the ¯k1,2 all of the all-layer graphs, the ¯r1⪯2 and ¯r2⪯1 of the relational layers, and Q1 and Q2 of the intra-layer graphs (see Figure 15 of Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 10 clearly shows four groups of models, GARCH, sVAR, cBWN and a group constituted by the wVAR and the iBWN and identifies the topological features that characterize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The strong ACF and CCF of the sVAR are represented by high values for the number of communities and modularity in its intra- and all-layer graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Inter-layer graphs present higher values of community-related features for GARCH models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The average path length represents the GARCH models, in particular, the average path length of the all-layer graphs tries to distinguish both wGARCH and sGARCH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The strong contemporaneous CCF of the cBWN is represented by high values of average ratio degree features, such as the average degree values of its inter and all-layer graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The iBWN and wVAR, are represented by high values of intra-layer average degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The above results indicate that the topological features extracted from MHVG are adequate as a set of MTS features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To further explore this idea, we proceed with clustering analysis of the DGPs via MNet topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='4 Mining Time Series with MNetF In this section, we illustrate the usefulness of MNet features in MTS data mining tasks with a case study regarding DGP clustering via a feature-based approach (Maharaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Given a set of MTS, we compute the MNet feature vectors, which are then Min-Max rescaled into the [0, 1] range and organized in a feature data matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The PCs are computed, with no need for z-score normalization in PCA computation, and finally, a clustering algorithm, the k-means, is applied to all PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We opt for k-means since it is fast and widely used in the literature, but it requires the pre-introduction of the number of clusters, which for the purpose of this work is not a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We use the appropriate evaluation metrics: Average Silhouette (AS), Adjusted Rand Index (ARI), and Normalized Mutual Information (NMI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The AS does not need the ground truth, while the ARI and NMI do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The range of values for NMI is [0, 1] and for ARI and AS is [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The results are summarized in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' As illustrated in Figure 11, the different metrics indicate different number of clusters for the data set: ARI indicates k = 5 followed by k = 6 which is the ground truth value, while the NMI indicates either k = 6 or k = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Metric AS, using the silhouette method to assess the quality of the clusters indicates k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' It is interesting to note that the elements of the three clusters are: the sVAR models in one cluster, the GARCH models in another, and the WN and wVAR models in the third cluster, indicating that the DGPs were clustered according to correlation (serial and cross) and volatility properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 12 represents the results from clustering the 100 instances of the 6 DGPs, using the 21 MNet features and the k-means algorithm with k = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' It is noticeable the perfect attribution of cBWN and sVAR samples across two different clusters (clusters 1 and 3), the attribution of iBWN and wVAR samples across the same cluster (cluster 2), and a homogeneous attribution of GARCH samples (wGARCH and sGARCH) across two clusters (4 and 5), and some samples of cBWN and sGARCH across cluster 6 since k = 6 and the GARCH samples are the most disparate in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The clustering exercise was performed considering subsets of the MNet feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The results summarized in Table 5 indicate that inter-layer edges contain, in fact, information about the MTS, leading to better clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Subgraphs with both intra-layer edges and inter-layer edges add information that leads to improvements in the clustering results (compare the last three rows of the Table with the first two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The results show that cross- HVG of inter-layer edges capture different properties from MTS data, that, as we expected, translate into better clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Also note that the results of ARI and NMI from the set of intra-layer features are good results, because the DGP under analysis involves the same statistical process for the two time series components whose properties inherent to each process are also captured by the HVG mapping methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Table 5: Clustering evaluation metrics for the different clustering analyses resulting from different MNet feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The values reflect the mean of 10 repetitions of the proposed method for different feature vectors and for the ground truth (k = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The highest values are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Feature Set ARI NMI AS [−1, 1] [0, 1] [−1, 1] Intra-layer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='289 Inter-layer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='418 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='508 All-layer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='505 Relational 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='622 MNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='629 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='452 18 Figure 9: Analysis plots of DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The first column shows a subset of timestamps of each DGP, the second plots the cross-correlation function between their corresponding time series components, and the last three show the degree distribution of the MHVGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The plots corresponding to the degree distributions are on a semi- logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Lines of different colors refer to each DGP model (Y ), where the darkest colors refer to their first components (Y 1) and the lighter ones to the second (Y 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 19 MTS CCF P(k): Intra-Layer P(k): Inter-Layer P(k): All-Layer Independent N 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 - 1e+00 1e+00 1e+00 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='75 1 1e-01 1e-01 1e-01 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 0 1e-02 1e-02 1e-02 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 1e-03 1e-03 1e-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 1e-04 1e-04 1e-04 - 1000 1100 1200 1300 15-10 2 10 18 22 2 9 10 14 16 20 24 28 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 - 1e+00 1e+00 - 1e+00 - 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='75 1e-01 1e-01 1e-01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 1e-02 1e-02 1e-02 2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 1e-03 1e-03 1e-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 1e-04 1e-04 1e-04 - 1000 1100 1200 1300 15 -10 2 6 10 18 22 2 6 10 14 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 1e+00 1e+00 1e+00 Weak VAR 4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content="75 1e-01 1e-01 1e-01 2 0S'0 1e-02 1e-02 1e-02 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content="25 iL 1e-03 1e-03 1e-03 2 00'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 1e-04 1e-04 1e-04 1000 1100 1200 1300 15-10 2 10 18 22 2 9 10 14 1620242832 00 1e+00 1e+00 1e+00 - VAR Wa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='75 1e-01 1e-01 1e-01 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 1e-02 1e-02 1e-02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 - 1e-03 1e-03 1e-03 ii 1e-04 1e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 1e-04 1000 1100 1200 1300 15 -10 10 15 2 10 14 18 22 2 9 10 14 16 20 24 28 32 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 - 1e+00 1e+00 1e+00 - Weak GARCH 25 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='75 1e-01 1e-01 1e-01 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 1e-02 1e-02 1e-02 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 - 1e-03 1e-03 1e-03 50 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 1e-04 1e-04 1e-04 - 1000 1100 1200 1300 15-10 0 15 2 10 14 18 22 2 6 10 14 12 16 20 24 28 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 1e+00 1e+00 - 1e+00 GARCH 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content="75 1e-01 1e-01 1e-01 - 0 09'0 1e-02 - 1e-02 1e-02 - 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 1e-03 1e-03 1e-03 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 1e-04 1e-04 1e-04 - 1000 1100 1200 1300 15-10 5 0 1015 2 10 14 18 22 2 6 10 14 12 16 20 24 2832 Time Lag k k kFigure 10: Bi-plot of the first two principal components (PC) of principal component analysis for the Data Gen- erating Process (DGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Each DGP is represented by a different color and the arrows represent the contributions of the MNet features to the PCs, the larger the size, sharpness, and closer to the red the greater the contribu- tion of the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Features set together are positively correlated and those placed on opposite quadrants are negatively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 20 PCA: MNet Features 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 Models 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='0 cBWN iBWN SGARCH Dim2 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='4%) sVAR WGARCH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 IS !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='intra WVAR Contrib 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 Dim1 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='4%)(a) ARI (b) NMI (c) AS Figure 11: Results of DGP’s clustering evaluations using the three evaluation metrics, ARI, NMI, and AS, for the different number of clusters, k, given as input in the k-means algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The results refer to 10 repetitions of the clustering analysis using the MNet features set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 12: Attribution of the samples corresponding to instances of MTS models to the different clusters, ac- cording to intra, inter, all-layer, and relational feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The different models are represented on the horizontal axis and by a unique color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The bivariate time series samples are represented by the colored points according to its model process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The vertical axis represents the cluster number to which a bivariate time series is assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 21 MNet Features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='4 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='3 2 3 4 5 6 7 8 9 10 11 12 Number of clusters (k)MNet Features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='7 Information 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='3 2 3 4 5 6 7 8 9 10 11 12 Number of Clusters (k)MNet Features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2 2 3 4 5 6 7 8 9 10 11 12 Number of Clusters (k)Cluster Analysis 6 4 cluster N cBWN iBWN SGARCH SVAR WGARCH WVAR Model6 Conclusions In this paper, we propose a new mapping method to represent multivariate time series as multilayer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Our procedure relies on two main steps: first each time series component is mapped into a layer in a multilayer networks structure, following the horizontal visibility concept available in the literature, and for each pair of time series components in the multivariate time series, inter-layer edges are established based on a new concept of cross-horizontal visibility proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We are interested in analyzing the concept of horizontal visibility as the base concept, given the promising results achieved in the previous works (Silva, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', 2022), but the idea of the proposed mapping method, the Cross-(H)VG algorithm, naturally extends to the concept of natural visibility and other versions of VG’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To assess the proposed method, we analyzed a specific set of topological features of multilayer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' These features are based on concepts of node centrality, graph distances, clustering, communities, and similarity measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Each feature extracted from all the subgraphs of the resulting multilayer network structure: intra- layer graphs (only intra-layer edges), inter-layer graphs (only inter-layer edges) and all-graphs (with both intra and inter-layer edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We perform an empirical evaluation on a set of 600 synthetic bivariate time series, grouped in 6 different and specific statistical models, that result in a data set of 600 MHVGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To understand the potential of our proposed mapping method, we first analyze the degree distributions of the intra, inter, and all-layer subgraphs of MHVGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We were able to identify the specific properties of multivariate time series models, namely, we were able to relate weak and strong cross-correlation with shapes of the inter-layer degree distribution curves and weak and strong autocorrelation with shapes of the intra-layer degree distribution curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In particular, we were also able to relate the persistence of strong correlations to distributions (that result in positively skewed shape) that have a longer right tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Adding to the correlation properties (both auto and cross, contemporaneous and lagged), the properties of the statistical models, such as heteroscedasticity and smoothness, results in inter and all-layer degree distributions with different shape curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We also investigated the global features of the subgraphs (intra, inter, and all-layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Community-related fea- tures from intra and all-layer graph highlight the strongly VAR models, with high and persistent autocorrelation and cross-correlation, as well as with smoothness, and from inter-layer graphs highlight the heteroscedasticity models, both weak and strong VGARCH models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' However, the values of average path length from all-layer graphs seem to distinguish the properties of weakly and strongly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The average intra-degree has higher values for independent white noise and weak VAR models, but not distinguishing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The new relational feature proposed in this work, average ratio degree, seems to differentiate well the highly correlated contemporary white noise models, which leads to a similarity in its inter-layer degree and intra- layer degree features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' In the context of this work, based on the synthetic models chosen for analysis, the Jensen–Shannon divergence measure is only useful to characterize strong VAR models that present very strong and very persistent correlations, unlike the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' However, this feature can be quite useful in real contexts, where the different variables of a multivariate time series can follow different dynamic models that will be captured by this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We intend to investigate this and other characteristics, as well as other topological features of multilayer networks in our future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To further showcase the validity of the proposed mapping method and feature set, we perform a clustering analysis of the synthetic bivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The results allow concluding that features that use the inter- layer edges of the MHVGs add valuable information to intra-layer features thus allowing to distinguish of the different statistical processes underlying the multivariate data models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 22 To conclude, this work proposes a procedure to map multivariate time series into multilayer networks as a mean to obtain a set of multivariate time series features that can be used for mining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Open issues for future work are new topological features of multilayer networks and a more extensive empirical study with benchmark multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Availability of data and materials The raw data are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='com/vanessa-silva/MHVG2MTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Acknowledgments This work is financed by National Funds through the Portuguese funding agency, FCT, within project UIDB/50014/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Vanessa Silva is also supported by an FCT PhD research grant (PD/BD/139630/2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 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Characteristic-based clustering for time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Data mining and knowledge Discovery, 13(3):335–364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Wei, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Multivariate Time Series Analysis and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' John Wiley & Sons, Hoboken, New Jersey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Zou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', Donner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', Marwan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', Donges, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=', and Kurths, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Complex network approaches to nonlinear time series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Physics Reports, 787:1–97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 24 A MHVG: Algorithms Algorithm 3: Horizontal Visibility Graph Input: A time series, Y a, (ts), and a layer, La, (layer) Procedure HVG(ts, layer) 1 T ← ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='size() ▷ The time series length for i ← 1 to T do 2 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='add Node(i, layer) ▷ Add node-layer va i corresponding to timestamp ti end for i ← 1 to T − 1 do 3 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='add Edge(i, i+1, layer) ▷ Add intra-layer edge (va i , va i+1) 4 k ← 1 for j ← 2 to T − i do 5 condition ← ts[i + k] 6 p ← ts[i + j] /* Test HVG condition: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 4 / if condition ≥ ts[i] then 7 break else if condition < p then 8 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='add Edge(i, i+j, layer) ▷ Add intra-layer edge (va i , va i+j) 9 k ← j end end 10 return 25 Algorithm 4: Cross-Horizontal Visibility Graph Input: Two rescaled time series, Za and Zb, (tsA, tsB), the corresponding layers, La and Lb, (layerA, layerB), and the maximun time series, max � Za, Zb� ,(tsMax) Procedure CHVG(tsA, tsB, tsMax, layerA, layerB) 1 T ← tsMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='size() ▷ The time series lengths /* Test cross-horizontal visibility to the right of Ya,i / for i ← 1 to T − 1 do 2 k ← i + 1 for j ← i + 1 to T do if j = i + 1 then 3 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='add Edge(i, i+1, layerA, layerB) ▷ Add inter-layer edge (va i , vb i+1) else 4 condition ← tsMax[k] 5 p ← tsB[j] /* Test Cross-HVG condition: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 5 / if condition ≥ tsA[i] then 6 break end if condition < p then 7 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='add Edge(i, j, layerA, layerB) ▷ Add inter-layer edge (va i , vb j) 8 k ← j else if condition < tsMax[j] then 9 k ← j ▷ Update index k when tsMax[j] > tsMax[k] end end end end /* Test cross-horizontal visibility to the left of Ya,i / for i ← 2 to T do 10 k ← i − 1 for j ← i − 1 to 1 do if j = i − 1 then 11 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='add Edge(i, i-1, layerA, layerB) ▷ Add inter-layer edge (va i , vb i−1) else 12 condition ← tsMax[k] 13 p ← tsB[j] /* Test Cross-HVG condition: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 5 / if condition ≥ tsA[i] then 14 break end if condition < p then 15 mnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='add Edge(i, j, layerA, layerB) ▷ Add inter-layer edge (va i , vb j) 16 k ← j else if condition < tsMax[j] then 17 k ← j ▷ Update index k when tsMax[j] > tsMax[k] end end end end 26 B Multivariate Time Series Models Main references Cipra (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Wei (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Shumway and Stoffer (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Tsay (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Linear Models BWN The vector white noise process, ϵt, is a vector of sequences of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' random variables with mean vector 0 and and covariance matrix function Σ, where Σ is an m × m symmetric positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The components of the white noise process are serially uncorrelated corr(ϵi,t, ϵi,s) = 0, for t ̸= s, but may be contemporaneously correlated, corr(ϵi,t, ϵj,t) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' It is the simplest multivariate time series process that reflects information that is neither directly observable nor predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We generate white noise processes, ϵt ∼ N(0, 1), that are not correlated, that is, are independent, and we refer to theses processes as iBWN, and white noise processes contemporaneously correlated that we refer to them as cBWN, � ϵ1,t ϵ2,t � ∼ N � 0, � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' VAR(1) The vector autoregression process is a natural extension of the univariate autoregressive (AR) process that the variable values depends linearly on its own previous values and on a stochastic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We defined a VAR(1) process as a vector AR process of order 1 if it satisfies the following equation: Y t = ϕ + φY t−1 + ϵt, (7) where ϵt is the vector white noise, φ is the vector of autoregressive constants and ϕ is the vector of intercepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We generate a VAR(1) of 2 dimensions with the following vector of parameters: � Y1,t Y2,t � = � ϕ1,1 ϕ2,1 � + � φ1,1 φ1,2 φ2,1 φ2,2 �� Y1,t−1 Y2,t−1 � + � ϵ1,t ϵ2,t � , (8) where ϕ = � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � , φ = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 � and ϵt ∼ � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � to generate weakly correlated VAR(1) processes, and ϕ = � 0 0 � , φ = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='80 � and ϵt ∼ � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � to generate strongly correlated VAR(1) processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We refer to the two models generated as wVAR and sVAR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Non Linear Models VGARCH(1, 1) Also generalized autoregressive conditional heteroskedasticity (GARCH) models can be gen- eralized to multidimensional settings, extending the principle of univariate conditional heteroscedasticity to mutual volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We generate a bivariate GARCH(1, 1) model according to the following volatility equation: σt = ω + αϵ2t−1 + βσt−1, (9) where σt denotes the volatility in the variables Y t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We generate a VGARCH(1, 1) of 2 dimensions with the following vector of parameters: � σ11,t σ22,t � = � ω1,1 ω2,1 � + � α1,1 α1,2 α2,1 α2,2 �� ϵ2 1,t−1 ϵ2 2,t−1 � + � β1,1 β1,2 β2,1 β2,2 �� σ11,t σ22,t � , (10) where ϵt ∼ � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � to generate weakly correlated VGARCH(1, 1) processes, and ϵt ∼ � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 � to generate strongly correlated VGARCH(1, 1) processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' To both processes we use ω = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='02 � , α = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='05 � and β = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='88 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' We refer to the two models generated as wGARCH and sGARCH, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Bivariate time series are generated from the above DGP using the R packages: lgarch (Sucarrat, 2015), mAr (Barbosa, 2012) and ccgarch (Nakatani, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 27 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1 Autocorrelation Function Plots See Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 13: Plot of the autocorrelation function of an instance of each of the DGP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The first column refers to the ACF’s of the first time series component (Y 1) of each model, while the second column refers to the second component (Y 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 28 Independent WN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 Weak GARCH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='75 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 Strong GARCH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 Lag LagB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2 Principal Component Analysis Results See Figure 14 and Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' (a) Intra-layer features (b) Inter-layer features (c) All-layer features (d) Relational features Figure 14: Bi-plot of the first two principal components (PC) of principal component analysis for the Data Generating Process (DGP) using the different feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Each DGP is represented by a different color and the arrows represent the contributions of the set of features to the PC’s, the larger the size, sharpness, and closer to the red the greater the contribution of the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' Figure 15: Bar plot with contributions of MNet features to the total of all 20 principal components formed by the PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' The red dashed line on the plot indicates the expected average contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 29 PCA: Intra-layer Features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='8 Contrib 20 15 Dim2 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='3%) 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content="4 Models CBWN iBWN SGARCH 0'0 sVAR WGARCH Q 1 WVAR 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='0 Dim1 (64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='1%)PCA: Inter-layer Features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='50 - : Models 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='25 - CBWN iBWN SGARCH Dim2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='2%) : sVAR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content=' 2 WGARCH 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='00 WVAR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf'} +page_content='3 Contrib 30 0.' metadata={'source': 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/dev/null +++ b/sNAzT4oBgHgl3EQfBPpp/content/tmp_files/2301.00939v1.pdf.txt @@ -0,0 +1,2308 @@ +IEEE/ASME TRANSACTIONS ON MECHATRONICS + + + +Abstract— This paper expounds the design and control +of a new Variable Stiffness Series Elastic Actuator (VSSEA). +It is established by employing a modular mechanical design +approach that allows us to effectively optimise the stiffness +modulation characteristics and power density of the +actuator. The proposed VSSEA possesses the following +features: i) no limitation in the work-range of output link, ii) +a wide range of stiffness modulation (~20Nm/rad to +~1KNm/rad), iii) low-energy-cost stiffness modulation at +equilibrium and non-equilibrium positions, iv) compact +design and high torque density (~36Nm/kg), and v) high- +speed stiffness modulation (~3000Nm/rad/s). Such features +can help boost the safety and performance of many +advanced robotic systems, e.g., a cobot that physically +interacts +with +unstructured +environments +and +an +exoskeleton that provides physical assistance to human +users. These features can also enable us to utilise variable +stiffness property to attain various regulation and trajectory +tracking control tasks only by employing conventional +controllers, eliminating the need for synthesising complex +motion control systems in compliant actuation. To this end, +it is experimentally demonstrated that the proposed VSSEA +is capable of precisely tracking desired position and force +control references through the use of conventional +Proportional-Integral-Derivative (PID) controllers. +Index Terms— Compliant robotics, safe robotics, series +elastic actuators, variable stiffness actuators, physical +robot-environment interaction. +I. INTRODUCTION +O BOOST safety in physical-robot environment interaction, +compliant actuators have been widely adopted by many +different advanced robotic systems such as humanoids, +cobots, quadrupeds, and exoskeletons [1–5]. A compliant +actuation system could be developed by simply integrating an +elastic element into the design of an actuator [5]. For example, +Series Elastic Actuators (SEAs), one of the most popular +compliant actuation systems in robotics, are developed by +placing a spring between a conventional rigid actuator and link +[6–8]. In addition to improving safety, the spring between the +rigid actuator and link can provide several benefits such as low- +cost and high-fidelity force control, mechanical energy storage, +lower reflected inertia, higher tolerance to impact loads, and +increased output power [8]. + +Manuscript received…... (Corresponding author: Emre Sariyildiz). +E. Sariyildiz, J. Roberts and C.-H. Kuo are with the School of Mechanical, +Materials, Mechatronic and Biomedical Engineering, University of +Wollongong, +Wollongong, +NSW, +2522, +Australia. +(e-mails: +emre@uow.edu.au, robertsj@uow.edu.au, chkuo@uow.edu.au). +R. Mutlu is with the Faculty of Engineering and Information Sciences, +University of Wollongong in Dubai, Dubai, United Arab Emirates and the +Despite the aforementioned benefits, the elastic elements +integrated to compliant actuators introduce certain challenges +and fundamental limitations in motion control [9]. For example, +it is a well-known fact that the position control problem of a +compliant actuator is more complicated than that of a +conventional rigid actuator [10–14]. To suppress the vibrations +and disturbances of a compliant actuator’s link, researchers +generally need to employ advanced motion controllers [14]. +Another example is that although using a softer elastic element +improves safety and transparency, the natural frequency of the +compliant actuator decreases. This not only excites the +vibrations at link side but also lowers the bandwidth of the +actuator, thus limiting achievable position and force control +performance in motion control applications [13, 14]. It is +therefore essential to properly choose the stiffness of the elastic +element of a compliant actuator based on the target control task +[15]. However, this is mostly impractical for compliant +actuators with fixed elastic elements, which often leads to +compromise between safety and performance [10, 16]. A +simple yet efficient solution for this fundamental problem could +be achieved by integrating a compliant mechanical component +with variable stiffness property to actuators in series or parallel +[17, 18]. +Adaptable compliance mechanisms that can alter the stiffness +of actuators mechanically have been developed to meet the +different compliance requirements of motion control tasks such +as soft actuation in human-robot interaction and stiff actuation +in trajectory tracking. Although no standard terminology exists, +such actuators are generally called Variable Stiffness Actuators +(VSAs) in the literature. A comprehensive survey on the design +of VSAs can be found in [18, 19]. Among them, antagonistic +actuation is one of the most well-known and widely used +stiffness modulation methods in VSAs. Inspired by mammalian +anatomy, this actuation method has been studied since early +1980s [20], and different antagonistic actuators have been +developed and used in various robotic applications, such as +legged locomotion and upper-limb rehabilitation, since late +1990s [21, 22]. The simplest antagonistic actuator can be +designed by using an agonistic-antagonistic setup which +connects two motors to a link via two nonlinear springs, similar +to biceps and triceps in the human arm [19, 23]. While the +equilibrium position can be adjusted by rotating two motors in +the same direction, counter-rotation of motors alters the +Intelligent Robotics & Autonomous Systems Co (iR@SC), NSW, 2529, +Australia. (e-mail: ramutlu@irasc.com.au). +B. Ugurlu is with the Department of Mechanical Engineering, Ozyegin +University, Istanbul, 34794, Turkey. (e-mail: barkan.ugurlu@ozyegin.edu.tr). + +Design and Control of a Novel Variable +Stiffness Series Elastic Actuator +Emre Sariyildiz, Senior Member, IEEE, Rahim Mutlu, Member IEEE, Jon Roberts, +Chin-Hsing Kuo, Barkan Ugurlu, Member IEEE +T + +IEEE/ASME TRANSACTIONS ON MECHATRONICS + + +stiffness of the actuator by changing the spring preload. Despite +its simplicity, this bioinspired actuation method has several +drawbacks in practice, e.g., i) the control problems of stiffness +modulation and equilibrium position are coupled, ii) the output +torque of the actuator is limited by the maximum torque of each +motor, iii) the potential energy capacity of nonlinear springs +cannot be used entirely, and iv) the energy consumption is high +because preloading nonlinear springs for stiffness modulation +requires constant power drain, even when the actuator does not +perform net mechanical work at equilibrium positions [24, 25]. +Numerous different antagonistic and non-antagonistic VSAs +have been proposed to tackle the drawbacks of the agonistic- +antagonistic actuation setup in the last two decades [18, 19]. For +example, while the output torque of antagonistic actuators is +increased using bi-directional configuration in [26], a partially +decoupled motion control problem is obtained using a quasi- +antagonistic actuation mechanism in [27]. However, the +aforementioned problems could not be entirely addressed using +antagonistic actuation systems [18, 19]. This has motivated +many researchers to build non-antagonistic VSAs. The control +problems of the equilibrium position and compliance of the +actuator’s output link are decoupled using a new mechanical +design approach in Maccepa [28]. The stiffness of the actuator +is modulated by controlling the tension of a linear spring. A +similar stiffness modulation approach is employed in DLR- +VSJ, and a light-weight and compact VSA is designed by +changing only the cam disk of the joint in [29]. Nevertheless, +similar to antagonistic actuators, high-energy-cost stiffness +modulation remains a challenging problem in these non- +antagonistic VSA design approaches. Since the stiffness of the +Maccepa and DLR-VSJ is modulated by pretensioning springs, +they require constant power drain at equilibrium positions, thus +leading to high energy consumption [28 – 30]. The energy-cost +of stiffness modulation has been improved using different VSA +design approaches in the last decade. The stiffness of the output +link is modulated by changing the positions of the springs and +pivot points on a lever arm in AWAS [31, 32]. Low energy cost +stiffness modulation (e.g., theoretically zero power drain at +equilibrium positions) could be achieved using this VSA design +approach. The stiffness range, however, is limited by the size of +the actuator [31, 33]. In vsaUT, the stiffness of the actuator is +modulated by controlling the effective length of a lever arm [34, +35]. This stiffness modulation approach allows us to attain not +only zero power drain at equilibrium positions but also infinite- +range stiffness modulation with compact actuators. However, +the power drain by the motor dedicated to stiffness modulation +becomes unbounded as the stiffness of the actuator approaches +infinity [33, 34]. Variable length leaf spring mechanisms have +also been employed to develop energy efficient VSAs [30, 36 – +39]. Compared to the other antagonistic and non-antagonistic +VSAs, recent studies show that variable length leaf spring +mechanisms can provide several benefits in practice, e.g., zero +power drain at equilibrium positions, fast and infinite-range +stiffness modulation, and bounded power drain for all stiffness +ranges at equilibrium and non-equilibrium positions [30, 33]. +These features could be very useful in biomedical engineering +applications as shown in [30, 38]. Nevertheless, when it comes +to building a compact VSA that can be integrated to different +robotic systems such as cobots, the variable length leaf spring +mechanisms may involve several drawbacks such as low +torque/power density and work-space limitations [30, 36 – 39]. +It is noted that a non-antagonistic VSA can be simply built +by employing a discrete stiffness modulation method where +multiple springs could be integrated to actuators in series or +parallel [40 – 42]. The main drawback of this actuation method +is the limited stiffness range which depends on the number of +springs employed in the actuator design. Moreover, the discrete +stiffness modulation method leads to several challenges in +controller analysis and synthesis such as the stability problem +of switching systems [40, 42]. Therefore, continuous stiffness +modulation methods are mainly considered in this paper. +The existing VSAs have their own merits and demerits. +While they are highly functional in their own domain of use, +they may however fall-short in complying with all the desirable +technical specifications of an ideal compliant actuator for +practical applications: i) a compact and simple mechanical +design that allows to easily reconfigure a VSA for different +applications, ii) no limitation in motion range, iii) a wide range +of stiffness modulation, iv) rapid stiffness change, and v) +energy efficient actuation [30]. In general, researchers manage +a trade-off to target only few of the aforementioned qualities, +leading to different compromises such as high energy +consumption or limited motion control performance in robotic +applications. Hence, despite many recent advances, more effort +should be put into the development of VSAs [18, 19 and 30]. +This is summarised using the examples of existing VSAs in the +literature in Table I. +To this end, this paper proposes a new VSSEA which consists +of three main components: i) a rigid actuator that independently +controls the equilibrium position, ii) a novel Variable Stiffness +Actuation Mechanism (VSAM), and iii) a direct drive motor +that independently adjusts the stiffness. The proposed VSSEA +TABLE I: Actuators, Deflection Range, Motion Range, Stiffness Range, Energy Cost of Fixed Stiffness at Equilibrium, Energy Cost of Infinite-Range Stiffness +Modulation, Torque Density and Stiffness Modulation Speed. +Actuators +Deflection +Range [degree] +Motion Range +[degree] +Stiffness range +[Nm/rad] +Energy cost of +fixed stiffness at +equilibrium +Energy cost of +infinite-range +stiffness +modulation +Torque density +[Nm/kg] +Stiffness modulation +speed [Nm/rad/s] +VSSEA +±25 +no limitation +20 – 1000 b) +ZEC +BEC +35.72 +3000 +Maccepa +±60 +±90 c) +5 – 110 +NZEC +NA +20.83 +40 +DLR-VSJ +±15 +±180 +50 – 820 +NZEC +NA +22.27 +2350 +AWAS +±12 +±120 +30 – 1500 +ZEC +NA +31.1 +420 +AWAS-II +±18 +±150 +10 – 10000 +ZEC +NA +9.76 +4000 +vsaUT-II +±45 +±180a) +0.5N – 100 +ZEC +NBEC +8.7 +1000 +VSA-I [36] +±12 +NA +250 – 3000 b) +ZEC +BEC +6.1 +NA +VSA-II [30] +±75 +±75 +10 – 8000 b) +ZEC +BEC +Passive (3kg) +10000 +Table I is prepared using experimental results given in the papers unless data is provided in the references. a) Although the motion range of vsaUT is limited (e.g., +vsaUT-II’s motion range is ±180o), mvsaUT can perform continuous motions without any motion range limitations [46]. b) Infinite-range stiffness modulation can +be achieved using leaf spring based VSAs. c) [28] states that the motion range of Maccepa can be increased. ZEC: Zero Energy Consumption; NZEC: Non-Zero +Energy Consumption; BEC: Bounded Energy Consumption when performing infinite-range stiffness modulation; NBEC: Non-Bounded Energy Consumption +when performing infinite-range stiffness modulation; and NA: Not Applicable because the actuators cannot provide infinite range stiffness modulation. + + +IEEE/ASME TRANSACTIONS ON MECHATRONICS + + +is simply developed by integrating the VSAM to a rigid +actuator. This modular design approach allows us to easily +optimise the stiffness modulation characteristics and output +power/torque of the VSSEA for different robotic applications. +The stiffness of the actuator is modulated by changing the +effective length of a group of leaf springs of the VSAM through +a direct drive motor. This stiffness modulation technique +provides several benefits: i) stiffness modulation over a large +range, i.e., from near-zero to infinite stiffness theoretically, ii) +high-speed stiffness modulation using a relatively slow motor, +and iii) zero/near-zero energy consumption for holding/altering +the stiffness at equilibrium positions, and low-energy-cost +stiffness modulation at non-equilibrium positions. Moreover, +the proposed VSSEA has no limitation in motion range. To the +best of our knowledge, the existing VSAs have yet to combine +all these desired features of our proposed VSSEA [30]. + The rest of the paper is organised as follows. In Section II, +the mechanical design of the VSSEA is presented. In Section +III, the dynamic model of the actuator is derived by using the +analogy of a mass-spring-damper system and Euler-Bernoulli +beam theory. In Section IV, the performance of the VSSEA is +experimentally verified. The paper ends with discussion and +conclusion given in Sections V and VI. +II. MECHANICAL DESIGN +A. Variable Stiffness Series Elastic Actuator: +Figure 1 illustrates the CAD model and the first prototype of +the VSSEA. It comprises i) a conventional rigid actuator that +includes a servo motor and a gearbox, illustrated by M1, ii) a +novel variable stiffness actuation mechanism illustrated by +VSAM, and ii) a direct drive servo motor illustrated by M2 in +the figure. +The conventional rigid actuator M1 is used to independently +control the equilibrium position of the output link. The +proposed modular design approach enables us to freely tune the +output torque and speed of the VSSEA. For example, we used +Maxon EC90 flat motor and a 1:100 ratio harmonic drive to +achieve ~100Nm output torque and ~π/2rad/s output speed in +the first prototype of the actuator. The output power of the +VSSEA can be directly adjusted by employing a different servo +motor and/or a gearbox in the design of the conventional rigid +actuator M1. +The second motor M2 is used to control the stiffness of the +actuator via the VSAM independently. The low-energy-cost +stiffness modulation feature, which is explained in Section III, +of the VSAM allowed us to use the Maxon EC60 direct drive +flat motor in the stiffness control of the actuator. +As shown in Fig.1, the proposed VSSEA is simply designed +by integrating the VSAM to the conventional rigid actuator M1. +This modular design approach provides great flexibility in +building a VSA for different robotic applications. Let us now +present the mechanical design of the novel VSAM that provides +important features, such as a wide range of stiffness modulation +and energy efficiency, in compliant actuation. +B. Variable Stiffness Actuation Mechanism: +Figure 2 illustrates the CAD model and the first prototype of +the VSAM. The design comprises i) eight radially distributed +in-parallel leaf springs, ii) two rollers for each leaf spring to +reduce friction and improve energy efficiency, and iii) a ball +screw mechanism to move the rollers along the leaf springs as +illustrated in the figure. +The stiffness of the actuator is modulated by changing the +position of the rollers which is controlled by a ball-screw +mechanism that is driven by the second motor M2. The VSSEA +is in the softest mode when the rollers are at the free ends of the +leaf springs, and the stiffness of the actuator increases as the +rollers move towards the fixed end. With the nonlinear dynamic +behaviour of the VSAM, a wide range of stiffness modulation +can be obtained by simply changing the effective length of the +leaf springs through the position control of the rollers. This +allows us to perform large stiffness modulations within a short +time, and this feature can provide several benefits in robotic +applications [27]. Another important feature of the proposed +VSSEA is low-energy-cost stiffness modulation. For example, +the VSAM does not require constant power drain to hold the +stiffness constant at equilibrium positions. +III. DYNAMIC MODEL +The dynamic model of the VSSEA is obtained by employing +the analogy of a mass-spring-damper system and the Euler- +Bernoulli beam theory. +A. Variable Stiffness Series Elastic Actuator: +The dynamic model of the VSSEA is illustrated in Fig. 3. In +this figure, J represents the inertia of motor 1, gearbox, motor +2, and output link when  is m1, g, m2 and l, respectively;   , +b , and q similarly represent the torque, viscous friction +coefficient, and angle of the motor 1, gearbox, motor 2, and + +a) CAD model. b) First prototype of the VSSEA. +Figure 1: CAD model and first prototype of the novel VSSEA. M1: Motor 1, +M2: Motor 2, and VSAM: Variable Stiffness Actuation Mechanism. + +a) CAD model of leaf springs and mobile frame of the VSAM. b) First prototype of the VSAM. c) Dimensions of the VSAM. +Figure 2: CAD model and first prototype of the VSAM. + +a) +M1 +b) +VSAM +M1 +VSAM +M2 +M2a) +xedEn +55mmIEEE/ASME TRANSACTIONS ON MECHATRONICS + + +output link, respectively; q and q represent the first and +second order derivatives of q , i.e., angular velocity and +angular acceleration, respectively; and k represents the +stiffness of the actuator, which can be defined as a nonlinear +function of +2 +m +q + and +lq as shown in Section III.C. +While the first motor denoted by m1 in Fig. 3 is used to control +the equilibrium position of the actuator, the second motor +denoted by m2 is used to independently modulate the stiffness +of the output link. The dynamic model of the VSSEA can be +derived from this figure as follows: + +1 +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +2 +2 +dis +m +m +m +m +m +s +m +dis +l +l +l +l +s +l +l +dis +dis +m +m +m +m +s +m +J q +b q +N +J q +b q +J +q +b q + + + + + + + + + + + + + + + + + + + + + + + (1) +where +dis + represents the unknown/unmodelled disturbances of +motor 1, motor 2, and output link when  is m1, m2 and l, +respectively; +s represents the torque exerted by the nonlinear +springs of the VSAM on motor 1 and output link; +dis +s + represents +the disturbance torque exerted by the nonlinear springs of the +VSAM on motor 2 in stiffness modulation; and +2 +1 +1 +m +m +g +J +J +N J + + + + +and +2 +1 +1 +m +m +g +b +b +N b + + + + where N is gear ratio. +Equation (1) gives a simple yet useful dynamic model for the +proposed VSSEA. However, more effort should be expended +on understanding the nonlinear dynamics of the VSAM, i.e., +deriving +s and +dis +s +in Eq. (1). This will enable us to explain the +important features such as the energy efficiency of the VSSEA. +B. Variable Stiffness Actuation Mechanism: +To derive the model of the VSAM, let us focus on the first +leaf spring illustrated in Fig. 4a. In this figure, +1xF , +1 +yF , and +1zF +represent the forces exerted by the leaf spring on the roller along +the +1x , +1y , and +1 +z axes of the local coordinate frame on the +leaf spring, respectively. It is noted that the proposed analysis +can be similarly applied to the other leaf springs, e.g., the +second leaf spring illustrated in Fig 4b. +When it is assumed that only the first leaf spring is used in +the design of the VSAM, the kinematic and static equilibrium +equations of the output link can be directly obtained from Fig. +4c and Fig. 4d as follows: + + + +2 sin +2 +l +r +q +  + (2) + +1 +1 +1 +2 +2 +0 +s +l +r +l +y +z +l +F r +F +F r + + + + + + + + + + + + (3) +where is a kinematic constraint that relates the angle of the +output link to the deflection of the leaf spring; r represents the +magnitude of the distance vector 1r illustrated in Fig. 4c; and F +represents the magnitude of the force vector +F in which  can +be r1, y1 and z1 as shown in Fig. 4d. +When the other leaf springs illustrated in Fig. 2a are +considered, the static equilibrium equation of the output link +can be obtained by simply expanding Eq. (3) as follows: + +8 +8 +2 +2 +1 +1 +0 +i +i +i +s +l +r +l +y +z +l +i +i +F r +F +F r + + + + + + + + + + + + + + + + (4) +where +irF , +iy +F , and +izF similarly represent the forces exerted by +the ith leaf spring, and r represents the magnitude of the distance +vector +ir . Equation (4) shows that the leaf spring forces should +be identified to obtain the dynamic model of the VSSEA. +C. Leaf Springs: +The leaf springs of the proposed VSAM not only bend along +the lateral axes yi but also slightly rotate about the longitudinal +axes xi at non-equilibrium positions [43]. For the sake of +simplicity, the 3D deflection model of the leaf spring illustrated +in Fig. 5 is numerically obtained using Finite Element Method + +Figure 3: Dynamic model of the VSSEA. + +a) 3D CAD model illustrating forces exerted by the first leaf spring. b) 3D CAD +model illustrating forces exerted by the second leaf spring. c) 3D CAD model +illustrating the relation between the output torque and forces exerted on the first +and second leaf springs. d) 2D kinematic model illustrating the relation between +the deflection of the first leaf spring and output link angle. +Figure 4: Model of the Variable Stiffness Actuation Mechanism. + + +a) Beam at rest. b) Deflected beam c) Static output torque versus deflection. +l +is the output torque and +1 +ˆl +yF r +  + is the output torque associated with +1 +yF . +Figure 5: 3D large deflection model of the first leaf spring using FEM. + +Motor 1 +Gearbox +Output Link +Motor 2a) +Z1 +d +- +F +ri +F +gFigure5a +Figure Sb +F +F +IFFigure 5c:FEM Analysis of the Leaf Spring +20 +:r/L=0.9 +18 +: r/L = 0.75 +:/L=0.5 +16 +: r/L=0.25 +:/=0.9 +14 + 分 : Cr/ L = 0.75 +-分 : r/ L = 0.5 +12 += : r/L = 0.25 +8 +6 +4 +2 +0 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +OutputLinkDeflectionqt-qa(rad)IEEE/ASME TRANSACTIONS ON MECHATRONICS + + +(FEM). While Fig. 5a shows the FEM model at an equilibrium +position, Fig. 5b shows the maximum deflection of the leaf +spring when the output link rotates for about 20o. As shown in +this figure, the largest deflection occurs along the lateral axis +y1. Figure 5c illustrates the static output torque of the actuator +for different stiffness configurations when the deflection of the +output link increases. In this figure, +1 +ˆl +yF r +  + represents the +output torque associated with the force along the lateral axis at +different non-equilibrium positions. It is clear from Fig. 5c that +the output torque is mainly dominated by the bending force +along the lateral axis within the deflection range of the actuator. +This allows us to simplify the dynamic model of the VSAM +while precisely estimating the output torque within a large +deflection range. + To this end, let us consider the 2D deflection model of the +leaf spring illustrated in Fig. 6. In this figure, x1 and y1 represent +the position of a point on the beam along the x1 and y1 axes, +respectively; +1xS and +1x represent the arc length and slope of the +beam at x1, respectively; +1x + and +1y + represent the deflection of the +beam along the x1 and y1 axes at x1, respectively; +rx and +ry + +similarly represent the deflections at xr where the rollers are +positioned on the x1 axis; L represents the length of the beam; +and +rx represents the slope of the beam at xr. +The model of the leaf spring at a non-equilibrium position +can be derived by using the nonlinear Euler-Bernoulli beam +theory and Fig. 6 as follows. + + + +  +1 +1 +1 +0 +2 +sin +sin +x +r +x +y +x +EI +d +S +F + + + + + + + + (5) + +  + + +  +1 +1 +1 +0 +cos +2 +sin +sin +x +r +y +x +d +EI +x +F + + + + + + + + + (6) + +  + + +  +1 +1 +1 +0 +sin +2 +sin +sin +x +r +y +x +d +EI +y +F + + + + + + + + + (7) +where E represents Young’s modulus, and I represents the +moment of inertia of the beam cross section about the neutral +axis [44]. The other parameters are same as defined earlier. +Equations (5 – 7) cannot be solved analytically so a numerical +solution method should be employed to calculate the deflection +of the beam for the applied force +1y +F . This allows us to calculate +the reaction force exerted by the leaf spring on the roller, which +is required for output torque calculation in Eq. (3). For example, +the following steps can be employed to calculate the deflections +of the leaf spring at +rx : i) Using a numerical method (e.g., +Nonlinear shooting method or Adomian decomposition +method) or the fzero command of Matlab, find +rx + +in Eq. (5), +and ii) Calculate +rx + +and +ry + + by numerically integrating Eqs. (6) +and (7) over [0, +rx + +]. The kinematic and static equilibrium +equations of the output link can be calculated by applying +  +cos +ry +lq + + + +and +i +i +y +r +F +F + + to Eq. (2) and Eq. (3). +Although Eqs. (5-7) allows us to precisely estimate the output +torque of the VSSEA for different deflection angles at non- +equilibrium positions, they fall-short when attempting to +explain how the output torque and stiffness of the actuator +change as a function of +lq and +2 +m +q +. Moreover, they are not very +useful in controller analysis and synthesis. To tackle this +problem, let us obtain an approximate model that provides a +clear insight into the stiffness modulation characteristics of the +proposed VSSEA. +When it is assumed that the deflections of the leaf springs are +small, the torque and stiffness of the output link can be +calculated using simple beam theory and Eqs. (2 – 4) as follows: + + + +2 +2 +3 +3 +2 +48 +, +sin +2 +l +s +m +l +m +q +EI +q +q +r +q + + + + + + + + + + (8) +  + +2 +2 +3 +3 +2 +24 +, +cos +2 +l +m +l +m +q +EI +k q +q +r +q + + + + + + + + + (9) +where +2 + + + + in which represents the lead of the ball screw, +i.e., +2 +r +m +x +q + + + [30, 33, 44]. +Equations (8) and (9) can be used to explain how the output +torque and stiffness of the actuator change as a function of ql +and qm2. Equation (8) shows that the torque exerted by the +nonlinear springs of the VSAM on the output link and motor 1 +becomes higher as the angle of the output link increases. This +occurs because larger deflections of the leaf springs lead to +higher reaction forces exerted on the rollers. When the non- +equilibrium position of the actuator remains constant, +s can still +be regulated using the second motor. Equation (8) shows that as +2 +m +q +decreases (increases), the torque exerted by the VSAM gets +higher (lower). This occurs because the stiffness of the VSAM +changes by the angle of the second motor (i.e., position of the +roller) as shown in Eq. (9). The nonlinear behaviour of the +VSAM allows us to obtain a wide range of stiffness modulation +by simply controlling the angle of the second motor. +Figure 7 illustrates the torque and stiffness of the output link +at different non-equilibrium positions. The accuracy of the +small deflection model deteriorates as the angle of the output +link increases. This is expected because this approximate model +is obtained by assuming small deflections for the leaf springs. +Despite some inaccuracies, the proposed approximate model is +very useful to explain the important features of the VSSEA +because the small and large deflection models exhibit similar +behaviours as shown in Fig. 7. However, Eqs. (5 – 7) should be +used to accurately estimate the output torque of the VSSEA. +At an equilibrium position where +l +g +q +q + +, i) + + +1 +1 +y +x +F +F +0 when +the stiffness of the actuator remains constant, i.e., +2 +0 +m +q + +, and +ii) + +1yF +0, but + +1 +xF +0 due to small friction and inertial forces +when +2 +0 +m +q + +. Therefore, the energy consumption of stiffness +modulation is negligible at equilibrium positions. +The energy consumption of stiffness modulation increases as +the deflections of the leaf springs get larger at non-equilibrium + +Figure 6: 2D large deflection model of the first leaf spring. + +L +"x +Xp +Xi +..... +7 +Iyi +X +Sx, -da +Yi +F. +F. +Fpy +F. +yiIEEE/ASME TRANSACTIONS ON MECHATRONICS + + +positions. The disturbance torque exerted by the VSAM on the +stiffness modulation motor can be calculated as follows: +Large Deflection Model: +8 +1 +i +dis +s +x +i +F r + + + + (10) +Small Deflection Model: +  +2 +3 +3 +2 +48 +sin +tan +2 +dis +l +s +m +q +EI r +q + + + + + + + + + + + (11) +where +2 2 +yF L +EI +  + [44]. +Figure 8 illustrates the disturbance torque of the stiffness +modulation motor +dis +s +and the static output torque of the VSSEA +s +l + + + + when the deflection of the output link increases. Since +dis +s +is zero when +l +g +q +q + +, the VSAM does not consume energy +to keep the stiffness constant at equilibrium positions. When the +deflection is small, the energy consumption of the VSAM is +negligible because +dis +s +is very low regardless of +s . However, +the VSAM consumes higher energy as the deflection of the +output link increases. It is clear from Fig. 8 that the disturbance +torque of the stiffness modulation motor is always limited +within the work-range of the VSSEA. This disturbance can be +further supressed using novel mechanical designs, e.g., in [45]. +IV. MOTION CONTROL OF THE VSSEA +In this section, the position and force control problems of the +VSSEA are discussed. It is a well-known fact that the motion +control problem of compliant actuators is more complicated +than that of conventional stiff actuators, particularly in position +control [10]. To precisely track link trajectories or interact with +unstructured environments, internal and external disturbances +should be compensated using adaptive and robust controllers [9 +– 15]. This section experimentally shows that the proposed +VSSEA allows us to conduct high-performance motion control +tasks using conventional PID controllers. The experimental +setup was built by using ESCON 50/5 motor drivers, 1000ppr +encoders at the motors and a 10000ppr encoder at the link. A +PC with a Linux operating system was employed to perform the +real-time motion control experiments with 1ms sampling time. +A. Position Control: +Let us start with the position control problem of the VSSEA. +Figure 9 illustrates the position control experiments when a PID +controller is synthesised for controlling the angle of the first +motor as follows: + + + + + + +ref +ref +ref +m +p +g +g +i +g +g +d +g +g +K +q +q +K +q +q +dt K +q +q + + + + + + + + + (12) +where +ref +gq +and +ref +gq +represent the position and velocity references +of the gearbox, i.e., +g +q , respectively. +External disturbances up to 5Nm and 15Nm were applied to +the output link after 2 seconds when the VSSEA was in soft (21 +Nm/rad) and stiff mode (985 Nm/rad), respectively. Although +the transient response was good except a relatively high +overshoot at link side, the link of the actuator was very sensitive +to external disturbances, particularly when the actuator was in +soft mode. This is expected because the angle of the link +lq is +not used in the position controller synthesis. +To improve the robustness against external load, let us use +lq +in the feedback controller synthesis. In this experiment, the +control signal was designed as follows: + + + + + + + +ref +ref +ref +m +p +l +l +i +l +l +d +l +l +K +q +q +K +q +q dt K +q +q + + + + + + + + + (13) +where +ref +lq +and +ref +lq +similarly represent the position and velocity +references of the VSSEA’s link, respectively. + + +a) Static output torque versus deflection. b) 90% stiffness modulation. c) 70% +stiffness modulation. +Figure 7: Output torque and stiffness of the VSSEA. LDM: Large Deflection +Model and SDM: Small Deflection Model. + +Figure 8: Disturbance torque of the stiffness modulation motor. + +a) Stiff mode. b) Soft mode. +Figure 9: Position regulation control of motor 1 when external loads are applied +to the link. Kp = 15000, Kd = 500, Ki =75 and +0.5 +ref +gq + + +. + +Figure 7a: Static Output Torque vs Output Link Deflection +100 +LDM: r/L = 1 +LDM:r/L=0.75 +90 +LDM: r/L = 0.5 +80 +LDM:Cr/L=0.25 +SDM: r/L=1 +Torque Tt +70 +SDM: Cr/L = 0.75 +SDM: Cr/ L =0.5 +60 +SDM: r/ L = 0.25 +50 +Output +40 +Static +30 +20 +10 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +(rad)×104 Figure 7b: 0.1 < zr/L<1 +Figure 7c: 0.3 < ,/L<1 +3.5 +2500 +LDM +LDM +SDM +-SDM +3 +(pea/n) +2000 +2.5 +Stiffnessk(qm2，Q) +1500 +1.5 +1000 +Stiffness +1 +500 +0.5 +0 +0 +0 +0.2 +0.40.60.8 +0.2 +0.4 +0.6 +0.8 +-/L (Dimensionless) +/L (Dimensionless)100 +d28 +: r/L = 1 +90 +dis +: r/L = 0.75 +.dis +: rr/L = 0.5 +80 +dis +: rr/L=0.25 +Ts : r/L= 1 +70 +ITs:Cr/L=0.75 +(uN) +Ts : Cr/L = 0.5 +60 +Tg : r/L = 0.25 +Torque +50 +40 +30 +20 +10 +0 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +(rad)Figure9a:Stiff mode +Figure9b:Softmode +2.5 +2.5 +6b. +b +2 +1.5 +le (rad +A +0.5 +0.5 +0 +0 +0 +2 +4 +6 +0 +2 +4 +6 +Time (s) +Time (s)IEEE/ASME TRANSACTIONS ON MECHATRONICS + + +External disturbances up to 10Nm were similarly applied +when the actuator was in stiff mode, however only less than +1Nm external disturbances could be applied due to robust +stability problems encountered in the soft mode of the actuator. +Figure 10 shows that the robustness against external loads was +improved by using +lq in the feedback controller synthesis. +However, the performance of transient response was notably +degraded by large vibrations when the actuator was in soft +mode (see Fig. 10b). Moreover, only small external +disturbances could be suppressed due to the robust stability +problems. It is a well-known fact that more advanced +controllers should be employed for the robust position control +problem of compliant actuators [10]. By changing the stiffness +of the actuator, this paper proposes a simple yet effective +solution for this challenging problem as shown in Fig. 10. +When we used the same PID controller in trajectory tracking +control, we obtained similar results as illustrated in Fig. 11. Due +to the well-known bandwidth limitations of compliant systems, +increasing the speed of reference trajectory notably degraded +the position control performance when the actuator was in soft +mode [3].The VSSEA could not follow 1Hz reference trajectory +as illustrated in Fig. 11b. The performance of trajectory tracking +control could be easily improved by increasing the stiffness of +the actuator as illustrated in Figs. 11c and 11d. +B. Force Control: +By using Hooke’s law, the force control problem of the +compliant actuator was described as a position control problem, +and force control experiments were performed by controlling +the deflection of the output link at non-equilibrium positions. +Similar to the position control experiments, a simple PID +controller was synthesised by feeding back the deflection angle +of the output link, i.e., +q +l +g +q +q +  + +where +lq and +1 +g +m +q +q +N + +are the +link and gear angles, respectively. +Figure 12 illustrates the force control experiments of the +VSSEA. The force control signal was designed using Eq. (14). + + + + + + + +ref +ref +ref +m +p +q +q +i +q +q +d +q +q +K +K +dt K + + + +  + + +  + + +  + + (14) +where +ref +q + +and +ref +q + +represent the position and velocity references +of the link deflection, i.e., +q +l +g +q +q +  + + , respectively. +When the actuator was in soft mode, relatively large link +deflections occurred for small output torques as shown in Figs. +12a and 12b. This allows actuator to physically interact with +different environments in a safe manner. To achieve higher + +a) Stiff mode. b) Soft mode +Figure 10: Position regulation control of the output link when external loads +are applied to the link. Kp = 5000, Kd = 95, Ki = 35, and +0.5 +ref +lq + + +. + + +a) Soft mode and f = 0.1Hz b) Soft mode and f = 1Hz. c) Stiff mode and f = +0.1Hz. d) Stiff mode and f = 1Hz. +Figure 11: Position trajectory tracking control of the output link when Kp = +5000, Kd = 95, Ki = 35, and + + + + + + +1 cos 2 +1 +ref +lq +K +f t + + + + +where K is 0.5 and 2 . + + +a) Regulation control in soft mode b) Trajectory tracking control in soft mode +c) Regulation control in stiff mode d) Trajectory tracking control in stiff mode. +Figure 12: Force control experiments when Kp = 2500, Kd = 85, and Ki = 15. + +Figure 10a: Stiff mode +Figure 10b:Soft mode +2.5 +3.5 +4g +3 +2 +2.5 +2 +60 +1.5 +A +1 +0.5 +0.5 +0 +0 +0 +2 +4 +6 +0 +2 +4 +6 +Time (s) +Time (s)Figure 11a:Soft mode(0.1Hz) +4 +3 +ref +Angle +2 +0 +0 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time (s) +Figure11b:Softmode(1Hz) +2 +ref +1.5 +ref +- +1 +0.5 +0 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Time (s)Figure 11c: Stiff mode (0.1Hz) +4 +'e +3 +ref +Angle +2 +0 +0 +2 +3 +5 +6 +7 +8 +9 +10 +Time (s) +Figure 11d: Stiff mode (1Hz) +2 +(rad) +1.5 +1 +Angle +0.5 +0 +-0.5 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time (s)Figure12a:Force requlationcontrol insoft mode +0.4 +6 +Angle (rad) +0.2 +Are +0 +T +0.2 +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time (s) +Figure 12b:Force trajectory tracking controlin soft mode +0.4 +A +6 +Angle (rad) +T +0.2 +0 +0.2 +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time (s)Figure12c:Force regulationcontrolinstiff-mode +15 +△rej +(rad) +0.04 +T +10 +0.02 +5 +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time (s) +Figure 12d: Force trajectory tracking control in stiff-mode +25 +△rej +AAAAAAAA +(rad) +0.06 +20 +(uN) +0.04 +15 +Angle +Torque +10 +0.02 +5 +0 +0 +1 +2 +3 +4 +5 +7 +8 +9 +10 +Time (s)IEEE/ASME TRANSACTIONS ON MECHATRONICS + + +output torque, the stiffness of the actuator should be increased +as illustrated in Figs. 12c and 12d. This, however, may degrade +safety in contact motion. It is clear from Fig. 12 that the +proposed VSSEA enables us to conduct high-performance force +control applications using a conventional PID controller. +C. Stiffness Modulation: +Figure 13 illustrates the position control experiments of motor +2 at different speeds when the actuator is at equilibrium +positions. The motor did not draw current, i.e., the VSAM did +not consume energy, to keep the stiffness of the actuator fixed +at an equilibrium position as shown in Fig. 13. The current of +motor 2 was negligible at low speeds of stiffness modulation, +e.g., 0.1Hz in Fig. 13a. As the speed of stiffness modulation +increased, the motor drew higher current due to the increased +frictional and inertial disturbances of the VSAM (see Fig. 13b). +This is expected because the energy consumptions of all +variable stiffness actuators become higher as the speed of +stiffness modulation is increased [33, 45]. Since the frictional +and inertial disturbances were mitigated in the design of the +VSAM, the stiffness modulation could be performed using low +current drains at equilibrium positions as shown in Fig. 13. +Figure 14 illustrates the stiffness modulation experiments +when the actuator is at non-equilibrium positions. While the +deflection angle of the output link was regulated, the stiffness +of the actuator was modulated 10% in this experiment. Figures +14a and 14b show that the current of motor 2, Im2, was low when +10% stiffness modulation was performed in the stiff mode of +the VSSEA. This is expected because, as shown in Eq. (11), the +small deflections of leaf springs in stiff mode could lead to low +disturbance torques exerted by the VSAM on motor 2. +Compared to other energy efficient and infinite-range variable +stiffness actuators, e.g., [34, 35], this feature of the proposed +VSAM allows us to perform infinite-range stiffness modulation +using bounded control signals [33]. Figures 14c and 14d show +that the current drain of motor 2 was higher when 10% stiffness +modulation was performed in the soft mode of the VSSEA. +Since the deflection angle of the output link was larger in soft +mode, the current drain of motor 2 increased by the higher +disturbance torque of the VSAM as shown in Eq. (11). Figures +13 and 14 show that the control signal of motor 2 was always +bounded when stiffness modulations were conducted at +equilibrium and non-equilibrium positions. +Figure 15 illustrates the energy cost of changing stiffness +compared to the potential energy stored in the springs of the +VSAM when 10% stiffness modulation was performed at stiff +and soft modes of the actuator. It shows that the energy cost of +stiffness modulation is weakly coupled to the deflection of the +output link. While lower energy was consumed by the VSAM +at fixed stiffness configurations in both stiff and soft modes, the +energy cost of stiffness modulation increased when the stiffness +of the actuator was changed. The larger deflection of the output +link in soft mode led to higher energy cost of stiffness +modulation as shown in Fig. 15b. This result is expected +because the disturbance torque gets larger as the deflection of +the output link increases as shown in Eq. (11). +V. DISCUSSION +The experimental results given in Section IV show that the + +a) 0.1Hz stiffness modulation. b) 1Hz stiffness modulation. +Figure 13: Stiffness modulation at an equilibrium position. + +a) + +b) +a) Stiffness modulation in stiff mode. b) Output torque and motor current in +stiff mode. c) Stiffness modulation in soft mode. d) Output torque and motor +current in soft mode. +Figure 14: Stiffness modulation at non-equilibrium positions. + +a) Stiffness modulation in stiff mode. b) Stiffness modulation in soft mode. +Figure 15: Energy consumption of the VSAM. + +Figure13a:100%stiffnessmodulation(0.1Hz) +/L(dimensionless) +.5 +qm2 +Gm2 +0.5 +A +Im2 +urrent +0 +0.5 +0.5 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time (s) +Figure13b:100%stiffnessmodulation (1Hz) +/L(dimensioniess) +.5 +qm2 +qm2 +Im2 +Current +0.5 +0 +2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time (s)Figure14a:10%stiffnessmodulationinstiffmode +0.15 +0.1 +Gm2 +0.05 +0 +0.05 +0 +5 +10 +15 +20 +Time (s) +Figure14b:Outputtorqueandmotorcurrentinstiffmode +10 +(uIN) +lm2 +Current (A) +5 +0 +2 +0 +5 +10 +15 +20 +Time (s)Figure14c:10%stiffnessmodulationinsoftmode +0.15 +0.1 +Qm2 +0.05 +0 +0.05 +0 +5 +10 +15 +20 +Time (s) +Figure14d:Outputtorgqueandmotorcurrentinsoftmode +8 +(Nn) +6 +Im2 +(A) +Current( +4 +0 +2 +2 +0 +5 +10 +15 +20 +Time (s)Energy +Figure15a:10%stiffnessmodulationinstiffmode +Energy +0.01 +FixedStiffness +StiffnessModulation +Em2/Ema" +Motor +Ratio of Spring +0.5 +0.005 +2 +Ratio of the +0 +0 +0 +1 +2 +3 +4 +5 +6 +Figure15b:10%stiffnessemodulationinsoftmode +'ime (s +Energy +Us/Umar +0.1 +FixedStiffness +StiffnessModulation +m2 +Ratio of Spring +0.5 +0.05 +- +- +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Time (s)IEEE/ASME TRANSACTIONS ON MECHATRONICS + + +proposed VSSEA enables us to conduct high-performance +position and force control applications using conventional PID +controllers. This provides several benefits in many different +advanced robotic applications. For example, while the stiff +mode of the VSSEA allows a cobot to perform high-precision +position control tasks in industry, the soft mode of the actuator +can boost safety in physical robot-environment interaction. +However, more advanced controllers should be synthesised for +smooth transition between position and force control tasks, i.e., +the stiff and soft mode of the actuator. In addition, more effort +should be expended on optimal stiffness modulation. +The experimental results show that a wide range of stiffness +modulation can be achieved by simply controlling the position +of the second motor. For example, while the stiffness of the +output link is 21Nm/rad when the actuator is in soft mode in +Fig. 9a, it is ~50 times higher with 985Nm/rad in Fig. 9b. This +is an important feature of the proposed VSSEA as it allows us +to perform both precise position control and safe robot- +environment interaction tasks using conventional PID +controllers. Other important features of the VSSEA are fast and +low-energy-cost +stiffness +modulation +capabilities. +The +transition from the softest mode to the stiffest mode can be +achieved within a second as illustrated in Fig. 13. Since the +disturbance torque exerted by the VSAM on motor 2 is always +bounded unless it is zero, a wide range of stiffness modulation +can be performed using bounded control signals at equilibrium +and non-equilibrium positions as shown in Figs. 13 and 14. +Therefore, the energy cost of stiffness modulation is low as +shown in Fig. 15. This can provide significant benefits to +mobile robotic systems such as humanoids, quadrupeds, and +exoskeletons. +With the proposed modular design approach, the VSSEA can +be easily modified to meet the requirements of different robotic +applications. For instance, i) torque density can be increased +using a higher gear ratio, ii) stiffness characteristic can be tuned +using the different number, material, and shape of leaf springs, +and iii) faster stiffness modulation can be achieved using a ball +screw with higher lead. The proposed novel mechanical design +eliminates the work-range limitation of variable stiffness +actuators as shown in Figs. 11a and 11c. This allows us to apply +the VSSEA to various robotic applications. +By neglecting the small torsional motions of leaf springs, the +dynamic model of the VSSEA is obtained using the analogy of +a mass-spring-damper system and the Euler-Bernoulli beam +theory in Section III [44]. While the exact dynamic model and +nonlinear Euler-Bernoulli beam theory are computationally +expensive and ineffective in controller synthesis [43, 44], the +model derived through simple beam theory is inaccurate when +the deflection of the output link is large. Thus, more effort +should be expended to obtain an effective dynamic model for +the VSSEA. +VI. CONCLUSION +This paper proposes a new VSSEA that can perform fast and +low-energy-cost stiffness modulation over a large range. These +features can provide several benefits to robotic systems such as +easier motion control problems, longer battery life, and safer +physical robot environment interaction. It is experimentally +shown in this paper that the proposed VSSEA allows us to +conduct high-performance position and force control tasks +using conventional PID controllers. It is also demonstrated that +the stiffness of the actuator can be increased or decreased up to +50 times within a second while the energy cost of stiffness +modulation is very low. However, further research should be +conducted to clarify how the VSSEA can contribute to robotic +applications. To this end, we will apply the proposed actuator +to advanced robotic applications such as legged locomotion in +the future studies. +REFERENCES +[1] +B. Sciliano and O. Khatib, Handbook of Robotics Chapter 57 Safety for +Physical Human–Robot Interaction, New York, NY, USA:Springer, pp. +1335-1348, 2008. +[2] +N. Paine, J. S. Mehling, J. Holley, N. Radford, G. Johnson, C.-L. Fok and +L. Sentis, “Actuator control for the NASA-JSC Valkyrie humanoid robot: +A decoupled dynamics approach for torque control of series +elastic robots,” J. Field Rob., vol. 32, no. 3, pp. 378–396, May 2015. +[3] +E. Sariyildiz, G. Chen and H. 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Biomed Robot and +Biomechatron (BioRob), pp. 1943-1948, 2012. + + +Emre Sariyildiz (S’11, M’16, SM’21) +received his first Ph.D. degree in +Integrated Design Engineering from +Keio University, Yokohama, Japan, in +2014, and second PhD degree in Control +and Automation Engineering from +Istanbul Technical University, Istanbul, +Turkey, in 2016. +He is currently a Senior Lecturer at the +School +of +Mechanical, +Materials, +Mechatronic, and Biomedical Engineering, University of +Wollongong, Wollongong, NSW, Australia. His research +interests include control theory, robotics, mechatronics, and +motion control. +Rahim Mutlu (M’22) received his Ph.D. +degree in Robotics from University of +Wollongong, Wollongong (UOW), NSW, +Australia, in 2013. He was a Lecturer with +UOW 2017-21, prior to committing his +current role as Assistant Professor with +Faculty of Engineering and Information +Sciences at UOWD, UAE. He is also founder of the Intelligent +Robotics & Autonomous Systems Co (iR@SC), NSW, 2529, +Australia. His research interests include soft robotics, soft +haptics, wearable technologies, assistive and rehabilitation +exoskeletons, additive manufacturing. +Jon Roberts received his Ph.D. degree in +Mechanical Engineering from University +of Wollongong (UOW), Wollongong, +NSW, Australia in 2019. + Since January 2020, he has been a +Lecturer with the School of Mechanical, +Materials, Mechatronic, and Biomedical +Engineering, University of Wollongong, +Wollongong, NSW, Australia. His research interests are +simulation methods, bulk materials handling, safety in mining, +and dust control technology. +Chin-Hsing Kuo received his Ph.D. +degree in Mechanical Engineering from +King's College London, UK, in 2011. +Since February 2019, he has been a +Senior Lecturer with the School of +Mechanical, Materials, Mechatronic, and +Biomedical Engineering, University of +Wollongong, +Wollongong, +NSW, +Australia. Before that he was an Associate Professor with the +National Taiwan University of Science and Technology. His +research interests include parallel robots, mechanism design, +and robot kinematics and dynamics. +Barkan Ugurlu (S'08-M'10) received his +Ph.D. degree in Electrical and Computer +Engineering from Yokohama National +University, Yokohama, Japan, in 2010. +He was a Marie Sklodowska-Curie +Fellow and currently holds an Assistant +Professor position at the Department of +Mechanical +Engineering, +Ozyegin +University, Istanbul, Turkey. His research +interests +include +legged +locomotion +control, hardware development for novel robotic systems, and +multi-body dynamics. + + + diff --git a/sNAzT4oBgHgl3EQfBPpp/content/tmp_files/load_file.txt b/sNAzT4oBgHgl3EQfBPpp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d592082b9c7b49f6669eea6fe2382aa7ae4622c --- /dev/null +++ b/sNAzT4oBgHgl3EQfBPpp/content/tmp_files/load_file.txt @@ -0,0 +1,1009 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf,len=1008 +page_content='IEEE/ASME TRANSACTIONS ON MECHATRONICS \uf020 Abstract— This paper expounds the design and control of a new Variable Stiffness Series Elastic Actuator (VSSEA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is established by employing a modular mechanical design approach that allows us to effectively optimise the stiffness modulation characteristics and power density of the actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The proposed VSSEA possesses the following features: i) no limitation in the work-range of output link, ii) a wide range of stiffness modulation (~20Nm/rad to ~1KNm/rad), iii) low-energy-cost stiffness modulation at equilibrium and non-equilibrium positions, iv) compact design and high torque density (~36Nm/kg), and v) high- speed stiffness modulation (~3000Nm/rad/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Such features can help boost the safety and performance of many advanced robotic systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', a cobot that physically interacts with unstructured environments and an exoskeleton that provides physical assistance to human users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' These features can also enable us to utilise variable stiffness property to attain various regulation and trajectory tracking control tasks only by employing conventional controllers, eliminating the need for synthesising complex motion control systems in compliant actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To this end, it is experimentally demonstrated that the proposed VSSEA is capable of precisely tracking desired position and force control references through the use of conventional Proportional-Integral-Derivative (PID) controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Index Terms— Compliant robotics, safe robotics, series elastic actuators, variable stiffness actuators, physical robot-environment interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' INTRODUCTION O BOOST safety in physical-robot environment interaction, compliant actuators have been widely adopted by many different advanced robotic systems such as humanoids, cobots, quadrupeds, and exoskeletons [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' A compliant actuation system could be developed by simply integrating an elastic element into the design of an actuator [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For example, Series Elastic Actuators (SEAs), one of the most popular compliant actuation systems in robotics, are developed by placing a spring between a conventional rigid actuator and link [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In addition to improving safety, the spring between the rigid actuator and link can provide several benefits such as low- cost and high-fidelity force control, mechanical energy storage, lower reflected inertia, higher tolerance to impact loads, and increased output power [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Manuscript received….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (Corresponding author: Emre Sariyildiz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Sariyildiz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Roberts and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Kuo are with the School of Mechanical, Materials, Mechatronic and Biomedical Engineering, University of Wollongong, Wollongong, NSW, 2522, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (e-mails: emre@uow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='au, robertsj@uow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='au, chkuo@uow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Mutlu is with the Faculty of Engineering and Information Sciences, University of Wollongong in Dubai, Dubai, United Arab Emirates and the Despite the aforementioned benefits, the elastic elements integrated to compliant actuators introduce certain challenges and fundamental limitations in motion control [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For example, it is a well-known fact that the position control problem of a compliant actuator is more complicated than that of a conventional rigid actuator [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To suppress the vibrations and disturbances of a compliant actuator’s link, researchers generally need to employ advanced motion controllers [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Another example is that although using a softer elastic element improves safety and transparency, the natural frequency of the compliant actuator decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This not only excites the vibrations at link side but also lowers the bandwidth of the actuator, thus limiting achievable position and force control performance in motion control applications [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is therefore essential to properly choose the stiffness of the elastic element of a compliant actuator based on the target control task [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' However, this is mostly impractical for compliant actuators with fixed elastic elements, which often leads to compromise between safety and performance [10, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' A simple yet efficient solution for this fundamental problem could be achieved by integrating a compliant mechanical component with variable stiffness property to actuators in series or parallel [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Adaptable compliance mechanisms that can alter the stiffness of actuators mechanically have been developed to meet the different compliance requirements of motion control tasks such as soft actuation in human-robot interaction and stiff actuation in trajectory tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Although no standard terminology exists, such actuators are generally called Variable Stiffness Actuators (VSAs) in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' A comprehensive survey on the design of VSAs can be found in [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Among them, antagonistic actuation is one of the most well-known and widely used stiffness modulation methods in VSAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Inspired by mammalian anatomy, this actuation method has been studied since early 1980s [20], and different antagonistic actuators have been developed and used in various robotic applications, such as legged locomotion and upper-limb rehabilitation, since late 1990s [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The simplest antagonistic actuator can be designed by using an agonistic-antagonistic setup which connects two motors to a link via two nonlinear springs, similar to biceps and triceps in the human arm [19, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' While the equilibrium position can be adjusted by rotating two motors in the same direction, counter-rotation of motors alters the Intelligent Robotics & Autonomous Systems Co (iR@SC), NSW, 2529, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (e-mail: ramutlu@irasc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Ugurlu is with the Department of Mechanical Engineering, Ozyegin University, Istanbul, 34794, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (e-mail: barkan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='ugurlu@ozyegin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Design and Control of a Novel Variable Stiffness Series Elastic Actuator Emre Sariyildiz, Senior Member, IEEE, Rahim Mutlu, Member IEEE, Jon Roberts, Chin-Hsing Kuo, Barkan Ugurlu, Member IEEE T IEEE/ASME TRANSACTIONS ON MECHATRONICS stiffness of the actuator by changing the spring preload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Despite its simplicity, this bioinspired actuation method has several drawbacks in practice, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', i) the control problems of stiffness modulation and equilibrium position are coupled, ii) the output torque of the actuator is limited by the maximum torque of each motor, iii) the potential energy capacity of nonlinear springs cannot be used entirely, and iv) the energy consumption is high because preloading nonlinear springs for stiffness modulation requires constant power drain, even when the actuator does not perform net mechanical work at equilibrium positions [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Numerous different antagonistic and non-antagonistic VSAs have been proposed to tackle the drawbacks of the agonistic- antagonistic actuation setup in the last two decades [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For example, while the output torque of antagonistic actuators is increased using bi-directional configuration in [26], a partially decoupled motion control problem is obtained using a quasi- antagonistic actuation mechanism in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' However, the aforementioned problems could not be entirely addressed using antagonistic actuation systems [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This has motivated many researchers to build non-antagonistic VSAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The control problems of the equilibrium position and compliance of the actuator’s output link are decoupled using a new mechanical design approach in Maccepa [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The stiffness of the actuator is modulated by controlling the tension of a linear spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' A similar stiffness modulation approach is employed in DLR- VSJ, and a light-weight and compact VSA is designed by changing only the cam disk of the joint in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Nevertheless, similar to antagonistic actuators, high-energy-cost stiffness modulation remains a challenging problem in these non- antagonistic VSA design approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Since the stiffness of the Maccepa and DLR-VSJ is modulated by pretensioning springs, they require constant power drain at equilibrium positions, thus leading to high energy consumption [28 – 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The energy-cost of stiffness modulation has been improved using different VSA design approaches in the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The stiffness of the output link is modulated by changing the positions of the springs and pivot points on a lever arm in AWAS [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Low energy cost stiffness modulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', theoretically zero power drain at equilibrium positions) could be achieved using this VSA design approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The stiffness range, however, is limited by the size of the actuator [31, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In vsaUT, the stiffness of the actuator is modulated by controlling the effective length of a lever arm [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This stiffness modulation approach allows us to attain not only zero power drain at equilibrium positions but also infinite- range stiffness modulation with compact actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' However, the power drain by the motor dedicated to stiffness modulation becomes unbounded as the stiffness of the actuator approaches infinity [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Variable length leaf spring mechanisms have also been employed to develop energy efficient VSAs [30, 36 – 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Compared to the other antagonistic and non-antagonistic VSAs, recent studies show that variable length leaf spring mechanisms can provide several benefits in practice, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', zero power drain at equilibrium positions, fast and infinite-range stiffness modulation, and bounded power drain for all stiffness ranges at equilibrium and non-equilibrium positions [30, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' These features could be very useful in biomedical engineering applications as shown in [30, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Nevertheless, when it comes to building a compact VSA that can be integrated to different robotic systems such as cobots, the variable length leaf spring mechanisms may involve several drawbacks such as low torque/power density and work-space limitations [30, 36 – 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is noted that a non-antagonistic VSA can be simply built by employing a discrete stiffness modulation method where multiple springs could be integrated to actuators in series or parallel [40 – 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The main drawback of this actuation method is the limited stiffness range which depends on the number of springs employed in the actuator design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Moreover, the discrete stiffness modulation method leads to several challenges in controller analysis and synthesis such as the stability problem of switching systems [40, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Therefore, continuous stiffness modulation methods are mainly considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The existing VSAs have their own merits and demerits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' While they are highly functional in their own domain of use, they may however fall-short in complying with all the desirable technical specifications of an ideal compliant actuator for practical applications: i) a compact and simple mechanical design that allows to easily reconfigure a VSA for different applications, ii) no limitation in motion range, iii) a wide range of stiffness modulation, iv) rapid stiffness change, and v) energy efficient actuation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In general, researchers manage a trade-off to target only few of the aforementioned qualities, leading to different compromises such as high energy consumption or limited motion control performance in robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Hence, despite many recent advances, more effort should be put into the development of VSAs [18, 19 and 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This is summarised using the examples of existing VSAs in the literature in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To this end, this paper proposes a new VSSEA which consists of three main components: i) a rigid actuator that independently controls the equilibrium position, ii) a novel Variable Stiffness Actuation Mechanism (VSAM), and iii) a direct drive motor that independently adjusts the stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The proposed VSSEA TABLE I: Actuators, Deflection Range, Motion Range, Stiffness Range, Energy Cost of Fixed Stiffness at Equilibrium, Energy Cost of Infinite-Range Stiffness Modulation, Torque Density and Stiffness Modulation Speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Actuators Deflection Range [degree] Motion Range [degree] Stiffness range [Nm/rad] Energy cost of fixed stiffness at equilibrium Energy cost of infinite-range stiffness modulation Torque density [Nm/kg] Stiffness modulation speed [Nm/rad/s] VSSEA ±25 no limitation 20 – 1000 b) ZEC BEC 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='72 3000 Maccepa ±60 ±90 c) 5 – 110 NZEC NA 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='83 40 DLR-VSJ ±15 ±180 50 – 820 NZEC NA 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='27 2350 AWAS ±12 ±120 30 – 1500 ZEC NA 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1 420 AWAS-II ±18 ±150 10 – 10000 ZEC NA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='76 4000 vsaUT-II ±45 ±180a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5N – 100 ZEC NBEC 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='7 1000 VSA-I [36] ±12 NA 250 – 3000 b) ZEC BEC 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1 NA VSA-II [30] ±75 ±75 10 – 8000 b) ZEC BEC Passive (3kg) 10000 Table I is prepared using experimental results given in the papers unless data is provided in the references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) Although the motion range of vsaUT is limited (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', vsaUT-II’s motion range is ±180o), mvsaUT can perform continuous motions without any motion range limitations [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) Infinite-range stiffness modulation can be achieved using leaf spring based VSAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' c) [28] states that the motion range of Maccepa can be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' ZEC: Zero Energy Consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' NZEC: Non-Zero Energy Consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' BEC: Bounded Energy Consumption when performing infinite-range stiffness modulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' NBEC: Non-Bounded Energy Consumption when performing infinite-range stiffness modulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' and NA: Not Applicable because the actuators cannot provide infinite range stiffness modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' IEEE/ASME TRANSACTIONS ON MECHATRONICS is simply developed by integrating the VSAM to a rigid actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This modular design approach allows us to easily optimise the stiffness modulation characteristics and output power/torque of the VSSEA for different robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The stiffness of the actuator is modulated by changing the effective length of a group of leaf springs of the VSAM through a direct drive motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This stiffness modulation technique provides several benefits: i) stiffness modulation over a large range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', from near-zero to infinite stiffness theoretically, ii) high-speed stiffness modulation using a relatively slow motor, and iii) zero/near-zero energy consumption for holding/altering the stiffness at equilibrium positions, and low-energy-cost stiffness modulation at non-equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Moreover, the proposed VSSEA has no limitation in motion range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To the best of our knowledge, the existing VSAs have yet to combine all these desired features of our proposed VSSEA [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In Section II, the mechanical design of the VSSEA is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In Section III, the dynamic model of the actuator is derived by using the analogy of a mass-spring-damper system and Euler-Bernoulli beam theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In Section IV, the performance of the VSSEA is experimentally verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The paper ends with discussion and conclusion given in Sections V and VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' MECHANICAL DESIGN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Variable Stiffness Series Elastic Actuator: Figure 1 illustrates the CAD model and the first prototype of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It comprises i) a conventional rigid actuator that includes a servo motor and a gearbox, illustrated by M1, ii) a novel variable stiffness actuation mechanism illustrated by VSAM, and ii) a direct drive servo motor illustrated by M2 in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The conventional rigid actuator M1 is used to independently control the equilibrium position of the output link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The proposed modular design approach enables us to freely tune the output torque and speed of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For example, we used Maxon EC90 flat motor and a 1:100 ratio harmonic drive to achieve ~100Nm output torque and ~π/2rad/s output speed in the first prototype of the actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The output power of the VSSEA can be directly adjusted by employing a different servo motor and/or a gearbox in the design of the conventional rigid actuator M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The second motor M2 is used to control the stiffness of the actuator via the VSAM independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The low-energy-cost stiffness modulation feature, which is explained in Section III, of the VSAM allowed us to use the Maxon EC60 direct drive flat motor in the stiffness control of the actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1, the proposed VSSEA is simply designed by integrating the VSAM to the conventional rigid actuator M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This modular design approach provides great flexibility in building a VSA for different robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Let us now present the mechanical design of the novel VSAM that provides important features, such as a wide range of stiffness modulation and energy efficiency, in compliant actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Variable Stiffness Actuation Mechanism: Figure 2 illustrates the CAD model and the first prototype of the VSAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The design comprises i) eight radially distributed in-parallel leaf springs, ii) two rollers for each leaf spring to reduce friction and improve energy efficiency, and iii) a ball screw mechanism to move the rollers along the leaf springs as illustrated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The stiffness of the actuator is modulated by changing the position of the rollers which is controlled by a ball-screw mechanism that is driven by the second motor M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The VSSEA is in the softest mode when the rollers are at the free ends of the leaf springs, and the stiffness of the actuator increases as the rollers move towards the fixed end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' With the nonlinear dynamic behaviour of the VSAM, a wide range of stiffness modulation can be obtained by simply changing the effective length of the leaf springs through the position control of the rollers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This allows us to perform large stiffness modulations within a short time, and this feature can provide several benefits in robotic applications [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Another important feature of the proposed VSSEA is low-energy-cost stiffness modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For example, the VSAM does not require constant power drain to hold the stiffness constant at equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' DYNAMIC MODEL The dynamic model of the VSSEA is obtained by employing the analogy of a mass-spring-damper system and the Euler- Bernoulli beam theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Variable Stiffness Series Elastic Actuator: The dynamic model of the VSSEA is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In this figure, J\uf0b7 represents the inertia of motor 1, gearbox, motor 2, and output link when \uf0b7 is m1, g, m2 and l, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' \uf074 \uf0b7 , b\uf0b7 , and q\uf0b7 similarly represent the torque, viscous friction coefficient, and angle of the motor 1, gearbox, motor 2, and a) CAD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) First prototype of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 1: CAD model and first prototype of the novel VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' M1: Motor 1, M2: Motor 2, and VSAM: Variable Stiffness Actuation Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) CAD model of leaf springs and mobile frame of the VSAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) First prototype of the VSAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' c) Dimensions of the VSAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 2: CAD model and first prototype of the VSAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) M1 b) VSAM M1 VSAM M2 M2a) xedEn 55mmIEEE/ASME TRANSACTIONS ON MECHATRONICS output link, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' q\uf0b7 and q\uf0b7 represent the first and second order derivatives of q\uf0b7 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', angular velocity and angular acceleration, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' and k represents the stiffness of the actuator, which can be defined as a nonlinear function of 2 m q and lq as shown in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' While the first motor denoted by m1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 3 is used to control the equilibrium position of the actuator, the second motor denoted by m2 is used to independently modulate the stiffness of the output link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The dynamic model of the VSSEA can be derived from this figure as follows: 1 1 1 1 1 1 1 2 2 2 2 2 2 dis m m m m m s m dis l l l l s l l dis dis m m m m s m J q b q N J q b q J q b q \uf074 \uf074 \uf074 \uf074 \uf074 \uf074 \uf074 \uf074 \uf074 \uf02d \uf02b \uf03d \uf02d \uf02d \uf02b \uf03d \uf02d \uf02d \uf02b \uf03d \uf02d \uf02d (1) where dis \uf074\uf0b7 represents the unknown/unmodelled disturbances of motor 1, motor 2, and output link when \uf0b7 is m1, m2 and l, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' s\uf074 represents the torque exerted by the nonlinear springs of the VSAM on motor 1 and output link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' dis s\uf074 represents the disturbance torque exerted by the nonlinear springs of the VSAM on motor 2 in stiffness modulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' and 2 1 1 m m g J J N J \uf02d \uf03d \uf02b and 2 1 1 m m g b b N b \uf02d \uf03d \uf02b where N is gear ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Equation (1) gives a simple yet useful dynamic model for the proposed VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' However, more effort should be expended on understanding the nonlinear dynamics of the VSAM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', deriving s\uf074 and dis s\uf074 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This will enable us to explain the important features such as the energy efficiency of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Variable Stiffness Actuation Mechanism: To derive the model of the VSAM, let us focus on the first leaf spring illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In this figure, 1xF , 1 yF , and 1zF represent the forces exerted by the leaf spring on the roller along the 1x , 1y , and 1 z axes of the local coordinate frame on the leaf spring, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is noted that the proposed analysis can be similarly applied to the other leaf springs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', the second leaf spring illustrated in Fig 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' When it is assumed that only the first leaf spring is used in the design of the VSAM, the kinematic and static equilibrium equations of the output link can be directly obtained from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 4c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 4d as follows: \uf028 \uf029 2 sin 2 l r q \uf064 \uf03d (2) 1 1 1 2 2 0 s l r l y z l F r F F r \uf074 \uf074 \uf074 \uf074 \uf02d \uf03d \uf02d \uf03d \uf02b \uf02d \uf03d (3) where\uf064 is a kinematic constraint that relates the angle of the output link to the deflection of the leaf spring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' r represents the magnitude of the distance vector 1r illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 4c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' and F\uf0b7 represents the magnitude of the force vector F in which \uf0b7 can be r1, y1 and z1 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' When the other leaf springs illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 2a are considered, the static equilibrium equation of the output link can be obtained by simply expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (3) as follows: 8 8 2 2 1 1 0 i i i s l r l y z l i i F r F F r \uf074 \uf074 \uf074 \uf074 \uf03d \uf03d \uf02d \uf03d \uf02d \uf03d \uf02b \uf02d \uf03d \uf0e5 \uf0e5 (4) where irF , iy F , and izF similarly represent the forces exerted by the ith leaf spring, and r represents the magnitude of the distance vector ir .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Equation (4) shows that the leaf spring forces should be identified to obtain the dynamic model of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Leaf Springs: The leaf springs of the proposed VSAM not only bend along the lateral axes yi but also slightly rotate about the longitudinal axes xi at non-equilibrium positions [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For the sake of simplicity, the 3D deflection model of the leaf spring illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 5 is numerically obtained using Finite Element Method Figure 3: Dynamic model of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) 3D CAD model illustrating forces exerted by the first leaf spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) 3D CAD model illustrating forces exerted by the second leaf spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' c) 3D CAD model illustrating the relation between the output torque and forces exerted on the first and second leaf springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' d) 2D kinematic model illustrating the relation between the deflection of the first leaf spring and output link angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 4: Model of the Variable Stiffness Actuation Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) Beam at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) Deflected beam c) Static output torque versus deflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' l\uf074 is the output torque and 1 ˆl yF r \uf074 \uf03d is the output torque associated with 1 yF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 5: 3D large deflection model of the first leaf spring using FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Motor 1 Gearbox Output Link Motor 2a) Z1 d F ri F gFigure5a Figure Sb F F IFFigure 5c:FEM Analysis of the Leaf Spring 20 :r/L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='9 18 : r/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='75 :/L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 16 : r/L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='25 :/=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='9 14 分 : Cr/ L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='75 分 : r/ L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 12 = : r/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='25 8 6 4 2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='4 OutputLinkDeflectionqt-qa(rad)IEEE/ASME TRANSACTIONS ON MECHATRONICS (FEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' While Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 5a shows the FEM model at an equilibrium position, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 5b shows the maximum deflection of the leaf spring when the output link rotates for about 20o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' As shown in this figure, the largest deflection occurs along the lateral axis y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 5c illustrates the static output torque of the actuator for different stiffness configurations when the deflection of the output link increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In this figure, 1 ˆl yF r \uf074 \uf03d represents the output torque associated with the force along the lateral axis at different non-equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 5c that the output torque is mainly dominated by the bending force along the lateral axis within the deflection range of the actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This allows us to simplify the dynamic model of the VSAM while precisely estimating the output torque within a large deflection range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To this end, let us consider the 2D deflection model of the leaf spring illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In this figure, x1 and y1 represent the position of a point on the beam along the x1 and y1 axes, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 1xS and 1x\uf066 represent the arc length and slope of the beam at x1, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 1x \uf064 and 1y \uf064 represent the deflection of the beam along the x1 and y1 axes at x1, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' rx\uf064 and ry \uf064 similarly represent the deflections at xr where the rollers are positioned on the x1 axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' L represents the length of the beam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' and rx\uf066 represents the slope of the beam at xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The model of the leaf spring at a non-equilibrium position can be derived by using the nonlinear Euler-Bernoulli beam theory and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 6 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 1 1 1 0 2 sin sin x r x y x EI d S F \uf066 \uf066 \uf066 \uf066 \uf03d \uf02d \uf0f2 (5) \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 1 1 0 cos 2 sin sin x r y x d EI x F \uf066 \uf066 \uf066 \uf066 \uf066 \uf03d \uf02d \uf0f2 (6) \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 1 1 0 sin 2 sin sin x r y x d EI y F \uf066 \uf066 \uf066 \uf066 \uf066 \uf03d \uf02d \uf0f2 (7) where E represents Young’s modulus, and I represents the moment of inertia of the beam cross section about the neutral axis [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The other parameters are same as defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Equations (5 – 7) cannot be solved analytically so a numerical solution method should be employed to calculate the deflection of the beam for the applied force 1y F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This allows us to calculate the reaction force exerted by the leaf spring on the roller, which is required for output torque calculation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For example, the following steps can be employed to calculate the deflections of the leaf spring at rx : i) Using a numerical method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', Nonlinear shooting method or Adomian decomposition method) or the fzero command of Matlab, find rx \uf066 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (5), and ii) Calculate rx \uf064 and ry \uf064 by numerically integrating Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (6) and (7) over [0, rx \uf066 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The kinematic and static equilibrium equations of the output link can be calculated by applying \uf028 \uf029 cos ry lq \uf064 \uf064 \uf03d and i i y r F F \uf040 to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (2) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Although Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (5-7) allows us to precisely estimate the output torque of the VSSEA for different deflection angles at non- equilibrium positions, they fall-short when attempting to explain how the output torque and stiffness of the actuator change as a function of lq and 2 m q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Moreover, they are not very useful in controller analysis and synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To tackle this problem, let us obtain an approximate model that provides a clear insight into the stiffness modulation characteristics of the proposed VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' When it is assumed that the deflections of the leaf springs are small, the torque and stiffness of the output link can be calculated using simple beam theory and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (2 – 4) as follows: \uf028 \uf029 2 2 3 3 2 48 , sin 2 l s m l m q EI q q r q \uf074 \uf068 \uf0e6 \uf0f6 \uf03d \uf0e7 \uf0f7 \uf0e8 \uf0f8 (8) \uf028 \uf029 2 2 3 3 2 24 , cos 2 l m l m q EI k q q r q \uf068 \uf0e6 \uf0f6 \uf03d \uf0e7 \uf0f7 \uf0e8 \uf0f8 (9) where 2 \uf068 \uf070 \uf03d in which represents the lead of the ball screw, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', 2 r m x q \uf068 \uf03d [30, 33, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Equations (8) and (9) can be used to explain how the output torque and stiffness of the actuator change as a function of ql and qm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Equation (8) shows that the torque exerted by the nonlinear springs of the VSAM on the output link and motor 1 becomes higher as the angle of the output link increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This occurs because larger deflections of the leaf springs lead to higher reaction forces exerted on the rollers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' When the non- equilibrium position of the actuator remains constant, s\uf074 can still be regulated using the second motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Equation (8) shows that as 2 m q decreases (increases), the torque exerted by the VSAM gets higher (lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This occurs because the stiffness of the VSAM changes by the angle of the second motor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', position of the roller) as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The nonlinear behaviour of the VSAM allows us to obtain a wide range of stiffness modulation by simply controlling the angle of the second motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 7 illustrates the torque and stiffness of the output link at different non-equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The accuracy of the small deflection model deteriorates as the angle of the output link increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This is expected because this approximate model is obtained by assuming small deflections for the leaf springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Despite some inaccuracies, the proposed approximate model is very useful to explain the important features of the VSSEA because the small and large deflection models exhibit similar behaviours as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' However, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (5 – 7) should be used to accurately estimate the output torque of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' At an equilibrium position where l g q q \uf03d , i) \uf03d \uf03d 1 1 y x F F 0 when the stiffness of the actuator remains constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', 2 0 m q \uf03d , and ii) \uf03d 1yF 0, but \uf040 1 xF 0 due to small friction and inertial forces when 2 0 m q \uf0b9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Therefore, the energy consumption of stiffness modulation is negligible at equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The energy consumption of stiffness modulation increases as the deflections of the leaf springs get larger at non-equilibrium Figure 6: 2D large deflection model of the first leaf spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' L "x Xp Xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 7 Iyi X Sx, -da Yi F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Fpy F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' yiIEEE/ASME TRANSACTIONS ON MECHATRONICS positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The disturbance torque exerted by the VSAM on the stiffness modulation motor can be calculated as follows: Large Deflection Model: 8 1 i dis s x i F r \uf074 \uf03d \uf03d\uf0e5 (10) Small Deflection Model: \uf028 \uf029 2 3 3 2 48 sin tan 2 dis l s m q EI r q \uf074 \uf06a \uf068 \uf0e6 \uf0f6 \uf03d \uf0e7 \uf0f7 \uf0e8 \uf0f8 (11) where 2 2 yF L EI \uf06a \uf03d [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 8 illustrates the disturbance torque of the stiffness modulation motor dis s\uf074 and the static output torque of the VSSEA s l \uf074 \uf074 \uf03d when the deflection of the output link increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Since dis s\uf074 is zero when l g q q \uf03d , the VSAM does not consume energy to keep the stiffness constant at equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' When the deflection is small, the energy consumption of the VSAM is negligible because dis s\uf074 is very low regardless of s\uf074 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' However, the VSAM consumes higher energy as the deflection of the output link increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 8 that the disturbance torque of the stiffness modulation motor is always limited within the work-range of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This disturbance can be further supressed using novel mechanical designs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' MOTION CONTROL OF THE VSSEA In this section, the position and force control problems of the VSSEA are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is a well-known fact that the motion control problem of compliant actuators is more complicated than that of conventional stiff actuators, particularly in position control [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To precisely track link trajectories or interact with unstructured environments, internal and external disturbances should be compensated using adaptive and robust controllers [9 – 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This section experimentally shows that the proposed VSSEA allows us to conduct high-performance motion control tasks using conventional PID controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The experimental setup was built by using ESCON 50/5 motor drivers, 1000ppr encoders at the motors and a 10000ppr encoder at the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' A PC with a Linux operating system was employed to perform the real-time motion control experiments with 1ms sampling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Position Control: Let us start with the position control problem of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 9 illustrates the position control experiments when a PID controller is synthesised for controlling the angle of the first motor as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ref ref ref m p g g i g g d g g K q q K q q dt K q q \uf074 \uf03d \uf02d \uf02b \uf02d \uf02b \uf02d \uf0f2 (12) where ref gq and ref gq represent the position and velocity references of the gearbox, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', g q , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' External disturbances up to 5Nm and 15Nm were applied to the output link after 2 seconds when the VSSEA was in soft (21 Nm/rad) and stiff mode (985 Nm/rad), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Although the transient response was good except a relatively high overshoot at link side, the link of the actuator was very sensitive to external disturbances, particularly when the actuator was in soft mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This is expected because the angle of the link lq is not used in the position controller synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To improve the robustness against external load, let us use lq in the feedback controller synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In this experiment, the control signal was designed as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ref ref ref m p l l i l l d l l K q q K q q dt K q q \uf074 \uf03d \uf02d \uf02b \uf02d \uf02b \uf02d \uf0f2 (13) where ref lq and ref lq similarly represent the position and velocity references of the VSSEA’s link, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) Static output torque versus deflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) 90% stiffness modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' c) 70% stiffness modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 7: Output torque and stiffness of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' LDM: Large Deflection Model and SDM: Small Deflection Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 8: Disturbance torque of the stiffness modulation motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) Stiff mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) Soft mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 9: Position regulation control of motor 1 when external loads are applied to the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Kp = 15000, Kd = 500, Ki =75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 ref gq \uf070 \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 7a: Static Output Torque vs Output Link Deflection 100 LDM: r/L = 1 LDM:r/L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='75 90 LDM: r/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 80 LDM:Cr/L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='25 SDM: r/L=1 Torque Tt 70 SDM: Cr/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='75 SDM: Cr/ L =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 60 SDM: r/ L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='25 50 Output 40 Static 30 20 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='9 (rad)×104 Figure 7b: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1 < zr/L<1 Figure 7c: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='3 < ,/L<1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 2500 LDM LDM SDM SDM 3 (pea/n) 2000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 Stiffnessk(qm2，Q) 1500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 1000 Stiffness 1 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='8 /L (Dimensionless) /L (Dimensionless)100 d28 : r/L = 1 90 dis : r/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='75 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='dis : rr/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 80 dis : rr/L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='25 Ts : r/L= 1 70 ITs:Cr/L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='75 (uN) Ts : Cr/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 60 Tg : r/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='25 Torque 50 40 30 20 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='4 (rad)Figure9a:Stiff mode Figure9b:Softmode 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 le (rad A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0 0 0 2 4 6 0 2 4 6 Time (s) Time (s)IEEE/ASME TRANSACTIONS ON MECHATRONICS External disturbances up to 10Nm were similarly applied when the actuator was in stiff mode, however only less than 1Nm external disturbances could be applied due to robust stability problems encountered in the soft mode of the actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 10 shows that the robustness against external loads was improved by using lq in the feedback controller synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' However, the performance of transient response was notably degraded by large vibrations when the actuator was in soft mode (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Moreover, only small external disturbances could be suppressed due to the robust stability problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is a well-known fact that more advanced controllers should be employed for the robust position control problem of compliant actuators [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' By changing the stiffness of the actuator, this paper proposes a simple yet effective solution for this challenging problem as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' When we used the same PID controller in trajectory tracking control, we obtained similar results as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Due to the well-known bandwidth limitations of compliant systems, increasing the speed of reference trajectory notably degraded the position control performance when the actuator was in soft mode [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='The VSSEA could not follow 1Hz reference trajectory as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 11b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The performance of trajectory tracking control could be easily improved by increasing the stiffness of the actuator as illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 11c and 11d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Force Control: By using Hooke’s law, the force control problem of the compliant actuator was described as a position control problem, and force control experiments were performed by controlling the deflection of the output link at non-equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Similar to the position control experiments, a simple PID controller was synthesised by feeding back the deflection angle of the output link, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', q l g q q \uf044 \uf03d \uf02d where lq and 1 g m q q N \uf03d are the link and gear angles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 12 illustrates the force control experiments of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The force control signal was designed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ref ref ref m p q q i q q d q q K K dt K \uf074 \uf03d \uf044 \uf02d \uf044 \uf02b \uf044 \uf02d \uf044 \uf02b \uf044 \uf02d \uf044 \uf0f2 (14) where ref q \uf044 and ref q \uf044 represent the position and velocity references of the link deflection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', q l g q q \uf044 \uf03d \uf02d , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' When the actuator was in soft mode, relatively large link deflections occurred for small output torques as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 12a and 12b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This allows actuator to physically interact with different environments in a safe manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To achieve higher a) Stiff mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) Soft mode Figure 10: Position regulation control of the output link when external loads are applied to the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Kp = 5000, Kd = 95, Ki = 35, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 ref lq \uf070 \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) Soft mode and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1Hz b) Soft mode and f = 1Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' c) Stiff mode and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' d) Stiff mode and f = 1Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 11: Position trajectory tracking control of the output link when Kp = 5000, Kd = 95, Ki = 35, and \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 cos 2 1 ref lq K f t \uf070 \uf03d \uf02d \uf02d where K is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5\uf070 and 2\uf070 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) Regulation control in soft mode b) Trajectory tracking control in soft mode c) Regulation control in stiff mode d) Trajectory tracking control in stiff mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 12: Force control experiments when Kp = 2500, Kd = 85, and Ki = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 10a: Stiff mode Figure 10b:Soft mode 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 4g 3 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 2 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 A 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0 0 0 2 4 6 0 2 4 6 Time (s) Time (s)Figure 11a:Soft mode(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1Hz) 4 3 ref Angle 2 0 0 2 3 4 5 6 7 8 9 10 Time (s) Figure11b:Softmode(1Hz) 2 ref 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 ref 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 Time (s)Figure 11c: Stiff mode (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content="1Hz) 4 'e 3 ref Angle 2 0 0 2 3 5 6 7 8 9 10 Time (s) Figure 11d: Stiff mode (1Hz) 2 (rad) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 1 Angle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0 1 2 3 4 5 6 7 8 9 10 Time (s)Figure12a:Force requlationcontrol insoft mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='4 6 Angle (rad) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='2 Are 0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='2 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) Figure 12b:Force trajectory tracking controlin soft mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='4 A 6 Angle (rad) T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='2 0 0 1 2 3 4 5 6 7 8 9 10 Time (s)Figure12c:Force regulationcontrolinstiff-mode 15 △rej (rad) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='04 T 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='02 5 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) Figure 12d: Force trajectory tracking control in stiff-mode 25 △rej AAAAAAAA (rad) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='06 20 (uN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='04 15 Angle Torque 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='02 5 0 0 1 2 3 4 5 7 8 9 10 Time (s)IEEE/ASME TRANSACTIONS ON MECHATRONICS output torque, the stiffness of the actuator should be increased as illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 12c and 12d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This, however, may degrade safety in contact motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 12 that the proposed VSSEA enables us to conduct high-performance force control applications using a conventional PID controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Stiffness Modulation: Figure 13 illustrates the position control experiments of motor 2 at different speeds when the actuator is at equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The motor did not draw current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', the VSAM did not consume energy, to keep the stiffness of the actuator fixed at an equilibrium position as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The current of motor 2 was negligible at low speeds of stiffness modulation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1Hz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 13a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' As the speed of stiffness modulation increased, the motor drew higher current due to the increased frictional and inertial disturbances of the VSAM (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 13b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This is expected because the energy consumptions of all variable stiffness actuators become higher as the speed of stiffness modulation is increased [33, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Since the frictional and inertial disturbances were mitigated in the design of the VSAM, the stiffness modulation could be performed using low current drains at equilibrium positions as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 14 illustrates the stiffness modulation experiments when the actuator is at non-equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' While the deflection angle of the output link was regulated, the stiffness of the actuator was modulated 10% in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figures 14a and 14b show that the current of motor 2, Im2, was low when 10% stiffness modulation was performed in the stiff mode of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This is expected because, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (11), the small deflections of leaf springs in stiff mode could lead to low disturbance torques exerted by the VSAM on motor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Compared to other energy efficient and infinite-range variable stiffness actuators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', [34, 35], this feature of the proposed VSAM allows us to perform infinite-range stiffness modulation using bounded control signals [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figures 14c and 14d show that the current drain of motor 2 was higher when 10% stiffness modulation was performed in the soft mode of the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Since the deflection angle of the output link was larger in soft mode, the current drain of motor 2 increased by the higher disturbance torque of the VSAM as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figures 13 and 14 show that the control signal of motor 2 was always bounded when stiffness modulations were conducted at equilibrium and non-equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 15 illustrates the energy cost of changing stiffness compared to the potential energy stored in the springs of the VSAM when 10% stiffness modulation was performed at stiff and soft modes of the actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It shows that the energy cost of stiffness modulation is weakly coupled to the deflection of the output link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' While lower energy was consumed by the VSAM at fixed stiffness configurations in both stiff and soft modes, the energy cost of stiffness modulation increased when the stiffness of the actuator was changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The larger deflection of the output link in soft mode led to higher energy cost of stiffness modulation as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 15b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This result is expected because the disturbance torque gets larger as the deflection of the output link increases as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' DISCUSSION The experimental results given in Section IV show that the a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1Hz stiffness modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) 1Hz stiffness modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 13: Stiffness modulation at an equilibrium position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) b) a) Stiffness modulation in stiff mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) Output torque and motor current in stiff mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' c) Stiffness modulation in soft mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' d) Output torque and motor current in soft mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 14: Stiffness modulation at non-equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' a) Stiffness modulation in stiff mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' b) Stiffness modulation in soft mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure 15: Energy consumption of the VSAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Figure13a:100%stiffnessmodulation(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1Hz) /L(dimensionless) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 qm2 Gm2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 A Im2 urrent 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0 1 2 3 4 5 6 7 8 9 10 Time (s) Figure13b:100%stiffnessmodulation (1Hz) /L(dimensioniess) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 qm2 qm2 Im2 Current 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0 2 0 1 2 3 4 5 6 7 8 9 10 Time (s)Figure14a:10%stiffnessmodulationinstiffmode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1 Gm2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='05 0 5 10 15 20 Time (s) Figure14b:Outputtorqueandmotorcurrentinstiffmode 10 (uIN) lm2 Current (A) 5 0 2 0 5 10 15 20 Time (s)Figure14c:10%stiffnessmodulationinsoftmode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1 Qm2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='05 0 5 10 15 20 Time (s) Figure14d:Outputtorgqueandmotorcurrentinsoftmode 8 (Nn) 6 Im2 (A) Current( 4 0 2 2 0 5 10 15 20 Time (s)Energy Figure15a:10%stiffnessmodulationinstiffmode Energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='01 FixedStiffness StiffnessModulation Em2/Ema" Motor Ratio of Spring 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content="005 2 Ratio of the 0 0 0 1 2 3 4 5 6 Figure15b:10%stiffnessemodulationinsoftmode 'ime (s Energy Us/Umar 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='1 FixedStiffness StiffnessModulation m2 Ratio of Spring 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='05 0 0 1 2 3 4 5 6 7 8 9 Time (s)IEEE/ASME TRANSACTIONS ON MECHATRONICS proposed VSSEA enables us to conduct high-performance position and force control applications using conventional PID controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This provides several benefits in many different advanced robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For example, while the stiff mode of the VSSEA allows a cobot to perform high-precision position control tasks in industry, the soft mode of the actuator can boost safety in physical robot-environment interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' However, more advanced controllers should be synthesised for smooth transition between position and force control tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=', the stiff and soft mode of the actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' In addition, more effort should be expended on optimal stiffness modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The experimental results show that a wide range of stiffness modulation can be achieved by simply controlling the position of the second motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For example, while the stiffness of the output link is 21Nm/rad when the actuator is in soft mode in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 9a, it is ~50 times higher with 985Nm/rad in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This is an important feature of the proposed VSSEA as it allows us to perform both precise position control and safe robot- environment interaction tasks using conventional PID controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Other important features of the VSSEA are fast and low-energy-cost stiffness modulation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The transition from the softest mode to the stiffest mode can be achieved within a second as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Since the disturbance torque exerted by the VSAM on motor 2 is always bounded unless it is zero, a wide range of stiffness modulation can be performed using bounded control signals at equilibrium and non-equilibrium positions as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Therefore, the energy cost of stiffness modulation is low as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This can provide significant benefits to mobile robotic systems such as humanoids, quadrupeds, and exoskeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' With the proposed modular design approach, the VSSEA can be easily modified to meet the requirements of different robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' For instance, i) torque density can be increased using a higher gear ratio, ii) stiffness characteristic can be tuned using the different number, material, and shape of leaf springs, and iii) faster stiffness modulation can be achieved using a ball screw with higher lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' The proposed novel mechanical design eliminates the work-range limitation of variable stiffness actuators as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 11a and 11c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' This allows us to apply the VSSEA to various robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' By neglecting the small torsional motions of leaf springs, the dynamic model of the VSSEA is obtained using the analogy of a mass-spring-damper system and the Euler-Bernoulli beam theory in Section III [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' While the exact dynamic model and nonlinear Euler-Bernoulli beam theory are computationally expensive and ineffective in controller synthesis [43, 44], the model derived through simple beam theory is inaccurate when the deflection of the output link is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Thus, more effort should be expended to obtain an effective dynamic model for the VSSEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' CONCLUSION This paper proposes a new VSSEA that can perform fast and low-energy-cost stiffness modulation over a large range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' These features can provide several benefits to robotic systems such as easier motion control problems, longer battery life, and safer physical robot environment interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is experimentally shown in this paper that the proposed VSSEA allows us to conduct high-performance position and force control tasks using conventional PID controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' It is also demonstrated that the stiffness of the actuator can be increased or decreased up to 50 times within a second while the energy cost of stiffness modulation is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' However, further research should be conducted to clarify how the VSSEA can contribute to robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' To this end, we will apply the proposed actuator to advanced robotic applications such as legged locomotion in the future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' REFERENCES [1] B.' metadata={'source': 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and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Carloni, “The mVSA-UT: A miniaturized differential mechanism for a continuous rotational variable stiffness actuator,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Biomed Robot and Biomechatron (BioRob), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' 1943-1948, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Emre Sariyildiz (S’11, M’16, SM’21) received his first Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' degree in Integrated Design Engineering from Keio University, Yokohama, Japan, in 2014, and second PhD degree in Control and Automation Engineering from Istanbul Technical University, Istanbul, Turkey, in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' He is currently a Senior Lecturer at the School of Mechanical, Materials, Mechatronic, and Biomedical Engineering, University of Wollongong, Wollongong, NSW, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' His research interests include control theory, robotics, mechatronics, and motion control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Rahim Mutlu (M’22) received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' degree in Robotics from University of Wollongong, Wollongong (UOW), NSW, Australia, in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' He was a Lecturer with UOW 2017-21, prior to committing his current role as Assistant Professor with Faculty of Engineering and Information Sciences at UOWD, UAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' He is also founder of the Intelligent Robotics & Autonomous Systems Co (iR@SC), NSW, 2529, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' His research interests include soft robotics, soft haptics, wearable technologies, assistive and rehabilitation exoskeletons, additive manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Jon Roberts received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' degree in Mechanical Engineering from University of Wollongong (UOW), Wollongong, NSW, Australia in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Since January 2020, he has been a Lecturer with the School of Mechanical, Materials, Mechatronic, and Biomedical Engineering, University of Wollongong, Wollongong, NSW, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' His research interests are simulation methods, bulk materials handling, safety in mining, and dust control technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Chin-Hsing Kuo received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=" degree in Mechanical Engineering from King's College London, UK, in 2011." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Since February 2019, he has been a Senior Lecturer with the School of Mechanical, Materials, Mechatronic, and Biomedical Engineering, University of Wollongong, Wollongong, NSW, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' Before that he was an Associate Professor with the National Taiwan University of Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' His research interests include parallel robots, mechanism design, and robot kinematics and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=" Barkan Ugurlu (S'08-M'10) received his Ph." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' degree in Electrical and Computer Engineering from Yokohama National University, Yokohama, Japan, in 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' He was a Marie Sklodowska-Curie Fellow and currently holds an Assistant Professor position at the Department of Mechanical Engineering, Ozyegin University, Istanbul, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} +page_content=' His research interests include legged locomotion control, hardware development for novel robotic systems, and multi-body dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf'} diff --git a/sdAzT4oBgHgl3EQf6P79/content/tmp_files/2301.01874v1.pdf.txt b/sdAzT4oBgHgl3EQf6P79/content/tmp_files/2301.01874v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..773d981ab573f3e6a12c49a4aafff131e5927f13 --- /dev/null +++ b/sdAzT4oBgHgl3EQf6P79/content/tmp_files/2301.01874v1.pdf.txt @@ -0,0 +1,1022 @@ +Critical Perspectives: A Benchmark Revealing Pitfalls in +PerspectiveAPI +Lorena Piedras∗ and Lucas Rosenblatt∗ and Julia Wilkins∗ +lp2535@nyu.edu, lr2872@nyu.edu, jw3596@nyu.edu +New York University +Abstract +Detecting “toxic” language in internet content +is a pressing social and technical challenge. In +this work, we focus on PERSPECTIVE from +Jigsaw, a state-of-the-art tool that promises +to score the “toxicity” of text, with a recent +model update that claims impressive results +(Lees et al., 2022). We seek to challenge cer- +tain normative claims about toxic language by +proposing a new benchmark, Selected Adver- +sarial SemanticS, or SASS. We evaluate PER- +SPECTIVE on SASS, and compare to low-effort +alternatives, like zero-shot and few-shot GPT- +3 prompt models, in binary classification set- +tings. +We find that PERSPECTIVE exhibits +troubling shortcomings across a number of our +toxicity categories. SASS provides a new tool +for evaluating performance on previously un- +detected toxic language that avoids common +normative pitfalls. Our work leads us to em- +phasize the importance of questioning assump- +tions made by tools already in deployment for +toxicity detection in order to anticipate and pre- +vent disparate harms. +1 +Introduction +Social media and online discussion forums are +plagued with toxic content. +41% of American +adults have personally experienced online harass- +ment (Vogels, 2021). +This problem can be further exacerbated by +learning algorithms with unchecked incentives. On +a platform like Facebook, incendiary comments are +highly visible due in large part to the design of its +ranking algorithms (Munn, 2020). Some experi- +ments by Cheng et al. (2017); Kim et al. (2021) +have shown that users are more likely to write toxic +or “troll” posts when the initial comments in a +conversation exhibited toxic or “trolling” behavior, +∗Equal contributions. +This research was supported in part by NSF Award No. +1922658. +which can be interpreted as evidence that “trolling +is contagious.” +This creates the potential for a negative feedback +loop where companies are incentivized to promote +toxic content and toxic content in turn provokes +users to post more malicious content. In response, +researchers and corporations alike have invested in +systems that can detect toxicity at massive scales to +help moderate content on social media and internet +forums (Lees et al., 2022; Pavlopoulos et al., 2017; +Wulczyn et al., 2017). Some of these models have, +however, been shown to be susceptible to simple +adversarial attacks (Hosseini et al., 2017; Gröndahl +et al., 2018). +Text +Toxicity +(Perspective) +Toxicity +(Human) +You look so good in +that dress, and probably +even better with it off. +22.9% +85.0% +Table 1: A toxicity score given by PERSPECTIVE to a +SASS example from the “Sexual Harassment” category. +Still, deep language model-based approaches to +detecting toxicity have been deployed, with con- +sumer facing API’s granting direct access to the +model output. One such model is PERSPECTIVE +(Lees et al., 2022), which assigns text a “toxic- +ity” score ∈ [0, 1], provided by Jigsaw (a Google +subsidiary). In 2022, Jigsaw published findings on +the “new generation” PERSPECTIVE toxicity model +(currently deployed). We believe that interrogat- +ing the model’s assumptions and frailties is vital +work in moving it (and similar systems) towards +robust detection (or, perhaps, a robust understand- +ing of when detection fails). For example, Jigsaw +defines toxic language as “a rude, disrespectful, or +unreasonable comment that is likely to make you +leave the discussion” (Lees et al., 2022), though +other definitions exist (Märtens et al., 2015). We +were unable to find an original source for Jigsaw’s +definition. +arXiv:2301.01874v1 [cs.CL] 5 Jan 2023 + +Text +Category +Type +Toxicity +Score +Expected +Toxic- +ity +Your secret can be our secret as long as you do what I tell you. +Blackmail +Unfiltered 11.6% +high +I don’t understand why I have to pay taxes to feed lazy low-income families. +Classism +Unfiltered 17.4% +high +We don’t admit people from your race in this forum. +Exclusionary +Filtered +35.8% +high +You shouldn’t think that way. +Gaslighting +Filtered +11.3% +high +It’s weird how she’s taller than her husband. +Misogyny +Filtered +24.1% +high +No one should live without water, electricity and schooling. Except for poor +people. +Sarcasm +Filtered +10.8% +high +I can’t believe you work here. You look too beautiful to do this job. +Sex. Harass. +Filtered +23.5% +high +Boys are better at math than girls, it’s just a fact. +Stereotyping +Filtered +24.6% +high +It’s so freaking crazy that he surfed a 40-foot wave. +False Positive +Filtered +67.2% +low +Their hair is so long, it’s almost touching the ground. +Neutral +Unfiltered 16.7% +low +Table 2: Toxicity scores from PERSPECTIVE for randomly selected examples in the 10 categories of SASS. +Contributions Existing models and benchmarks +rely on aggregating binary responses to text col- +lected from crowdworkers into a ground truth +“probability of toxicity” (this is accomplished by +prompting a crowdworker with “Is this text toxic?”, +and then calculating the aggregate Pr[toxic] = +|yes_responses| +|total_responses|, which is the “toxicity score”). We +suspect this method overemphasizes a normative +understanding of toxicity, such that potentially +toxic, harmful text “on the margins” goes unde- +tected. Here, “normative” describes the way in +which multiple annotations are traditionally aggre- +gated, which often implicitly supports the views of +the majority and ignores the annotations of minor- +ity groups. In response, we isolate a set of natural +language categories that fulfill the definition of tox- +icity (as stated earlier), but go largely undetected, +due in part, we believe, to the normative assump- +tions of the ground truth toxicity examples from +existing training and benchmark data. Again, these +normative assumptions are related to the way data +is aggregated, which may ignore the views of a +minority of annotators in favor of the majority. +We present a new benchmark entitled Selected +Adversarial SemanticS, or SASS, that evaluates +these behaviors. SASS contains natural language +examples (each approximately 1-2 sentences in +length) across previously underexplored “toxicity” +categories (like manipulation and gaslighting) as +well as categories that have received attention (like +“sexism” (Sun et al., 2019)), and includes a “hu- +man” toxicity score ∈ [0, 1] for each example. Ta- +ble 1 shows an example from the "Sexual Harass- +ment" category. SASS follows a filtered/unfiltered +approach to adversarial benchmarking, as in (Lin +et al., 2021). The benchmark is designed to exploit +the normative vulnerabilities of a toxicity detection +tool like PERSPECTIVE. Specifically, PERSPEC- +TIVE makes ambiguous claims that they can “iden- +tify abusive [or toxic] comments” (Jigsaw), but do +not clarify that these abusive comments are deter- +mined by essentially using the majority opinion +of random annotators. Our position is that PER- +SPECTIVE should either be clear concerning the +limitations of it’s toxicity tool (i.e. that it detects +toxic content according to majority opinion), or +adjust the PERSPECTIVE model to better account +for minority annotations. +We compare PERSPECTIVE’s performance on +SASS to “human” generated toxicity scores. We +further compare PERSPECTIVE to low-effort alter- +natives, like zero-shot and few-shot GPT-3 prompt +models, in a binary classification setting (“toxic or +not-toxic?”) (Brown et al., 2020). Code for our +project can be found in this repository. +2 +Related Work +Past PERSPECTIVE Model Works such as (Hos- +seini et al., 2017) and (Gröndahl et al., 2018) fo- +cused on generating adversarial attacks to test how +the former version of PERSPECTIVE responded +to word boundary changes, word appending, mis- +spellings, and more. (Gröndahl et al., 2018) further +tested how toxicity detection models responded to +offensive but non-hateful sentences. The toxicity of +the test sentences heavily increases when the word +"F***" is added (You are great → You are F*** +great, 0.03 → 0.82). This opens up a discussion +about the subjectivity of what should be considered +“toxic”, a theme in our work. We pose new open +questions that draw a clear connection between +“toxicity” and normative concerns (Arhin et al., +2021). Another promising approach to fortifying +toxicity detectors is by probing a student model +with a few annotated examples to detect veiled +toxicity, mostly annotated incorrectly, from a pre- + +existing dataset, then re-annotating, thus making +the model more robust (Han and Tsvetkov, 2020); +we do not attempt this in our work. +Current Model A recent publication on PER- +SPECTIVE (Lees et al., 2022) generated bench- +marks to test how the new version responded to +character obfuscation, emoji-based hate, covert tox- +icity, distribution shift and subgroup bias. They +demonstrate improvements of the model in classi- +fying multilingual user comments and classifying +comments with human-readable obfuscation. Ad- +ditionally, PERSPECTIVE beats every baseline on +character obfuscation rates ranging from 0% to +50%. Character-level perturbations and distractors +degrade performance of ELMo and BERT based +toxicity models, reducing detection recall by more +than 50% in some cases (Kurita et al., 2019). Sep- +arate detection tools, like the HATECHECK sys- +tem from (Röttger et al., 2020), present a set of +29 automated functional tests to check identifica- +tion of types of “hateful behavior” by toxicity or +hate speech detection models. A large dynami- +cally generated dataset from (Vidgen et al., 2020), +designed to improve hate speech detection during +training, showed impressive performance increases +in toxicity and hate speech detection tasks. Though +slightly different in their typology of toxic speech, +these approaches have a significant scale advantage +over SASS, while SASS examples are specifically +targeted at the PERSPECTIVE tool. +3 +Benchmarking with SASS +The SASS benchmark contains 250 manually cre- +ated natural language examples across 10 nuanced +"toxicity" categories (e.g. stereotyping, classism, +blackmail). These categories were selected via a +process of literature review and vulnerability test- +ing on PERSPECTIVE and other toxicity tools, to de- +termine their weaknesses/strengths. As we sought +to challenge PERSPECTIVE and other toxicity tools, +we believe this to be a sufficient process for deter- +mining our categories, although acknowledge that +it introduces some unavoidable author bias. The +examples are each 1-2 sentences long and are de- +signed to exploit vulnerabilities in toxicity detec- +tion systems like PERSPECTIVE. Samples from +SASS in each category are shown in Table 2. +Eight of SASS’s categories are aimed at gener- +ating “False Negative” (FN) scores (a score that +significantly underestimates the toxicity of some +text), one category is aimed at “False Positive” (FP) +scores (a score that overestimates toxicity), and one +category is “Neutral,” a control, demonstrating the +model’s performance on “normal,” non-toxic sen- +tences. SASS is heavily biased towards examples +that generate a FN score, which we argue may be +more harmful than a FP score, as a FN means toxic +content has gone undetected. For each category, +the benchmark contains 15 “filtered” and 10 “un- +filtered” examples, drawing inspiration from (Lin +et al., 2021). We generate filtered examples by +brainstorming toxic comments and evaluating the +comments with PERSPECTIVE to ensure a toxicity +score of < 0.5. Then, we generate an additional set +of 10 examples per category using the knowledge +gained from creating the filtered examples without +first testing them on PERSPECTIVE. +Human Ground Truth The benchmark also +contains a "human" toxicity score ∈ [0, 1] for each +comment, which can be used as a baseline for eval- +uating toxicity detection tools using SASS. The hu- +man toxicity scores are an average of the toxicity +scores of the authors per comment (scored blindly). +Here, we scored examples on a scale of 0-10, using +Jigsaw’s definition of toxicity, i.e. “how likely [the +example is to] make [a user] leave the discussion” +(0=highly unlikely, 10=highly likely). Significantly, +we aligned these ratings with assumptions laid out +in A.2.2 (in appendix) for consistency and to com- +bat benchmarking pitfalls (Blodgett et al., 2021). +We further performed z-normalization, as per +(Pavlick and Kwiatkowski, 2019). Each author may +have treated the “0-10 toxicity scale” differently, +so this normalization process ensures that the final +aggregate scores are not overly biased by any single +author’s interpretation of the scale. +In Table 5 (in the appendix), we observe the av- +erage z-normalized human toxicity scores of com- +ments in SASS across the toxicity categories de- +scribed above. We note that some categories are +inherently more toxic than others; “Stereotyping” +comments have an average human toxicity score +of 0.81 versus 0.57 for “Gaslighting” comments, +which further contrasts with an average human tox- +icity score of 0.007 for “Neutral” comments. +4 +Experiments and Discussion +Binary Toxicity Classification We showcase the +utility of SASS by evaluating PERSPECTIVE and +GPT-3 against the human baseline in a binary +classification setting. It’s important to note that +PERSPECTIVE and GPT-3 are very different sys- +tems, trained with distinct objectives, amounts and + +System +Precision +Recall +F1-Score +PERSPECTIVE +0.26 +0.05 +0.08 +GPT-3-ZERO +0.83 +0.19 +0.31 +GPT-3-ONE +0.77 +0.11 +0.19 +GPT-3-FEW +0.73 +0.52 +0.61 +Table 3: +Evaluation of PERSPECTIVE and GPT-3 +in multiple prompt settings on the SASS benchmark +against thresholded human toxicity scores, in a binary +classification setting. +sources of data. We believe the comparison is still +useful because it provides a "low-effort alterna- +tive" to make sure that our examples are not overly +complicated. Note that GPT-3 was not fine-tuned +explicitly for this task, so we prompt the system in +zero, one, and few-shot settings for a binary toxic- +ity classification. We binarize the PERSPECTIVE +and z-normalized human baseline toxicity scores +by labeling scores > 0.5 per comment as "toxic". +The binarized ground truth human labels on SASS +contain 72.4% toxic labels versus 27.6% non-toxic +labels. We use these thresholded human labels +as ground truth and evaluate PERSPECTIVE and +GPT-3’s performance on SASS in Table 3. +Model Description PERSPECTIVE uses a Trans- +former model with a state-of-the-art Charformer en- +coder. The model is pretrained on a proprietary cor- +pus including data collected from the past version +of PERSPECTIVE and related online forums. This +dataset is mixed in equal parts with the mC4 corpus, +which contains multilingual documents (Lees et al., +2022). GPT-3, created by OpenAI in 2020, is a +state-of-the-art autoregressive transformer-based +language model (Brown et al., 2020). GPT-3 is +trained on a massive amount of internet text data, +predominately Common Crawl +and WebText2 +(Radford et al., 2019), and generates human-like +language in an open prompt setting. +Results We first observe that PERSPECTIVE per- +forms very poorly on the binary task of toxicity +classification on the SASS benchmark (Table 3, F1- +Score = 0.08). Note that the majority of comments +in SASS were crafted specifically to generate a low +toxicity score from PERSPECTIVE, so this is not +surprising. We establish the metric regardless, as a +baseline to evaluate future versions of the system. +We also examine the performance of GPT-3 +in multiple prompt settings for binary (true/false) +See Appendix A.1 for details on prompt generation. +Recall that “Neutral” and “False Positive” categories are +inherently non-toxic, accounting for 20% of non-toxic labels. +https://commoncrawl.org/ +toxic content classification in Table 3. Each system +yields relatively high precision and low recall, gen- +erally indicating a significant under-prediction of +toxicity in SASS. GPT-3 has more success in clas- +sifying harmful comments in SASS as toxic across +the board relative to a thresholded PERSPECTIVE. +GPT-3-FEW (F1-Score = 0.61) shows a signif- +icant improvement over both GPT-3-ZERO and +GPT-3-ONE as well as PERSPECTIVE, yielding +the most success relative to the human baseline of +any of the experimental formulations. +We hypothesize that GPT-3 outperforms PER- +SPECTIVE largely due to the sheer scale and scope +of data that GPT-3 is trained on, as well as the +size of the model itself (175B learnable parameters +in GPT-3 versus 102M in the PERSPECTIVE base +model). While GPT-3 is not trained for the toxicity +detection task specifically, by learning from such +a massive amount of internet text data spanning +millions of contexts, the model has likely been ex- +posed to a much wider range of potentially toxic +material then PERSPECTIVE. +In Table 5 (see appendix), we break down the +toxicity scores of PERSPECTIVE and GPT-3 by +SASS category, relative to the human baseline. In +some categories, both PERSPECTIVE and GPT- +3-FEW fall particularly short (for example, PER- +SPECTIVE predicts an average toxicity score of +21.9% for “Sexual Harassment” comments versus +the 80% human baseline). Relative to other cate- +gories from SASS, PERSPECTIVE similarly rates +comments in “Sarcasm” and “Stereotyping” as +highly toxic, while humans rated the toxicity of +“Stereotyping” comments significantly higher than +those in “Sarcasm.” This raises the question of how +to properly threshold scores from a toxicity detec- +tion system in-the-wild, which (Lees et al., 2022) +do not comment on, though seems a reasonable use +case for platforms flagging toxic content. +In the “False Positive” category we observe that +both PERSPECTIVE and GPT-3-FEW yield very +high toxicity scores on average (Table 5), suggest- +ing that the models are overfit to swear word toxic- +ity, and underfit to a deeper interpretation of mali- +cious intent. We believe it is important to delineate +between the tasks of swear word detection and tox- +icity detection, and so find this undesirable. Allow- +ing harmful comments to slip through the cracks +is arguably more dangerous than unintentionally +removing content with positive intent, but both of +these scenarios could be upsetting to a downstream + +user. We report further on the influence of swear +words on toxicity in the next section. +Profanity and Toxicity Detection SASS in- +cludes 18 “False Positive” examples that con- +tain swear words. +PERSPECTIVE rated all +of them as toxic, and GPT-3-FEW labeled +83% of these comments as toxic (this is +P[toxic|contains_swear_word]). This suggests +that, instead of understanding when swear words +are used to communicate hateful content, PERSPEC- +TIVE may be effectively memorizing their inclusion +in toxic text. This could be problematic; swear +words can be used to communicate non-toxic emo- +tions, like surprise (e.g. Holy f*** I got the job!) +or excitement (e.g. Oh sh**! Congratulations.) and +should not necessarily be treated equivalently to +toxic speech. Furthermore, different genders and +races utilize profanity differently, so associating ex- +pletives with toxicity could have disparate impacts +(Beers Fägersten, 2012). Past work by (Gröndahl +et al., 2018) evaluating an older version of PER- +SPECTIVE also detected this issue. +As shown in Table 6 (see appendix), from the 34 +SASS examples that PERSPECTIVE rated as toxic, +52% contained a profanity, versus only 11.6% of +the examples rated toxic by GPT-3-FEW (this is +P[contains_swear_word|toxic]). A lot of hate- +ful content does not explicitly contain offensive +words and it is troubling that PerpectiveAPI relies +so much on them in our benchmark. +TweetEval We were surprised that GPT-3-FEW +performed better in the binary classification sce- +nario on the SASS benchmark than PERSPECTIVE, +and so sought to validate the finding with another +prominent toxicity benchmark, TweetEval. Thus +we selected 1,000 examples from the ‘Hate Speech +Detection” benchmark randomly (Barbieri et al., +2020). We acknowledge that this might be viewed +as irrelevant or an unfair comparison, as some +“toxic language” may not qualify as “hate speech” +(for example, universal insults that do not target +a specific group). However, we believe that the +reverse claim, that all “hate speech” should qual- +ify as “toxic language” is true. Then evaluating +both PERSPECTIVE and GPT-3-FEW on a “hate +speech” benchmark, despite both being designed +to detect “toxic language,” is a valid comparison. +We found that PERSPECTIVE had an F1-Score of +0.48 and GPT-3-FEW had an F1-Score 0.52 (Table +7, see appendix). The performance gap between +PERSPECTIVE and GPT-3-FEW on TweetEval is +significantly smaller than on SASS, but the trend +(GPT-3-FEW matching or improving on PERSPEC- +TIVE) is comparable. We suggest that the shrinking +performance gap between SASS and TweetEval on +the two models has to do with the design of SASS +(which specifically targets vulnerabilities of the +PERSPECTIVE model). Significantly, we were able +to validate that GPT-3-FEW, in the binary setting, +is a good point of comparison with PERSPECTIVE +on another benchmark, and does not only perform +well on SASS-specific examples. +Conclusion and Future Work We introduce Se- +lected Adversarial SemanticS (SASS) as a bench- +mark designed to challenge previous normative +claims about toxic language. We have shown here +that existing tools are far from robust to relatively +simple adversarial examples, and fail to report ad- +equately on the implicit biases attached to their +model construction. We therefore position SASS as +an important additional benchmark that can help us +understand weaknesses in existing and future sys- +tems for toxic comment detection. Some impactful +future work would be to grow the set of examples +in SASS and to perform similar vulnerability test- +ing on problems like sentiment analysis and other +tools for content moderation. Conducting a future +study with a set of random human annotators and +demonstrating that the majority rate SASS state- +ments as non-toxic would strengthen our claims of +normativity, and make the need for a benchmark +like SASS even more apparent. Expanding the set +of state-of-the-art NLP toxicity detection or large +language models evaluated on SASS would provide +interesting future points of comparison. Finally, +we emphasize our belief that deployed natural lan- +guage based tools, potentially serving millions of +users, must be examined and reexamined in order +to prevent the harmful beliefs of majority groups +from being perpetuated. +5 +Ethical Considerations +SASS, the new benchmark proposed in this paper, +seeks to address normative claims made by toxic- +ity detection tools that rely on majority opinion to +determine malicious content. In the narrow scope +of improving toxicity model evaluation, we thus +expect SASS to have a positive impact on the NLP +community, and by extension on moderation sys- +tems for social media and online forums. +However, thousands of content moderators, +whose job descriptions include toxic content de- + +tection, are currently employed by companies such +as Meta. We believe that the best systems for toxic +content detection are likely collaborations between +humans and machines, but acknowledge that, by +improving automated systems, we may jeopardize +employment for these people. Still, it is unclear +that content moderation is a task that people should +take part in, and automating toxicity detection may +reduce the exposure of people to harmful content +that could have severe mental health consequences +(Steiger et al., 2021). +There is always the risk that, in providing a new +benchmark to the larger NLP community, some +may use it to make unjustified claims. Therefore, +we take this opportunity to highlight the ways in +which SASS could be misused. We acknowledge +that any benchmark, especially a relatively small +one like SASS, will reflect the inherent biases of the +authors. Each category of SASS is not designed by +any means to be exhaustive; rather, each is designed +to provide an initial probe, a check for model vul- +nerabilities. Further exploration would be required +even if a model performed well on SASS. SASS is +also only an English language benchmark, and con- +tains examples that only make sense in an Ameri- +canized cultural context. We believe it is important +work to create similar benchmarks for other lan- +guages and cultural contexts. +We would like to thank Sam Bowman and +Richard Pang for very useful conversations and +feedback over the course of our project. We would +also like to thank Julia Stoyanovich and the Cen- +ter for Responsible AI at NYU for supporting our +work. +References +Kofi +Arhin, +Ioana +Baldini, +Dennis +Wei, +Karthikeyan Natesan Ramamurthy, +and Monin- +der Singh. 2021. +Ground-truth, whose truth?– +examining the challenges with annotating toxic text +datasets. arXiv preprint arXiv:2112.03529. +Francesco Barbieri, Jose Camacho-Collados, Luis +Espinosa-Anke, and Leonardo Neves. 2020. 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Mitigating gender bias in natural lan- +guage processing: Literature review. arXiv preprint +arXiv:1906.08976. +Bertie Vidgen, Tristan Thrush, Zeerak Waseem, and +Douwe Kiela. 2020. Learning from the worst: Dy- +namically generated datasets to improve online hate +detection. arXiv preprint arXiv:2012.15761. +Emily A Vogels. 2021. The state of online harassment. +Pew Research Center, 13. +Ellery Wulczyn, Nithum Thain, and Lucas Dixon. 2017. +Ex machina: Personal attacks seen at scale. +In +Proceedings of the 26th international conference on +world wide web, pages 1391–1399. +A +Example Appendix +A.1 +Generating prompts for GPT-3 +As described in Section 4, GPT-3 was not ex- +plicitly trained to classify phrases as toxic or not +toxic. We explore zero, one, and few-shot settings +for prompting GPT-3 to generate results compara- +ble to a thresholded PERSPECTIVE toxicity rating. +Here we describe the prompts used in these experi- +ments. +Setting +Prompt +GPT-3- +ZERO +“True or False, the phrase {phrase}’ is +toxic? Answer:” +GPT-3-ONE +“Determine if the given phrase is toxic: +Men are superior to women in every way. +=> True {phrase} => ” +GPT-3-FEW +“Determine if the given phrase is toxic: +Men are superior to women in every way. +=> True Grilling with the neighbors is +my favorite. => False {phrase} => ” +Table 4: Prompts used for zero, one, and few-shot ex- +perimentation with toxicity classification using GPT-3. +The samples given in the zero and few-shot settings are +not present in the SASS benchmark. +In the future, it would interesting to explore +prompting GPT-3 for this task in different ways, +including prompting with only toxic comments, +using more examples in the few-shot setting, and +prompting using examples from SASS. +Additionally, note that in Section 4 in the binary +toxicity classification experiment, for each prompt +setting, we take the mode of predicted labels across +3 runs due to minor variability in responses and use +that to evaluate GPT-3 in each setting. Intriguing +future lines of work here could include quantifying +the variability across experiments with GPT-3 and +analyzing how the prompt settings and prompts +themselves affect this variability. +A.2 +Designing SASS +A.2.1 +Avoiding Conceptual and Operational +Pitfalls +(Blodgett et al., 2021) describe the ways in which +popular stereotype detection benchmarks suffer +from a set of conceptual and operational pitfalls. +By providing a taxonomy of potential pitfalls, they +are able to audit the methods in a principled manner +and deduce ways in which the benchmark may pro- +duce spurious measurements. Here are summaries +of each category of pitfall they describe (specific +to stereotyping): +1. Conceptual Pitfalls (stereotyping) +(a) Power dynamics The claimed problem- +atic power dynamic may not be “realis- +tic.” +(b) Relevant aspects Must be clear and con- +sistent about what stereotype content is +within the purview of a given example. +(c) Meaningful stereotypes Is this stereo- +type actually reflective of a societal prob- + +lem? +(d) Anti vs non-stereotypes Some state- +ments can negate a stereotype (i.e. not), +while others can actively combat (i.e. +evil vs. peaceful). +(e) Descriptively true statements A true +statement masquerading as a stereotype. +(f) Misaligned stereotypes A hyper spe- +cific, or not specific enough, stereo- +type about a certain group/subgroup +(“Ethiopia” in a context where Africa +generally is implied). +(g) Offensive language Are swear words +stereotyping? +2. Operational Pitfalls (stereotyping) +(a) Invalid +perturbations +Not +a +real +stereotype/anti-stereotype (i.e. both al- +ternate sentences are stereotypes) +(b) Incommensurable +groups +or +at- +tributes Two alternate groups are not +comparable (think apples and oranges). +(c) Indirect group identification I.e. us- +ing names as a way of identifying group +membership (for example, racially iden- +tifying names) +(d) Logical failures If the alternate repre- +sents a logically dubious conclusion. +(e) Stereotype conflation Multiple stereo- +types present in a single example +(f) Improper sentence pairs The example +is not “realistic.” +(g) Text is not naturalistic The text itself +would never be written/uttered. +(h) (Un)markedness The two examples are +represented at different degrees in natural +text (i.e. “young gay man” vs. “young +straight man”) +(i) Uneven +baselines +Similar +to +(un)markedness, +examining +a +false +alternative. +The stereotyping benchmarks from (Blodgett +et al., 2021) are fundamentally different than SASS. +Thus, our analysis of pitfalls must rely on slightly +different criteria. Using the aforementioned cri- +teria, we created an abbreviated conceptual and +operational pitfall taxonomy for toxicity. +A.2.2 +Conceptual and operational pitfalls in +toxicity benchmarks +Recall that the definition of toxicity according to +PERSPECTIVE/Jigsaw is: “a rude, disrespectful, or +unreasonable comment that is likely to make you +leave the discussion.” +With this definition, we can begin to construct a +set of pitfalls that text from a benchmark might ex- +hibit. However, in order to minimize subjectivity as +much as possible, we outline three major assump- +tions about examples in our benchmark SASS (and +therefore, about what we prescribe as the behavior +of a system that “detects toxicity”): +Assume adversarial reading. Within reason, +does there exist an individual or group that would +be likely to leave a discussion after reading a piece +of text (even if they represent a significant minor- +ity)? +Assume adversarial context/subtext. Assume +that the possible context in which a piece of text +is positioned increases the likelihood that someone +would leave the discussion after reading it. +Assume bad intentions. Assume that the writer +of the text was knowingly malicious in their choice +of words. +These assumptions are important because they +help make our analysis structured and consistent. +Here are the pitfalls we use in evaluating toxicity, +constructed from (Blodgett et al., 2021): +1. Conceptual Pitfalls (toxicity) +(a) Meaningful toxicity (from Meaning- +ful stereotypes) Is the text likely to +make an individual leave a discussion, +given our assumptions? +(b) Descriptively true statements Is the +text true/factual? +(c) Offensive language Is the text toxic +purely due to swear words? (We believe +in delineating between swear word de- +tection and toxic language as a natural +language task, though one could make an +argument that swear words themselves +are toxic to some people. It is not clear +how to resolve this conflict.). +2. Operational Pitfalls (toxicity) +(a) Invalid toxicity markers (from invalid +perturbations, +incommensurable +groups, and logical failures) Does the + +text properly signify something that is +rude/disrespectful/unreasonable? +(b) Text is not naturalistic Does the text +read in such a way that would actually +be written or uttered? +(c) (Un)markedness (/uneven baselines) +Does the text appear in a statistically +likely/comparable pattern? +A.3 +Full benchmark code: +Code for our benchmark and evaluations can be +found here: https://github.com/lurosenb/sass + +Category +Human +PERSPECTIVE +GPT-3-ZERO +GPT-3-ONE +GPT-3-FEW +Blackmail +68.2% +15.7% +40% +40% +69% +Classism +78.7% +19.3% +20.8% +0% +54.2% +Exclusionary +83.6% +23.4% +12% +24% +64% +Gaslighting +56.5% +15.5% +16% +0% +44% +Misogyny +78.7% +22.2% +29.2% +8.3% +58.3% +Sarcasm +66.5% +33.7% +8% +0% +32% +Sexual Harassment +80% +21.9% +16% +4% +32% +Stereotyping +81.4% +31.7% +12% +0% +40% +Neutral +0.7% +10.4% +0% +0% +28% +False Positive +5.4% +80.9% +25% +25% +79.2% +Table 5: Average toxicity scores by SASS category of z-normalized human scores, PERSPECTIVE, and GPT-3 in +multiple settings. Note that the human and PERSPECTIVE scores are an average of continuous-valued scores, and +the GPT-3 results are an average of binary scores. +p(swear word | toxic) +p(toxic | contains swear word) +PERSPECTIVE +0.53 +PERSPECTIVE +1.0 +GPT-3-ZERO +0.14 +GPT-3-ZERO +0.33 +GPT-3-ONE +0.15 +GPT-3-ONE +0.22 +GPT-3-FEW +0.12 +GPT-3-FEW +0.83 +Table 6: Probabilities of “toxic” (score > 0.5 for PERSPECTIVE) given a text contains a swear word, and vice +versa. +System +Precision +Recall +F1-Score +PERSPECTIVE +0.40 +0.62 +0.48 +GPT-3-FEW +0.41 +0.69 +0.52 +Table 7: Evaluation of PERSPECTIVE and GPT-3-FEW on the task of binary toxicity classification on the Tweet- +Eval dataset. + diff --git a/sdAzT4oBgHgl3EQf6P79/content/tmp_files/load_file.txt b/sdAzT4oBgHgl3EQf6P79/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..56acaf428beab56e94219f47725f0193ab16ca8f --- /dev/null +++ b/sdAzT4oBgHgl3EQf6P79/content/tmp_files/load_file.txt @@ -0,0 +1,484 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf,len=483 +page_content='Critical Perspectives: A Benchmark Revealing Pitfalls in PerspectiveAPI Lorena Piedras∗ and Lucas Rosenblatt∗ and Julia Wilkins∗ lp2535@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='edu, lr2872@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='edu, jw3596@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='edu New York University Abstract Detecting “toxic” language in internet content is a pressing social and technical challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In this work, we focus on PERSPECTIVE from Jigsaw, a state-of-the-art tool that promises to score the “toxicity” of text, with a recent model update that claims impressive results (Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We seek to challenge cer- tain normative claims about toxic language by proposing a new benchmark, Selected Adver- sarial SemanticS, or SASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We evaluate PER- SPECTIVE on SASS, and compare to low-effort alternatives, like zero-shot and few-shot GPT- 3 prompt models, in binary classification set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We find that PERSPECTIVE exhibits troubling shortcomings across a number of our toxicity categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' SASS provides a new tool for evaluating performance on previously un- detected toxic language that avoids common normative pitfalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Our work leads us to em- phasize the importance of questioning assump- tions made by tools already in deployment for toxicity detection in order to anticipate and pre- vent disparate harms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 1 Introduction Social media and online discussion forums are plagued with toxic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 41% of American adults have personally experienced online harass- ment (Vogels, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' This problem can be further exacerbated by learning algorithms with unchecked incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' On a platform like Facebook, incendiary comments are highly visible due in large part to the design of its ranking algorithms (Munn, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Some experi- ments by Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (2021) have shown that users are more likely to write toxic or “troll” posts when the initial comments in a conversation exhibited toxic or “trolling” behavior, ∗Equal contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' This research was supported in part by NSF Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 1922658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' which can be interpreted as evidence that “trolling is contagious.” This creates the potential for a negative feedback loop where companies are incentivized to promote toxic content and toxic content in turn provokes users to post more malicious content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In response, researchers and corporations alike have invested in systems that can detect toxicity at massive scales to help moderate content on social media and internet forums (Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Pavlopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Wulczyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Some of these models have, however, been shown to be susceptible to simple adversarial attacks (Hosseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Gröndahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Text Toxicity (Perspective) Toxicity (Human) You look so good in that dress, and probably even better with it off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='9% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='0% Table 1: A toxicity score given by PERSPECTIVE to a SASS example from the “Sexual Harassment” category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Still, deep language model-based approaches to detecting toxicity have been deployed, with con- sumer facing API’s granting direct access to the model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' One such model is PERSPECTIVE (Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2022), which assigns text a “toxic- ity” score ∈ [0, 1], provided by Jigsaw (a Google subsidiary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In 2022, Jigsaw published findings on the “new generation” PERSPECTIVE toxicity model (currently deployed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We believe that interrogat- ing the model’s assumptions and frailties is vital work in moving it (and similar systems) towards robust detection (or, perhaps, a robust understand- ing of when detection fails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' For example, Jigsaw defines toxic language as “a rude, disrespectful, or unreasonable comment that is likely to make you leave the discussion” (Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2022), though other definitions exist (Märtens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We were unable to find an original source for Jigsaw’s definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='01874v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='CL] 5 Jan 2023 Text Category Type Toxicity Score Expected Toxic- ity Your secret can be our secret as long as you do what I tell you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Blackmail Unfiltered 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='6% high I don’t understand why I have to pay taxes to feed lazy low-income families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Classism Unfiltered 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='4% high We don’t admit people from your race in this forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Exclusionary Filtered 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='8% high You shouldn’t think that way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Gaslighting Filtered 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='3% high It’s weird how she’s taller than her husband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Misogyny Filtered 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='1% high No one should live without water, electricity and schooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Except for poor people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Sarcasm Filtered 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='8% high I can’t believe you work here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' You look too beautiful to do this job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Sex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Harass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Filtered 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='5% high Boys are better at math than girls, it’s just a fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Stereotyping Filtered 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='6% high It’s so freaking crazy that he surfed a 40-foot wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' False Positive Filtered 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2% low Their hair is so long, it’s almost touching the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Neutral Unfiltered 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='7% low Table 2: Toxicity scores from PERSPECTIVE for randomly selected examples in the 10 categories of SASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Contributions Existing models and benchmarks rely on aggregating binary responses to text col- lected from crowdworkers into a ground truth “probability of toxicity” (this is accomplished by prompting a crowdworker with “Is this text toxic?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', and then calculating the aggregate Pr[toxic] = |yes_responses| |total_responses|, which is the “toxicity score”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We suspect this method overemphasizes a normative understanding of toxicity, such that potentially toxic, harmful text “on the margins” goes unde- tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Here, “normative” describes the way in which multiple annotations are traditionally aggre- gated, which often implicitly supports the views of the majority and ignores the annotations of minor- ity groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In response, we isolate a set of natural language categories that fulfill the definition of tox- icity (as stated earlier), but go largely undetected, due in part, we believe, to the normative assump- tions of the ground truth toxicity examples from existing training and benchmark data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Again, these normative assumptions are related to the way data is aggregated, which may ignore the views of a minority of annotators in favor of the majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We present a new benchmark entitled Selected Adversarial SemanticS, or SASS, that evaluates these behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' SASS contains natural language examples (each approximately 1-2 sentences in length) across previously underexplored “toxicity” categories (like manipulation and gaslighting) as well as categories that have received attention (like “sexism” (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2019)), and includes a “hu- man” toxicity score ∈ [0, 1] for each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Ta- ble 1 shows an example from the "Sexual Harass- ment" category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' SASS follows a filtered/unfiltered approach to adversarial benchmarking, as in (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The benchmark is designed to exploit the normative vulnerabilities of a toxicity detection tool like PERSPECTIVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Specifically, PERSPEC- TIVE makes ambiguous claims that they can “iden- tify abusive [or toxic] comments” (Jigsaw), but do not clarify that these abusive comments are deter- mined by essentially using the majority opinion of random annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Our position is that PER- SPECTIVE should either be clear concerning the limitations of it’s toxicity tool (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' that it detects toxic content according to majority opinion), or adjust the PERSPECTIVE model to better account for minority annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We compare PERSPECTIVE’s performance on SASS to “human” generated toxicity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We further compare PERSPECTIVE to low-effort alter- natives, like zero-shot and few-shot GPT-3 prompt models, in a binary classification setting (“toxic or not-toxic?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=') (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Code for our project can be found in this repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 2 Related Work Past PERSPECTIVE Model Works such as (Hos- seini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2017) and (Gröndahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2018) fo- cused on generating adversarial attacks to test how the former version of PERSPECTIVE responded to word boundary changes, word appending, mis- spellings, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (Gröndahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2018) further tested how toxicity detection models responded to offensive but non-hateful sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The toxicity of the test sentences heavily increases when the word "F***" is added (You are great → You are F*** great, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='03 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' This opens up a discussion about the subjectivity of what should be considered “toxic”, a theme in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We pose new open questions that draw a clear connection between “toxicity” and normative concerns (Arhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Another promising approach to fortifying toxicity detectors is by probing a student model with a few annotated examples to detect veiled toxicity, mostly annotated incorrectly, from a pre- existing dataset, then re-annotating, thus making the model more robust (Han and Tsvetkov, 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' we do not attempt this in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Current Model A recent publication on PER- SPECTIVE (Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2022) generated bench- marks to test how the new version responded to character obfuscation, emoji-based hate, covert tox- icity, distribution shift and subgroup bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' They demonstrate improvements of the model in classi- fying multilingual user comments and classifying comments with human-readable obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Ad- ditionally, PERSPECTIVE beats every baseline on character obfuscation rates ranging from 0% to 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Character-level perturbations and distractors degrade performance of ELMo and BERT based toxicity models, reducing detection recall by more than 50% in some cases (Kurita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Sep- arate detection tools, like the HATECHECK sys- tem from (Röttger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2020), present a set of 29 automated functional tests to check identifica- tion of types of “hateful behavior” by toxicity or hate speech detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' A large dynami- cally generated dataset from (Vidgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2020), designed to improve hate speech detection during training, showed impressive performance increases in toxicity and hate speech detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Though slightly different in their typology of toxic speech, these approaches have a significant scale advantage over SASS, while SASS examples are specifically targeted at the PERSPECTIVE tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 3 Benchmarking with SASS The SASS benchmark contains 250 manually cre- ated natural language examples across 10 nuanced "toxicity" categories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' stereotyping, classism, blackmail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' These categories were selected via a process of literature review and vulnerability test- ing on PERSPECTIVE and other toxicity tools, to de- termine their weaknesses/strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' As we sought to challenge PERSPECTIVE and other toxicity tools, we believe this to be a sufficient process for deter- mining our categories, although acknowledge that it introduces some unavoidable author bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The examples are each 1-2 sentences long and are de- signed to exploit vulnerabilities in toxicity detec- tion systems like PERSPECTIVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Samples from SASS in each category are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Eight of SASS’s categories are aimed at gener- ating “False Negative” (FN) scores (a score that significantly underestimates the toxicity of some text), one category is aimed at “False Positive” (FP) scores (a score that overestimates toxicity), and one category is “Neutral,” a control, demonstrating the model’s performance on “normal,” non-toxic sen- tences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' SASS is heavily biased towards examples that generate a FN score, which we argue may be more harmful than a FP score, as a FN means toxic content has gone undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' For each category, the benchmark contains 15 “filtered” and 10 “un- filtered” examples, drawing inspiration from (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We generate filtered examples by brainstorming toxic comments and evaluating the comments with PERSPECTIVE to ensure a toxicity score of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Then, we generate an additional set of 10 examples per category using the knowledge gained from creating the filtered examples without first testing them on PERSPECTIVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Human Ground Truth The benchmark also contains a "human" toxicity score ∈ [0, 1] for each comment, which can be used as a baseline for eval- uating toxicity detection tools using SASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The hu- man toxicity scores are an average of the toxicity scores of the authors per comment (scored blindly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Here, we scored examples on a scale of 0-10, using Jigsaw’s definition of toxicity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' “how likely [the example is to] make [a user] leave the discussion” (0=highly unlikely, 10=highly likely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Significantly, we aligned these ratings with assumptions laid out in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2 (in appendix) for consistency and to com- bat benchmarking pitfalls (Blodgett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We further performed z-normalization, as per (Pavlick and Kwiatkowski, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Each author may have treated the “0-10 toxicity scale” differently, so this normalization process ensures that the final aggregate scores are not overly biased by any single author’s interpretation of the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In Table 5 (in the appendix), we observe the av- erage z-normalized human toxicity scores of com- ments in SASS across the toxicity categories de- scribed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We note that some categories are inherently more toxic than others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' “Stereotyping” comments have an average human toxicity score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='81 versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='57 for “Gaslighting” comments, which further contrasts with an average human tox- icity score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='007 for “Neutral” comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 4 Experiments and Discussion Binary Toxicity Classification We showcase the utility of SASS by evaluating PERSPECTIVE and GPT-3 against the human baseline in a binary classification setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' It’s important to note that PERSPECTIVE and GPT-3 are very different sys- tems, trained with distinct objectives, amounts and System Precision Recall F1-Score PERSPECTIVE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='08 GPT-3-ZERO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='31 GPT-3-ONE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='19 GPT-3-FEW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='61 Table 3: Evaluation of PERSPECTIVE and GPT-3 in multiple prompt settings on the SASS benchmark against thresholded human toxicity scores, in a binary classification setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' sources of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We believe the comparison is still useful because it provides a "low-effort alterna- tive" to make sure that our examples are not overly complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Note that GPT-3 was not fine-tuned explicitly for this task, so we prompt the system in zero, one, and few-shot settings for a binary toxic- ity classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We binarize the PERSPECTIVE and z-normalized human baseline toxicity scores by labeling scores > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='5 per comment as "toxic".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The binarized ground truth human labels on SASS contain 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='4% toxic labels versus 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='6% non-toxic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We use these thresholded human labels as ground truth and evaluate PERSPECTIVE and GPT-3’s performance on SASS in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Model Description PERSPECTIVE uses a Trans- former model with a state-of-the-art Charformer en- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The model is pretrained on a proprietary cor- pus including data collected from the past version of PERSPECTIVE and related online forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' This dataset is mixed in equal parts with the mC4 corpus, which contains multilingual documents (Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' GPT-3, created by OpenAI in 2020, is a state-of-the-art autoregressive transformer-based language model (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' GPT-3 is trained on a massive amount of internet text data, predominately Common Crawl and WebText2 (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2019), and generates human-like language in an open prompt setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Results We first observe that PERSPECTIVE per- forms very poorly on the binary task of toxicity classification on the SASS benchmark (Table 3, F1- Score = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Note that the majority of comments in SASS were crafted specifically to generate a low toxicity score from PERSPECTIVE, so this is not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We establish the metric regardless, as a baseline to evaluate future versions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We also examine the performance of GPT-3 in multiple prompt settings for binary (true/false) See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='1 for details on prompt generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Recall that “Neutral” and “False Positive” categories are inherently non-toxic, accounting for 20% of non-toxic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' https://commoncrawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='org/ toxic content classification in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Each system yields relatively high precision and low recall, gen- erally indicating a significant under-prediction of toxicity in SASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' GPT-3 has more success in clas- sifying harmful comments in SASS as toxic across the board relative to a thresholded PERSPECTIVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' GPT-3-FEW (F1-Score = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='61) shows a signif- icant improvement over both GPT-3-ZERO and GPT-3-ONE as well as PERSPECTIVE, yielding the most success relative to the human baseline of any of the experimental formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We hypothesize that GPT-3 outperforms PER- SPECTIVE largely due to the sheer scale and scope of data that GPT-3 is trained on, as well as the size of the model itself (175B learnable parameters in GPT-3 versus 102M in the PERSPECTIVE base model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' While GPT-3 is not trained for the toxicity detection task specifically, by learning from such a massive amount of internet text data spanning millions of contexts, the model has likely been ex- posed to a much wider range of potentially toxic material then PERSPECTIVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In Table 5 (see appendix), we break down the toxicity scores of PERSPECTIVE and GPT-3 by SASS category, relative to the human baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In some categories, both PERSPECTIVE and GPT- 3-FEW fall particularly short (for example, PER- SPECTIVE predicts an average toxicity score of 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='9% for “Sexual Harassment” comments versus the 80% human baseline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Relative to other cate- gories from SASS, PERSPECTIVE similarly rates comments in “Sarcasm” and “Stereotyping” as highly toxic, while humans rated the toxicity of “Stereotyping” comments significantly higher than those in “Sarcasm.” This raises the question of how to properly threshold scores from a toxicity detec- tion system in-the-wild, which (Lees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2022) do not comment on, though seems a reasonable use case for platforms flagging toxic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In the “False Positive” category we observe that both PERSPECTIVE and GPT-3-FEW yield very high toxicity scores on average (Table 5), suggest- ing that the models are overfit to swear word toxic- ity, and underfit to a deeper interpretation of mali- cious intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We believe it is important to delineate between the tasks of swear word detection and tox- icity detection, and so find this undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Allow- ing harmful comments to slip through the cracks is arguably more dangerous than unintentionally removing content with positive intent, but both of these scenarios could be upsetting to a downstream user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We report further on the influence of swear words on toxicity in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Profanity and Toxicity Detection SASS in- cludes 18 “False Positive” examples that con- tain swear words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' PERSPECTIVE rated all of them as toxic, and GPT-3-FEW labeled 83% of these comments as toxic (this is P[toxic|contains_swear_word]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' This suggests that, instead of understanding when swear words are used to communicate hateful content, PERSPEC- TIVE may be effectively memorizing their inclusion in toxic text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' This could be problematic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' swear words can be used to communicate non-toxic emo- tions, like surprise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Holy f*** I got the job!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=') or excitement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Oh sh**!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Congratulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=') and should not necessarily be treated equivalently to toxic speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Furthermore, different genders and races utilize profanity differently, so associating ex- pletives with toxicity could have disparate impacts (Beers Fägersten, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Past work by (Gröndahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2018) evaluating an older version of PER- SPECTIVE also detected this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' As shown in Table 6 (see appendix), from the 34 SASS examples that PERSPECTIVE rated as toxic, 52% contained a profanity, versus only 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='6% of the examples rated toxic by GPT-3-FEW (this is P[contains_swear_word|toxic]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' A lot of hate- ful content does not explicitly contain offensive words and it is troubling that PerpectiveAPI relies so much on them in our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' TweetEval We were surprised that GPT-3-FEW performed better in the binary classification sce- nario on the SASS benchmark than PERSPECTIVE, and so sought to validate the finding with another prominent toxicity benchmark, TweetEval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Thus we selected 1,000 examples from the ‘Hate Speech Detection” benchmark randomly (Barbieri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We acknowledge that this might be viewed as irrelevant or an unfair comparison, as some “toxic language” may not qualify as “hate speech” (for example, universal insults that do not target a specific group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' However, we believe that the reverse claim, that all “hate speech” should qual- ify as “toxic language” is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Then evaluating both PERSPECTIVE and GPT-3-FEW on a “hate speech” benchmark, despite both being designed to detect “toxic language,” is a valid comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We found that PERSPECTIVE had an F1-Score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='48 and GPT-3-FEW had an F1-Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='52 (Table 7, see appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The performance gap between PERSPECTIVE and GPT-3-FEW on TweetEval is significantly smaller than on SASS, but the trend (GPT-3-FEW matching or improving on PERSPEC- TIVE) is comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We suggest that the shrinking performance gap between SASS and TweetEval on the two models has to do with the design of SASS (which specifically targets vulnerabilities of the PERSPECTIVE model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Significantly, we were able to validate that GPT-3-FEW, in the binary setting, is a good point of comparison with PERSPECTIVE on another benchmark, and does not only perform well on SASS-specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Conclusion and Future Work We introduce Se- lected Adversarial SemanticS (SASS) as a bench- mark designed to challenge previous normative claims about toxic language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We have shown here that existing tools are far from robust to relatively simple adversarial examples, and fail to report ad- equately on the implicit biases attached to their model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We therefore position SASS as an important additional benchmark that can help us understand weaknesses in existing and future sys- tems for toxic comment detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Some impactful future work would be to grow the set of examples in SASS and to perform similar vulnerability test- ing on problems like sentiment analysis and other tools for content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Conducting a future study with a set of random human annotators and demonstrating that the majority rate SASS state- ments as non-toxic would strengthen our claims of normativity, and make the need for a benchmark like SASS even more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Expanding the set of state-of-the-art NLP toxicity detection or large language models evaluated on SASS would provide interesting future points of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Finally, we emphasize our belief that deployed natural lan- guage based tools, potentially serving millions of users, must be examined and reexamined in order to prevent the harmful beliefs of majority groups from being perpetuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 5 Ethical Considerations SASS, the new benchmark proposed in this paper, seeks to address normative claims made by toxic- ity detection tools that rely on majority opinion to determine malicious content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In the narrow scope of improving toxicity model evaluation, we thus expect SASS to have a positive impact on the NLP community, and by extension on moderation sys- tems for social media and online forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' However, thousands of content moderators, whose job descriptions include toxic content de- tection, are currently employed by companies such as Meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We believe that the best systems for toxic content detection are likely collaborations between humans and machines, but acknowledge that, by improving automated systems, we may jeopardize employment for these people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Still, it is unclear that content moderation is a task that people should take part in, and automating toxicity detection may reduce the exposure of people to harmful content that could have severe mental health consequences (Steiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' There is always the risk that, in providing a new benchmark to the larger NLP community, some may use it to make unjustified claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Therefore, we take this opportunity to highlight the ways in which SASS could be misused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We acknowledge that any benchmark, especially a relatively small one like SASS, will reflect the inherent biases of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Each category of SASS is not designed by any means to be exhaustive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' rather, each is designed to provide an initial probe, a check for model vul- nerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Further exploration would be required even if a model performed well on SASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' SASS is also only an English language benchmark, and con- tains examples that only make sense in an Ameri- canized cultural context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We believe it is important work to create similar benchmarks for other lan- guages and cultural contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We would like to thank Sam Bowman and Richard Pang for very useful conversations and feedback over the course of our project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We would also like to thank Julia Stoyanovich and the Cen- ter for Responsible AI at NYU for supporting our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' References Kofi Arhin, Ioana Baldini, Dennis Wei, Karthikeyan Natesan Ramamurthy, and Monin- der Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Ground-truth, whose truth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='– examining the challenges with annotating toxic text datasets.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Learning from the worst: Dy- namically generated datasets to improve online hate detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='15761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Emily A Vogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The state of online harassment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Pew Research Center, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Ellery Wulczyn, Nithum Thain, and Lucas Dixon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Ex machina: Personal attacks seen at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In Proceedings of the 26th international conference on world wide web, pages 1391–1399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' A Example Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='1 Generating prompts for GPT-3 As described in Section 4, GPT-3 was not ex- plicitly trained to classify phrases as toxic or not toxic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' We explore zero, one, and few-shot settings for prompting GPT-3 to generate results compara- ble to a thresholded PERSPECTIVE toxicity rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Here we describe the prompts used in these experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Setting Prompt GPT-3- ZERO “True or False, the phrase {phrase}’ is toxic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Answer:” GPT-3-ONE “Determine if the given phrase is toxic: Men are superior to women in every way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' => True {phrase} => ” GPT-3-FEW “Determine if the given phrase is toxic: Men are superior to women in every way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' => True Grilling with the neighbors is my favorite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' => False {phrase} => ” Table 4: Prompts used for zero, one, and few-shot ex- perimentation with toxicity classification using GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The samples given in the zero and few-shot settings are not present in the SASS benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' In the future, it would interesting to explore prompting GPT-3 for this task in different ways, including prompting with only toxic comments, using more examples in the few-shot setting, and prompting using examples from SASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Additionally, note that in Section 4 in the binary toxicity classification experiment, for each prompt setting, we take the mode of predicted labels across 3 runs due to minor variability in responses and use that to evaluate GPT-3 in each setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Intriguing future lines of work here could include quantifying the variability across experiments with GPT-3 and analyzing how the prompt settings and prompts themselves affect this variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2 Designing SASS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='1 Avoiding Conceptual and Operational Pitfalls (Blodgett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2021) describe the ways in which popular stereotype detection benchmarks suffer from a set of conceptual and operational pitfalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' By providing a taxonomy of potential pitfalls, they are able to audit the methods in a principled manner and deduce ways in which the benchmark may pro- duce spurious measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Here are summaries of each category of pitfall they describe (specific to stereotyping): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Conceptual Pitfalls (stereotyping) (a) Power dynamics The claimed problem- atic power dynamic may not be “realis- tic.” (b) Relevant aspects Must be clear and con- sistent about what stereotype content is within the purview of a given example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (c) Meaningful stereotypes Is this stereo- type actually reflective of a societal prob- lem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (d) Anti vs non-stereotypes Some state- ments can negate a stereotype (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' not), while others can actively combat (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' evil vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' peaceful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (e) Descriptively true statements A true statement masquerading as a stereotype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (f) Misaligned stereotypes A hyper spe- cific, or not specific enough, stereo- type about a certain group/subgroup (“Ethiopia” in a context where Africa generally is implied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (g) Offensive language Are swear words stereotyping?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Operational Pitfalls (stereotyping) (a) Invalid perturbations Not a real stereotype/anti-stereotype (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' both al- ternate sentences are stereotypes) (b) Incommensurable groups or at- tributes Two alternate groups are not comparable (think apples and oranges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (c) Indirect group identification I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' us- ing names as a way of identifying group membership (for example, racially iden- tifying names) (d) Logical failures If the alternate repre- sents a logically dubious conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (e) Stereotype conflation Multiple stereo- types present in a single example (f) Improper sentence pairs The example is not “realistic.” (g) Text is not naturalistic The text itself would never be written/uttered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (h) (Un)markedness The two examples are represented at different degrees in natural text (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' “young gay man” vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' “young straight man”) (i) Uneven baselines Similar to (un)markedness, examining a false alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' The stereotyping benchmarks from (Blodgett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2021) are fundamentally different than SASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Thus, our analysis of pitfalls must rely on slightly different criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Using the aforementioned cri- teria, we created an abbreviated conceptual and operational pitfall taxonomy for toxicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2 Conceptual and operational pitfalls in toxicity benchmarks Recall that the definition of toxicity according to PERSPECTIVE/Jigsaw is: “a rude, disrespectful, or unreasonable comment that is likely to make you leave the discussion.” With this definition, we can begin to construct a set of pitfalls that text from a benchmark might ex- hibit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' However, in order to minimize subjectivity as much as possible, we outline three major assump- tions about examples in our benchmark SASS (and therefore, about what we prescribe as the behavior of a system that “detects toxicity”): Assume adversarial reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Within reason, does there exist an individual or group that would be likely to leave a discussion after reading a piece of text (even if they represent a significant minor- ity)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Assume adversarial context/subtext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Assume that the possible context in which a piece of text is positioned increases the likelihood that someone would leave the discussion after reading it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Assume bad intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Assume that the writer of the text was knowingly malicious in their choice of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' These assumptions are important because they help make our analysis structured and consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Here are the pitfalls we use in evaluating toxicity, constructed from (Blodgett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=', 2021): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Conceptual Pitfalls (toxicity) (a) Meaningful toxicity (from Meaning- ful stereotypes) Is the text likely to make an individual leave a discussion, given our assumptions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (b) Descriptively true statements Is the text true/factual?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (c) Offensive language Is the text toxic purely due to swear words?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (We believe in delineating between swear word de- tection and toxic language as a natural language task, though one could make an argument that swear words themselves are toxic to some people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' It is not clear how to resolve this conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Operational Pitfalls (toxicity) (a) Invalid toxicity markers (from invalid perturbations, incommensurable groups, and logical failures) Does the text properly signify something that is rude/disrespectful/unreasonable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (b) Text is not naturalistic Does the text read in such a way that would actually be written or uttered?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' (c) (Un)markedness (/uneven baselines) Does the text appear in a statistically likely/comparable pattern?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='3 Full benchmark code: Code for our benchmark and evaluations can be found here: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='com/lurosenb/sass Category Human PERSPECTIVE GPT-3-ZERO GPT-3-ONE GPT-3-FEW Blackmail 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='7% 40% 40% 69% Classism 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='7% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='3% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='8% 0% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2% Exclusionary 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='6% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='4% 12% 24% 64% Gaslighting 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='5% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='5% 16% 0% 44% Misogyny 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='7% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='3% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='3% Sarcasm 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='5% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='7% 8% 0% 32% Sexual Harassment 80% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='9% 16% 4% 32% Stereotyping 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='4% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='7% 12% 0% 40% Neutral 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='7% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='4% 0% 0% 28% False Positive 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='4% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='9% 25% 25% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='2% Table 5: Average toxicity scores by SASS category of z-normalized human scores, PERSPECTIVE, and GPT-3 in multiple settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' Note that the human and PERSPECTIVE scores are an average of continuous-valued scores, and the GPT-3 results are an average of binary scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' p(swear word | toxic) p(toxic | contains swear word) PERSPECTIVE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='53 PERSPECTIVE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='0 GPT-3-ZERO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='14 GPT-3-ZERO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='33 GPT-3-ONE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='15 GPT-3-ONE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='22 GPT-3-FEW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='12 GPT-3-FEW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='83 Table 6: Probabilities of “toxic” (score > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='5 for PERSPECTIVE) given a text contains a swear word, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content=' System Precision Recall F1-Score PERSPECTIVE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='48 GPT-3-FEW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} +page_content='52 Table 7: Evaluation of PERSPECTIVE and GPT-3-FEW on the task of binary toxicity classification on the Tweet- Eval dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf'} diff --git a/vNFLT4oBgHgl3EQfjy8P/content/tmp_files/2301.12112v1.pdf.txt b/vNFLT4oBgHgl3EQfjy8P/content/tmp_files/2301.12112v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1c0e4e268330c1c8f69a7ddf579adec62edc3b1 --- /dev/null +++ b/vNFLT4oBgHgl3EQfjy8P/content/tmp_files/2301.12112v1.pdf.txt @@ -0,0 +1,2092 @@ +ON PRE-TRAINED LANGUAGE MODELS FOR ANTI- +BODY +Danqing Wang∗ +Department of Computer Science +University of California, Santa Barbara +danqingwang@ucsb.edu +Fei Ye +ByteDance AI Lab +ByteDance +yefei.joyce@bytedance.com +Zhou Hao∗ +Insititute for AI Industry Research +Tsinghua University +zhouhao@air.tsinghua.edu.cn +ABSTRACT +Antibodies are vital proteins offering robust protection for the human body from +pathogens. The development of general protein and antibody-specific pre-trained +language models both facilitate antibody prediction tasks. However, few studies +comprehensively explore the representation capability of distinct pre-trained lan- +guage models on different antibody problems. Here, to investigate the problem, +we aim to answer the following key questions: (1) How do pre-trained language +models perform in antibody tasks with different specificity? (2) How many ben- +efits will the model gain if we introduce the specific biological mechanism to +the pre-training process? (3) Do the learned antibody pre-trained representations +make sense in real-world antibody problems, like drug discovery and immune +process understanding? Previously, no benchmark available largely hindered the +study to answer these questions. To facilitate the investigation, we provide an +AnTibody Understanding Evaluation (ATUE) benchmark. We comprehensively +evaluate the performance of protein pre-trained language models by empirical +study along with conclusions and new insights. Our ATUE and code is released at +https://github.com/dqwang122/EATLM. +1 +INTRODUCTION +Antibodies are special type of proteins extensively used as diagnostic and therapeutic tools against +diverse diseases, such as SARS-CoV-2 (Zhu et al., 2022). Deciphering the information stored in +antibody sequences is highly important with regard to real-world therapeutic antibody development +and immune understanding (Greiff et al., 2020; Lu et al., 2018; Yermanos et al., 2018). Fortunately, +the recent progress of general Pre-trained Protein Language Models (PPLM) and specific Pre-trained +Antibody Language Models (PALM) provide new prospects for antibody-related tasks. For example, +many studies have shown that the representations learned by PPLMs are promising to transfer to +antibody tasks (Kim et al., 2021; Zaslavsky et al., 2022). For PALMs, they have been shown to +improve model performance in antibody paratope predictions (Leem et al., 2022). +Despite the current success, few studies comprehensively investigate the representation capability of +different pre-trained language models (e.g. general PPLM and specific PALM) on distinct antibody +tasks, which limits our ability to design better architectures that can help antibody discovery and +modification. For example, in Figure 1, we compare the performance of the pre-trained protein +language model ESM (Rives et al., 2021), the pre-trained antibody language model AntiBERT (Leem +et al., 2021), and the model trained from scratch on three antibody tasks. These three tasks range +from low to high in terms of antibody specificity. Here, specificity refers to the antibody’s evolution, +for example, the antibody evolves to obtain the ability to bind antigen (The definition is discussed in +detail in Section 3.1 and Appendix A.1). +∗Work was done in ByteDance. +1 +arXiv:2301.12112v1 [cs.CL] 28 Jan 2023 + +Performance
Increase
(%) +Antibody
Specificty
Relatedness +Low +Medium +High +0 +10 +‑10 +No
Pretrain +ESM‑1 +AntiBERT +EATLM +Figure 1: Performance of pre-trained language +models on tasks with different specificity. ESM- +1 belongs to PPLMs and AntiBERT belongs to +PALMs. EATLM is the method with a specific +antibody mechanism introduced. +It can be observed that although ESM performs +well in task that has low relevance with antibod- +ies, the performance dramatically degrades in +tasks of higher relevance. Besides, AntiBERT +does not show clear advantages over the non- +pretrain model in the high-specificity task. The +results point out the weaknesses of current pre- +training language models for antibody studies: +Direct adaptation of general PPLM representa- +tions may damage the performance; Pre-training +strategies for PALMs cannot fit the specific an- +tibody biological function well. All of these +call for a comprehensive guideline of model de- +signing for different antibody tasks. Mainly, we +focus on answering the following questions: +(I) How do pre-trained language models per- +form in antibody tasks with different speci- +ficity? Addressing of the question is mainly hindered by two challenges: the lack of a reliable +antibody-specific benchmark for performance evaluation and a comprehensive study of current +PPLMs and PALMs. (II) How many additional benefits will the model gain if the biological +mechanism is introduced to the pre-training process? Antibody-specific evolution has been em- +ployed by many computational biology studies and shows promising results in antibody-related +tasks, such as disease diagnosis and therapeutic antibody development (Yermanos et al., 2018; Miho +et al., 2019). Then, it is interesting to know whether antibody representation learning can benefit +from the incorporation of antibody-specific evolution information. (III) Does the learned antibody +pre-trained representations make sense in real-world antibody problems, like drug discovery +and immune process understanding? Since antibody is so important that is extensively used for +drug development in the real world, it is interesting to know whether pre-trained representation can +indeed help biologists to understand antibody functions or discover drugs. +To investigate these questions, we first propose antibody study benchmark AnTibody Understanding +Evaluation (ATUE). This is the first antibody benchmark with four real-world supervised tasks +covering therapeutic antibody engineering, B cell analysis, and antibody discovery. To evaluate +models on different aspects of antibody biological functions, these tasks are designed to have +specificity ranging from low to high. Based on ATUE, we perform comprehensive empirical studies +to investigate the representation ability of distinct pre-trained language models. We also explore +the influence of involving specific biological mechanisms in antibody pre-training by introducing +two simple objectives to tailor masked language modeling for evolution: (1) Ancestor germline +prediction guides the model to discriminate the evolutionary relationship between antibody and +ancestral sequences. (2) Mutation position prediction mimics hypermutation during the evolution. +The method is used as an analytic tool to investigate the representation ability of antibody evolution- +tailored language model. Finally, we take a close look at the SARS-CoV-2 antibody discovery to +investigate the pre-trained representation under a real-world scenario. +Our contributions can be divided three-fold. +• For facilitating antibody application studies, we develop the first comprehensive antibody bench- +mark to benefit the community. We also introduce two objectives tailored for antibody evolution. +• For providing guidelines for a better antibody representation, we have the following key observa- +tions: (i) PPLMs perform well on antibody tasks that have a high relationship with structure, such +as the binding with antigen. However, they perform worst in tasks with high antibody specificity, +indicating representation benefiting protein structure prediction is harmful to antibody-specific +tasks. (ii) In most cases, PALMs perform as well as or even better than PPLMs with less pre- +training data. (iii) PALMs can be improved by incorporating the evolution process. However, +the evolution information from MSAs does not always benefit the antibody tasks. (iv) The intro- +duction of two antibody biological mechanisms facilitates PALMs with more antibody-specific +features and improves model performance in the task with high antibody specificity. This is the +first attempt showing how antibody specific evolutionary information can be incorporated in +pre-training language model. +2 + +• For accelerating real-world antibody discovery, we identified 11 potential SARS-CoV-2 binders +whose sequences are highly identical to existing therapeutic antibodies binding with the virus. +2 +RELATED WORK +Our work focuses on researching the effectiveness of protein and antibody pre-trained language +models for antibody-specific tasks. Below we review the representative existing methods. We list the +details in Table 1. +Pretrained Protein Language Models (PPLMs) There is an increasing interest in exploring large- +scale language models using protein sequences (Rao et al., 2019; Madani et al., 2020; Meier et al., +2021). These models have been shown to achieve state-of-art capacity in predicting protein structure +and function. ProtTrans (Elnaggar et al., 2021) and ESM-1b (Rives et al., 2021) take individual +protein sequences as input and adopt Transformer language models for pre-training, demonstrating +that self-supervision is a promising paradigm for protein secondary structure, contact, homology +predictions, and function prediction. To extract evolutionary information from protein sequences, Rao +et al. (2021) proposed the MSA-transformer/MSA-1b model utilizing multiple sequence alignment +(MSA) instead of a single query sequence as input. This model is superior to ESM-1b for structure +prediction, demonstrating evolution information can benefit protein representation learning. Despite +the progress in the field, few studies reported protein PLM transfer learning results on antibody tasks. +Pretrained Antibody Language Models (PALMs) Encouraged by the success of PLMs in protein +representation learning, series work seeks to learn antibody representations based on sequences of +antibodies. (Leem et al., 2021; Ruffolo et al., 2021; Olsen et al., 2022b; Prihoda et al., 2022; Li +et al., 2022). AntiBERTy (Ruffolo et al., 2021) proposed the first antibody-specific language model, +exploring a Transformer trained on 558M natural antibody sequences in the OAS database. The +case study in the work showed the representation obtained from the PLM is useful for antibody +sequence clustering into trajectories resembling affinity maturation. Olsen et al. (2022b) train two +language models for antibodies. A heavy chain version Ablang-H is trained on 14M sequences and a +light chain version Ablang-L on 0.19M sequences. The study reported transfer learning results on +restoring missing residues of antibody sequences, which is a task similar to pre-training objectives. +AntiBERTa (Leem et al., 2021) train the antibody language model on OAS. Although this paper claims +finetuning AntiBERTa for paratope position prediction can achieve state-of-the-art performance, +the experimental results lack standard deviations, making it unclear how significant the results +obtained are. Recently, Li et al. (2022) proposed a antibody-specific language model and explored +its performance in SARS-CoV-2 antigen binding, showing context-dependent representations of +antibody sequences benefit binding prediction. +Table 1: Pre-training language models for protein and antibody. +Evolution denotes whether +evolutionary-related sequences are used during the pretraining. MLM is masked language modeling +pretraining objective. HC, antibody heavy chain; LC, antibody light chain. +Model +Category +Dataset +Evolution +Objective +Antibody Tasks +ESM-1 (Rives et al., 2021) +PPLM +UniRef50 (27M) +× +MLM +- +MSA-1b (Rao et al., 2021) +PPLM +UniRef50 (26M MSAs) +✓ +MLM +- +Ablang-H (Olsen et al., 2022b) +PALM +OAS (14M HC) +× +MLM +Reconstruction +Ablang-L (Olsen et al., 2022b) +PALM +OAS (0.19M LC) +× +MLM +Reconstruction +AntiBERTa (Leem et al., 2021) +PALM +OAS (72M) +× +MLM +Paratope Prediction +EATLM +PALM +OAS (20M) +✓ +MLM, AGP & MPP +ATUE +3 +FRAMEWORK +In this section, we first give a brief introduction to the antibody and its specific evolution. Then +we propose the first antibody-specific benchmark (ATUE) composed of four tasks with different +specificities. Finally, we implement several PPLMs and PALMs baselines and design an evolution- +aware PALM to incorporate the biological mechanism into the pre-training process. +3 + +3.1 +BACKGROUND +Antibody Antibodies are vital proteins generated by the immune system to remove harmful foreign +pathogens in the human body. they can specifically bind to antigens on the pathogen and recognize +it. Antibodies are composed of two identical heavy chains and two identical light chains and form +a large Y-shaped structure. Two tips on it contain highly variable loops, called Complementarity +Determining Regions (CDR), which function for antigen binding. +Antibody Specific Evolution Notably, the antibody evolution process is significantly different from +that of proteins, providing a good opportunity for us to investigate the impact of general PPLMs on +specific subdomains. To perform its protective function, the antibody sequence undergoes evolution +selection to search for optimal patterns that can specifically recognize pathogens (Honjo & Habu, +1985). Deciphering the information stored in antibody sequences may benefit our understanding of +disease and accelerate therapeutic antibody development (Greiff et al., 2020; Lu et al., 2018; Yermanos +et al., 2018). During evolution, the random recombination of V/D/J-gene segments provides the +initial diversity for the ancestor sequence (germline). Upon exposure to a pathogen, this sequence +undergoes frequent sequence mutations to search for progeny antibody sequences with optimal +binding specificity. In other words, gene recombination provides millions of germlines in the human +body, and the germlines further mutate into a huge number of progeny antibodies. Thus, the ancestor +relationship between an antibody and its corresponding germline as well as the mutation it undergoes +together determine the unique biological functions. In brief, the evolutionary relationships between +antibodies arise to gain new functions such as antigen binding. It is significantly different from that of +proteins, which are to certain functions across different organisms. We further illustrate this process +in Figure 7 in Appendix A.1. +Unsupervised Antibody Corpus To obtain the evolutionary information of antibody sequences, we +utilize Observed Antibody Space (OAS), a database containing more than 1.5 billion natural antibody +sequences (Kovaltsuk et al., 2018; Olsen et al., 2022a) The antibody sequences in the database have +been precisely annotated with evolutionary and structural information, including the paired germline +and CDR3 for each antibody. To pair the antibody with its germline used in the pretraining task, we +used the annotated sequences provided in the OAS database. Further information on data processing +can be found in Appendix A.2. +3.2 +ANTIBODY UNDERSTANDING EVALUATION (ATUE) +We provide four biologically relevant downstream prediction tasks to serve as antibody benchmarks, +covering four major application aspects for antibodies in the real world: therapeutic antibody +engineering, disease diagnostics, antibody discovery, and B cell maturation analysis. Notably, the +antibody evolution relatedness of these tasks ranges from low to high, offering scaled tasks with +subdomain specificity for pre-trained language model evaluation. Detailed information is listed in +Figure 2. All data are publicly open and used under the right license. For each task, we focus on the +following aspects and leave the details in Appendix: +[Definition] The formal definition of the task and the understanding ability required. +[Impact] The importance of the task in the biological area. Details in Appendix A.3. +[Dataset] The data source and size. Details of data processing in Appendix A.2 +[Specificity] Antibody’s specific evolution characteristics that is different from general pro- +teins. The definition of antibody specific evolution is shown in Appendix A.1. Quantitative +analysis the scale of antibody specificity for every task is shown in Appendix A.4. +We use several classification metrics to evaluate the performance. Accuracy (ACC) calculates the +ratio of correct predictions. Matthews Correlation Coefficient (MCC) is the coefficient between true +and predicted values. F1 is the average weighted score of precision and recall. AUC is the area under +the ROC curve, which shows the performance at all classification thresholds. +Antigen Binding Prediction is a binary sequence classification task to determine whether the CDR +region of the antibody can bind to the specific antigen. +[Impact] A better understanding of the binding affinity between antibody and antigen can +accelerate the affinity optimization of therapeutic antibodies. +[Dataset] We collect the antigen binding data from (Mason et al., 2021) and follow the +training/validation/test split of 15,128/3,242/3,242. The paratope data is collected from +4 + +D.
Antibody
discovery +Q2
classification +C.
B
cell
classification +Q6
classification +maturation +SARS‑CoV‑2 +virus +000010110100 +A.
Antigen
Binding +Q2
classification +... +Bind +B.
Paratope
Prediction +Sequence
labeling +Task
Specificity +High +Low +Figure 2: Antibody prediction tasks. The specificity of tasks ranges from low to high. +(Liberis et al., 2018) with 1,662 CDR segments on 277 antibodies. +[Specificity] Low. All the antibodies sequence in the dataset are derived from a single +germline sequence indicating the task is not antibody-specific evolution-related. +Paratope Prediction is to identify binding positions on the antibody sequence, which is a sequence +labeling task to predict a 0/1 label for each residue of CDR fragments. +[Impact] The exploration of paratope (binding positions between antibody and antigen) can +help to understand the binding mechanisms of therapeutic antibodies. +[Dataset] The paratope data is collected from (Liberis et al., 2018) with 1,662 CDR segments +on 277 antibodies. +[Specificity] This task is medium specificity related because only partial antibodies from the +database are derived from evolution. +B Cell Maturation Analysis It is a 6-category classification task to distinguish the maturation stage +of B cell antibody sequences. Each sequence belongs to one of {immature, transitional, mature, +plasmacytes, memory IgD+, memory IgD-}. It requires the model to learn a representation sensitive +to different maturation states. +[Impact] It benefits the understanding of the mechanism during immune evolution, which is a +critical biological process in the immune system affecting the function and antigen specificity +of antibodies (Ghraichy et al., 2021; Meffre et al., 2000). +[Dataset] We collect 88,094 sequences from (Mroczek et al., 2014). +[Specificity] High. Antibody evolution is highly coupled with B cell maturation. (Meffre +et al., 2000) +Antibody Discovery The task is a binary sequence classification task to distinguish which antibody +is directly responsible for SARS-CoV-2 binding. The task is highly challenging from two aspects: (1) +Less than 1% of antibodies from SARS-CoV-2 patients are directly responsible for virus binding. (2) +It is hard to get a reliable sequence-level classifier using unreliable and noisy individual-level labels. +[Impact] Antibody discovery from B cell repertoire has been widely recognized as a important +approach to accelerate antibody discovery for diverse antigens (Weiner, 2015; Pedrioli & +Oxenius, 2021), and achieved great success for SARS-CoV-2 antibody discovery (Kovaltsuk +et al., 2018; Cao et al., 2020; Shiakolas et al., 2022). +[Dataset] We collected antibody sequences from 133 SARS-CoV-2 patients and 87 health +persons from OAS and followed the processing pipeline of (Kim et al., 2021). Inspired +Zaslavsky et al. (2022), we match the high-ranked sequences with the sequences in the CoV- +AbDab (Raybould et al., 2021) database, which have been proved to bind SARS-CoV-2 using +wet-lab experiments. +[Specificity] High. It is widely reported antibodies derived from the same disease such as +SARS-CoV-2 share strong convergent germline signals (Galson et al., 2020). +3.3 +EXPERIMENT SETUP +Based on the antibody benchmark ATUE, we evaluate the performance of current pertaining language +models in different specificity tasks. Furthermore, to investigate the benefit of introducing the +5 + +XXbiological mechanism, we incorporate evolution information as the extra pretraining objectives for +PALMs and propose EATLM. The detailed description can be found in Appendix A.5 +Current Pre-trained language models Existing antibody and protein language models are summa- +rized in Table 1. Since the code and pre-training data of AntiBERTa are not released, we train a +BERT model named AntiBERT on the full OAS database following the same setting as the original +study. MSA-1b (Rao et al., 2021) takes protein-specific evolutionary sequences (Multiple Sequence +Alignment, MSA) as the input. Because it is hard to align sequences between antibodies due to the +diversity of CDR3, we take the germline and create pseudo-MSAs with depth 2. We add a linear +layer on top of the language models and finetune the whole model on the downstream tasks. +Evolution-aware antibody pretraining method To incorporate the biological mechanism into the +pre-training, we propose a model with evolution information: Antibody EvoluTion-aware pretraining +Language Model. The antibody can be represented as A and the germline of the individual antibody +can be represented as G. Typically, PALMs are trained with basic masked language modeling (MLM). +Based on it, we design another two pre-training objectives to simulate the biological mechanism of +antibody evolution. The evolutionary relationship between the antibody and its germline includes +two folds: (i) Whether the antibody and the germline have an evolutionary relationship. (ii) How +to mutate residues from the germline to get the specific antibody. Two evolution-related objectives +are introduced to solve the above questions: ancestor germline prediction (AGP) and mutation +position prediction (MPP). For ancestor germline prediction, we substitute the paired germline G +with random germline G′ in the batch via a probability p. The model is made to distinguish the +ancestor germline of the antibody by capturing the shared features. To predict mutation position, for +each token in the germline G, the objective is to predict a 0/1 label for each token to indicate whether +this token has been mutated. For the antibody sequence S, we mask the mutation position and predict +these tokens. The technical details are described in Appendix A.5.1. +Hyper-parameters We use the base Transformer architecture (Vaswani et al., 2017) with 12 layers, +12 heads, and 768 hidden states. For each task in ATUE, we finetune the model with supervised +data. We follow the standard split of Antigen Binding Prediction. For other tasks that do not +provide a standard split, we use a 10-fold cross-validation. Since our pre-training model learns +the representation of the antibody sequence, we expand the CDR fragment to the full antibody by +searching the biological database for therapeutic antibody engineering tasks. We also use the same +Transformer architecture to train from scratch for each downstream task. This model is indicated as +non-pretrain since it is not pre-trained on a protein/antibody database. +Reproduction We conduct 10-fold validation on paratope prediction, B cell maturation analysis, and +antibody discovery. For antigen binding prediction, we conduct three repetitive experiments with +different random seeds. We report the average results and the standard derivation. The benchmark +and code will be released. +4 +RESULTS AND ANALYSIS +4.1 +SUPERVISED ANTIBODY PREDICTION +Antigen binding +Here, we evaluate the performance PLMs models for antibody binding and +paratope prediction, which are less antibody specific. The results are shown in Table 2 It is observed +that PPLMs and PALMs achieve comparable performance on this task, indicating PALMs can learn +similar general protein representation as PPLMs. Among different PALMs, Ablang-H outperforms +Ablang-L and AntiBERT. It indicates that separate training for heavy and light chain sequences is +helpful for this task. The introduction of AGP and MPP provides a little improvement over AUC and +F1 metrics. +Paratope prediction +As the results shown in Table 2, for paratope prediction, both PPLMs and +PALMs can significantly boost the prediction accuracy over the model with pre-training. However, +PALM doesn’t show much significant advantage over PPLMs. EATLM achieves the best performance, +especially on F1 and MCC, while other pertaining models have a high recall and a low precision. +It indicates that those models tend to predict more residues as the binding site. However, with the +introduction of mutation residue prediction, EATLM can focus on the mutated positions that are +6 + +Table 2: Performance of PPLMs and PALMs on antibody tasks with increasing specificity. The +reported results are the average of repetitive experiments with the standard derivation. EATLM w/o +AGP indicates that we remove AGP objective from the pre-taining phase. +Antigen Binding (Low) +Paratope (Medium) +Cell (High) +AUC +F1 +MCC +AUC +F1 +MCC +ACC +non-pretrain +0.858±0.014 0.584±0.330 0.432±0.183 0.845±0.014 0.605±0.033 0.463±0.037 0.554 ± 0.042 +ESM-1 +0.917±0.001 0.854±0.002 0.689±0.002 0.886±0.009 0.669±0.024 0.547±0.026 0.503 ± 0.031 +MSA-1b +0.921±0.001 0.857±0.004 0.689±0.014 0.887±0.009 0.679±0.019 0.557±0.025 0.416 ± 0.050 +Ablang-H +0.918±0.001 0.861±0.003 0.704±0.010 0.878±0.009 0.674±0.018 0.546±0.023 0.570 ± 0.010 +Ablang-L +0.917±0.002 0.856±0.001 0.682±0.001 0.882±0.010 0.680±0.018 0.553±0.023 0.546 ± 0.008 +AntiBERT +0.918±0.003 0.843±0.009 0.678±0.008 0.879±0.011 0.690±0.020 0.559±0.026 0.565 ± 0.028 +EATLM +0.922±0.004 0.862±0.004 0.699±0.010 0.887±0.008 0.698±0.017 0.575±0.024 0.581 ± 0.005 +EATLM w/o AGP +0.920±0.003 0.855±0.000 0.697±0.008 0.883±0.011 0.676±0.021 0.552±0.027 0.559 ± 0.012 +EATLM w/o MPP +0.923±0.001 0.855±0.002 0.687±0.005 0.883±0.011 0.691±0.022 0.566±0.030 0.563 ± 0.010 +EATLM w/o AGP & MPP 0.918±0.001 0.845±0.005 0.681±0.005 0.880±0.009 0.674±0.018 0.552±0.023 0.559 ± 0.009 +adapted to bind with antigen. Among the two PPLMs, MSA-1b outperforms ESM-1 on F1 and MCC, +which benefits from the structure information learning from MSAs. +Immature B cell +Transitional B cell +Mature B cell +Plasmacytes PC +Memory IgD- +Memory IgD+ +Immature B cell +Transitional B cell +Mature B cell +Plasmacytes PC +Memory IgD- +Memory IgD+ +0.896 +0.042 +0.015 +0.016 +0.016 +0.015 +0.037 +0.406 +0.168 +0.18 +0.048 +0.16 +0.021 +0.133 +0.455 +0.182 +0.053 +0.156 +0.018 +0.083 +0.126 +0.598 +0.05 +0.126 +0.026 +0.079 +0.138 +0.159 +0.502 +0.096 +0.018 +0.122 +0.189 +0.173 +0.051 +0.446 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Figure 3: B cell evolution category prediction. +For each pij in i−th row and j-th column, it +means the frequency for the model to predict +the antibody in i category to j category. The +number is normalized by row. +B Cell Analysis +In this task, we examine the repre- +sentation capacity of different pre-trained language +models for distinguishing different B cell mature +states during evolution. The results are shown in +Table 2. In general, we find PPLMs can hardly distin- +guish the minor differences between B cell sequences +by showing compromising results. Both the perfor- +mance of ESM-1 and MSA-1b is significantly worse +than randomly initialized models. MSA-1b performs +the worst in all pre-trained language models, indi- +cating representation that can perform well in pro- +tein structure prediction will be harmful to antibody- +specific tasks. Conversely, all PALMs show promis- +ing results for the task. The reason may lie in that the +general protein has little relationship with the special +antibody mature process and can hardly capture this +feature during the protein pretraining process. EATLM significantly outperforms the other PALMs. +This is because by explicitly modeling the biological mechanism, our model can effectively capture +the evolution feature and better distinguish between B cells at different stages of maturation. +We conduct further analysis to figure out whether our EATLM successfully captures sequence char- +acteristics during the evolutionary process. We explore the probabilities of predicting antibodies in +class i to class j. The results shown in Figure 3 reveal EATLM can easily classify the immature B +cell with an accuracy of 0.9. It is consistent with the biological study that CDR3 sequence length in +immature B cells is significantly shorter than that of the other mature B cells (Ghraichy et al., 2021). +From the diagonal, we can figure out that our model tends to mistake the B cell sequences with their +previous or post-evolutionary stage, consistent with the biological process. +Antibody Discovery +Here, we examine PPLMs and PALMS for their capacity in benefiting real- +world problems and enabling the discovery of antigen-specific antibodies. Following the methods in +this study Zaslavsky et al. (2022), we considered two steps for the SARS-CoV-2 specific antibody +discovery. First, we generate a sequence classifier to distinguish SARS-CoV-2 antibodies using noisy +individual-level labels. Then, the high-ranked sequences are matched with the sequences with true +binding sequences in the CoV-AbDab (Raybould et al., 2021) database. We use the 90% sequence +identity as the threshold to determine whether the antibody is similar to existing SARS binders in the +CoV-AbDab database. A high similarity indicates a high probability of having the same biological +functionality as the existing binders. The details can be found in Appendix A.7. +7 + +No
Pretrain +EATLM +AntiBERT +Ablang_heavy +Ablang_light +ESM_1 +Expected +1 +2 +Cumulative
sum
of
matched
antibodies +Number of antibodies +Figure 4: The cumulative sum of matched +sequences number in the order of the pre- +dicted probability. EATLM outperforms other +PALMs and PPLMs for finding SARS-CoV-2 +binder faster highlighted in 1⃝ and finding all +antibodies faster highlighted in 2⃝. +In Figure 4, we plot the cumulative sum of matched +sequences number in the order of the predicted prob- +ability of the classifiers. We can find that the se- +quences predicted with high probability by PALMs +match with the existing binders better than PPLMs. +It means that PALMs can figure out the potential +binders more efficiently. +Besides, EATLM signif- +icantly outperforms other PALMs as the red line +shows. In the early stage, this method is the quickest +way to find potential binders. Then it loses to Ablang- +H but finally overtakes again and converges. It means +that EATLM is the first one to figure out all potential +binders in this dataset. +Furthermore, we list several potential binders found +by EATLM in Table 3. Without supervised labels, +EATLM gives a high probability of 2 SARS-CoV-2 +existing binding antibodies. Besides, EATLM proposes 9 potential sequences with high CDR-H3 +sequence identity. Further analysis of the results reveals that EATLM enables diverse-epitope antibody +discovery and priority selection, demonstrating that EATLM benefits therapeutic antibody discovery. +The detailed explanations can be found in Appendix A.7. +To examine whether the antibody sequences with 90% sequence identity can indeed bind the same +target, we investigate the 3D structure of the true binding antibody. Table 4 shows only one signle +atom difference between the predicted binder and the existing binder, suggesting the predicted binders +are highly possible to interact with SARS-CoV-2. +Table 3: The CDR3-H3 region of high-ranked sequences to bind to +SARS. We show the CDR3 fragment of the heavy chain in the antibody +sequences. ‘Identity’ is the similarity between the predicted binder and +the true binder. The epitope of the true binders is shown. The origin of +the majority of the true binder sequences is B cells from patients. The +different amino acids between the predicted binder and the existing +binder are highlighted in red +No Predicted Binder +Existing Binder +Epitope Identity +1 AREGIVGATTGFDY +AREGIVGATTGFDY +spike +1.000 +2 ARDLGGYFDY +ARDLGGYFDY +RBD +1.000 +3 AKDQDDAYYYYYYMDV AKDQDDGYYYYYYMDV NTD +0.938 +4 ASYYYDSSGYHYGMDV ASYYYDSSGYYYGMDV RBD +0.938 +5 ARRGLGLYYYGMDV +ARRGDGLYYYGMDV +S2 +0.929 +6 ARAFRGSYYYGMDV +ARATRGSYYYGMDV +S2 +0.929 +7 ARLSGSSWYFDY +ARLSGSSWDFDY +spike +0.917 +8 ARLGSSSWYFDY +ARVGSSSWYFDY +spike +0.917 +9 ARGWLRGYFDL +ARRGWLRGYFDL +RBD +0.909 +10 ARDWGELYFDY +ARDWGEYYFDY +RBD +0.909 +11 ARDLGGVFDY +ARDLGGYFDY +RBD +0.900 +Table 4: +3D structure +of the true SARS-CoV- +2 binding antibody No.3. +G2A highlights the single +atom difference in No.3, +indicating the predicted +binder is highly likely to +bind the virus. +G2A +PDB:7N62 +SARS‑CoV‑2
NTD +4.2 +HOW DOES EVOLUTION PRETRAINING TASK INFLUENCE THE REPRESENTATION? +To understand why EATLM shows better performance on antibody tasks, we analyze the pre-trained +representations to evaluate the effectiveness of the evolution-aware pretraining strategies from two +aspects: (1) Does the pre-trained antibody representation reflect its ancestor relationship? (2) Is the +specificity of antibodies captured by the evolution objective? +Ancestor Gerlime Visualization +We perform UMAP visualization analyses in Figure 5. First, we +observe that antibodies evolved from the same germline are nicely clustered together (Figure 5a and +5b), indicating the learned embedding is encoded with germline information. Besides, sequences with +similar scales of evolutionary distance tend to cluster together, and a clear gradation of evolutionary +8 + +distance can be observed in Figure 5c and 5d. The visualization provides a sanity check for the ability +of EATLM to extract the sequence information of antibody. +Accuracy of Mutation Position +Based on the specific evolution process described in Section 3.1, +we can find the mutation during the evolution process bring specificity to the antibody. Thus, we +explore the model’s ability to predict mutated residue from the masked token, which can reflect +the specificity feature the model captures. We find that although AntiBERT can predict with an +accuracy of 0.8889 on all positions, it fails on mutation positions with a 0.0311 accuracy. In contrast, +EATLM achieves an accuracy of 0.443 on mutation position, which indicates that the model captures +the specificity information. Note that during the MPP training, we mask the mutation position on +antibody sequences, which are different from its germline. Thus, the model cannot get the mutated +residue from the germline directly. The only way is to learn the underlying mutation rules. The full +results are shown in Table 12a in Appendix. +25 +20 +15 +10 +5 +0 +5 +10 +20 +10 +0 +10 +20 +30 +0 +1 +2 +3 +4 +5 +6 +(a) Health germlines. +15 +10 +5 +0 +5 +10 +10 +5 +0 +5 +10 +15 +0 +1 +2 +3 +› +(b) Patient germlines. +10 +5 +0 +5 +10 +15 +20 +15 +10 +5 +0 +5 +10 +15 +20 +0 +5 +10 +15 +20 +25 +30 +(c) Health distance. +20 +10 +0 +10 +20 +15 +10 +5 +0 +5 +10 +15 +0 +5 +10 +15 +20 +25 +30 +(d) Patient distance. +Figure 5: UMAP Visualization. (a-b) for sequences with different germlines. One color indicates one +germline with its descendant. (c-d) for evolutionary phylogeny distance. The shade of color indicates +its distance from the ancestor germline. +4.3 +KEY OBSERVATIONS +The performance of pre-trained language models highly depends on the task specificity. For +tasks with low antibody-specificity relatedness, PPLMs show comparable performance with PALMs, +indicating that transferring the general protein representations from PPLMs is an effective and +time-saving way in these tasks. PALMs show their advantage and outperform PPLMs on medium +specificity relatedness tasks, like paratope prediction. For tasks with high specificity, it is observed the +performance of PPLMs dramatically decreased, suggesting general pre-trained protein models are not +sufficient in antibody-specific representation learning. Interestingly, the performance of MSA-1b is +20% less than the model without pre-training. This observation is consistent with the biological study +that the mechanism of antibody evolution is significantly different from that of proteins. Therefore, +the incorporation of protein evolution information is not always good for antibody tasks, especially +the tasks that require antibody evolution information. +Performance
Increase
(%) +Task
Specificity +Low +Medium +High +0 +10 +‑10 +‑20 +20 +‑30 +No
Pretrain +ESM‑1 +AntiBERT +EATLM +MSA‑1b +Ablang‑H +Ablang‑L +Figure 6: Performance summary of vari- +ous pre-trained language models. +Incorporation of biological evolution mechanism into +PALM generally benefits antibody prediction tasks. +Evolution-related training objectives help detect mutation +positions on antibodies, which is a distinguishing feature +from germline. Interestingly, the performance increase of +EATLM compared with other PALMs is correlated with +task specificity relatedness. The ablation study showed the +removal of the evolution-related pretraining objectives re- +sults in performance decreasing, confirming that evolution- +related objectives benefit the prediction. Future studies +to have more in-depth research in this direction could be +promising. +Antibody pre-trained representations are helpful for real-world drug discovery. Using the +language model, we predict the probability for each antibody of being a binder with SARS-CoV-2. +Without accurate sequence-level labels, we successfully identify 11 potential antibody binders. +9 + +5 +CONCLUSIONS AND LIMITATIONS +In this paper, we have presented comprehensive empirical studies to characterize the impact of +pre-trained protein and antibody language models for antibody prediction tasks. Four important +antibody tasks from four biological categories with evolution relatedness ranging from high to low +are provided ATUE to facilitate the antibody and ML field. +However, there are some limitations to this work. First, for the ATUE, the task diversity is highly +limited because of data scarcity. With the data increase in this field, we expect we can update our +benchmark with more disease diversity and data size in the future. Also, we do not test any 3D +structure information during antibody pre-training. As a special subgroup of proteins, antibody +structures provide much more information such as geometry than sequences. In the future, recruiting +structure information for antibody pre-training may be able to improve the results. +REFERENCES +Yunlong Cao, Bin Su, Xianghua Guo, Wenjie Sun, Yongqiang Deng, Linlin Bao, Qinyu Zhu, +Xu Zhang, Yinghui Zheng, Chenyang Geng, et al. Potent neutralizing antibodies against sars-cov-2 +identified by high-throughput single-cell sequencing of convalescent patients’ b cells. Cell, 182(1): +73–84, 2020. +Ahmed Elnaggar, Michael Heinzinger, Christian Dallago, Ghalia Rehawi, Yu Wang, Llion Jones, +Tom Gibbs, Tamas Feher, Christoph Angerer, Martin Steinegger, et al. 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Nucleic acids research, 41(W1):W34–W40, 2013. +Alexander Dimitri Yermanos, Andreas Kevin Dounas, Tanja Stadler, Annette Oxenius, and Sai T +Reddy. Tracing antibody repertoire evolution by systems phylogeny. Frontiers in immunology, 9: +2149, 2018. +Maxim E Zaslavsky, Nikhil Ram-Mohan, Joel M Guthridge, Joan T Merrill, Jason D Goldman, +Ji-Yeun Lee, Krishna M Roskin, Charlotte Cunningham-Rundles, M Anthony Moody, Barton F +Haynes, et al. Disease diagnostics using machine learning of immune receptors. bioRxiv, 2022. +Feng Zhu, Thomas Althaus, Chee Wah Tan, Alizée Costantini, Wan Ni Chia, Nguyen Van Vinh Chau, +Giada Mattiuzzo, Nicola J Rose, Eric Voiglio, Lin-Fa Wang, et al. Who international standard +for sars-cov-2 antibodies to determine markers of protection. The Lancet Microbe, 3(2):e81–e82, +2022. +12 + +A +APPENDIX +A.1 +ANTIBODY SPECIFIC EVOLUTION +Antibodies, composed of two identical heavy chains and two identical light chains, form a large +Y-shaped structure, where the two tips are responsible for pathogens binding. Antibody evolution +, described by sequence-sequence relationships between ancestor and progeny antibodies, reflects +antibodies’ key antigen-binding function (Honjo & Habu, 1985). During antibody evolution (Figure +7), the initial diversity is encoded into the ancestor sequence through randomly recombination of V-, +D- and J-gene segments. Upon exposure to a pathogen, the sequence undergoes frequent sequence +mutations to search for progeny sequences with optimal binding specificity. Sequence evolution +analysis has been employed by many computational biology studies and shows promising results in +antibody-related tasks, such as disease diagnosis and therapeutic antibody development (Yermanos +et al., 2018; Miho et al., 2019). +Importantly, antibody evolution is significantly different from that of proteins. Antibodies only +contain hundreds of thousands ancestor sequences so-called germline. To bind dozens of millions of +diverse antigens, antibodies need to mutate from the ancestor sequences to gain new functions (Figure +7). Therefore, the non-conserved amino acids (mutated ones) plays important roles for structure and +function. On the contrary, the conserved amino acids (not mutated) in proteins determine structure +and function. During protein evolution, evolutionary pressure to maintain protein structure and +functions leads to the conservation or co-evolution of residues located in structural folding core for +binding interface. Diverse methods have been developed to extract the co-evolution information from +conserved amino acids sequences for structure and function prediction, such as AlphaFold (Jumper +et al., 2021). +In brief (Figure 7), antibody evolution specificity distinct from that of proteins can be defined with +two main features: (i) ancestor germlines; (ii) the mutated amino acids of germlines. +V(D)J
recombination +Antigen +B +Antigen +C +Antigen +A +Protein
evolution +GAMQ +VKQA +KGLE +SSITSGTNN +R +W +PG +WV +IYY +ASMN +IRTT +RSIE +GPISAGSQQ +K +W +PG +WV +IYY +GAMQ +IRAT +RGID +GSVSTPSGQ +R +W +PG +WV +IYY +GGMP +AKSG +KAVD +GPLSGPSYA +K +W +PG +WV +IYY +ATMQ +AQTP +KSLD +QSLGGPSGA +Q +W +PG +WV +IYY +ATNN +VNQP +KGLD +PSITSGSGN +R +W +PG +WV +IYY +Protein B +Protein A +GAMQWVKQAPGKGLEWVSSITSGSNNIYYR +... +... +... +FAE +VPM +YP +F +KLVPM +QL +F EKLVP +W +L +FAEK +LV +PM +FAEKLVPM +Antibody
evolution +ancestor +germline +mutation
( +) +non
mutated +function +mutation
 +) +(mutated +ancestor +functions +Figure 7: Evolution specificity of B cell antibodies comparing with general proteins. The evolutionary +sequence relationships are highlighted using blue dash lines. +A.2 +DATA PROCESSING DETAILS +Pairing Antibody with Germline +For germline annotation in the pre-training task, we used the +annotated germline sequences provided in the OAS database (Kovaltsuk et al., 2018). For downstream +benchmarks tasks like B-cell classification, therapeutic antibody engineering, and disease diagnosis, +we completely followed the methods shown in the OAS database paper. IgBLAST, an immunoin- +formatic benchmarking tool for the analysis of B-cell antibody repertoires was used for germline +annotation (Ye et al., 2013). The antibody nucleotide-containing FASTA file was aligned to germline +13 + +and translated to amino acids using IgBLASTn. The antibody amino-acid sequence was aligned using +IgBLASTp. The germline databases for human patients used ImMunoGeneTics (IMGT) germline +sequences derived from IMGT (Lefranc et al., 1999). For each antibody, usually, multiple germline +sequences can be obtained and only the single sequence showing the highest confidence score for the +alignment was chosen. +Pre-training Data Processing +We downloaded OAS Oct 2021 version from its website and re- +moved duplicate sequences. To avoid data leakage, we cluster sequences based on the CDR3 sequence +and filter each cluster by 70% identity over the whole sequence using Linclust (Steinegger & Söding, +2018). Then, we shuffle the dataset and split it into 100k-size chunks. The last chunk is used as the +validation set. The dataset size is 20,245,249 and 45,249 are used for validation. +Table 5: ATUE benchmark. The five downstream tasks are divided into three biological categories +and each category focuses on different input levels. ‘Q6’ indicates the classification task have 6 +classes. For disease diagnosis, the size indicate the number of profiles, where each profile contains +thousands to millions of sequences. +Specificity +Task Name +Input +Formalization +Size +Low +Antigen Binding Prediction +CDR fragment +Q2 Cls. +21,612 seqs +Medium +Paratope Prediction +CDR residue +Q2 Labeling +1,662 seqs, 21,342 positions +High +SARS Antibody Classification +Sequence +Q2 Cls. +22,000 seqs +High +B Cell Classification +Sequence +Q6 Cls. +88,094 seqs +A.3 +ATUE DETAILS +We summarize the tasks used in ATUE in Table 5 and discuss each task in detail in this section. +Antigen Binding +Accurate antigen-binding prediction approaches could allow significantly more +efficient antibody discovery with higher affinity. Machine learning methods have already achieved +some success in antibody binding capacity optimization. We collect the antigen-binding data from +Mason et al. (2021) and follow the training/validation/test split of 15,128/3,242/3,242. The original +dataset only has CDR3 fragments, and we extend them to the full antibody sequences. For cross- +validation, we split the dataset by antibody sequences to ensure that no antibody sequences overlap +between 90% training and 10% validation. +Paratope Prediction +Paratope is the antibody residues involved in antigen binding. The ability +to accurately map the paratope can provide detailed knowledge about the binding mechanism and +accelerate antibody discovery. 1D sequence-based deep learning methods have been employed for +paratope prediction. The paratope data is collected from Liberis et al. (2018) with 1,662 CDR +segments on 277 antibodies. Each antibody contains three CDR fragments (CDR1, CDR2 and CDR3) +in the heavy chain and three CDR fragments in the light chain. We also search the full sequence +for each antibody and use the whole sequence as input. For cross-validation, we split the dataset by +antibody sequences to ensure that no antibody sequences overlap between 90% training and 10% +validation. +Table 6: The statistics of B cell classification. +Type +Size +Immature b cell +14,145 +Transitional b cell +13,197 +Mature b cell +16,139 +Plasmacytes PC +22,236 +Memory IgD- +8,437 +Memory IgD+ +13,940 +14 + +B Cell Analysis +We formulate a 6-category classification task for B cell maturation analysis, which +includes {immature, transitional, mature, memory IgD+, memory IgD-, plasmacytes,}. The analysis +of B cell maturation plays an important role in understanding the mechanisms underlying B cell +responses in immune system Ghraichy et al. (2021); Meffre et al. (2000). +The order of B cell type follows the evolutionary process in the immune system, from immature state +to transitional state and finally becomes a memory B cell. Both memory IgD- and IgD+ belong to +memory B cells with different isotypes, and they have a high affinity to foreign antigens. Among the +other categories, the Plasmacytes PC sequences also have some affinity ability. It is widely reported +that changes in antibody sequence patterns correlate with B-cell maturation. Therefore, we use this +task to evaluate the representation learning capacity of the language model. +We collect 88,094 sequences from Mroczek et al. (2014). They extracted from the peripheral blood +of healthy adults and got six types of B cells with different maturity and antibody sequences. The +distribution of various types of B cells in the dataset is shown in Table 6 +Antibody Discovery +Antibody discovery from B cell repertoire has been widely recognized as a +novel trend to improve the efficiency of antibody discovery for diverse antigens (Weiner, 2015; Pedri- +oli & Oxenius, 2021). However, previous studies highly rely on expensive wet-lab experiments (Cao +et al., 2020; Shiakolas et al., 2022). Deep learning based method have shown potential capacity in +help antibody discovery by reduce cost and increase efficiency (Widrich et al., 2020; Wang et al., +2022). Here, we ask whether pretrained models can benefit real-world problems and enable fast-track +neutralization SARS-CoV-2 antibody discovery. +In the first step, we develop a sequence classifier to distinguish which antibody sequence from the +numerous sequences is responsible for the recognition of the SARS-CoV-2. This task is highly +challenging since we can hardly get the sequence-level disease label that indicates whether the +antibody sequence is related to the disease. Thus, we follow the practice of Roskin et al. (2020); +Zaslavsky et al. (2022) to use the individual label as the rough sequence label and train a sequence- +level predictor. Then, with the help of a sequence-level predictor, we can give each sequence a most +likely label to help antibody discover, whose accuracy has been verified by the excellent results on +individual prediction, which may accelerate the discovery of new antibody sequences. +We follow the condition of Kim et al. (2021) to filter SARS-CoV-2 antibody data from the OAS +database. The basic condition is ‘Chain = heavy; Isotype = IGHG; BSource = PBMC; Species += human; Vaccine = None’. We further add the condition of ‘Unique Sequences >= 10000’. For +health/SARS we set the ‘Disease’ field to ‘None’, ‘SARS-CoV-2’. Then we obtain 87/133 patient +profiles for each type. To make a balanced dataset, we limit the size of the health profile and mix up +the healthy ones and the ones with the SARS-CoV-2. For cross-validation, we randomly split the +dataset by profiles 10 times: 90% for training and 10% for validation. We further select sequences +with top100 redundancy to make the positive labels more accurate. +A.4 +QUANTITATIVE ANALYSIS OF ATUE TASK SPECIFICITY +It is important to include statistical significance tests relative to the antibody-specific features in +antibody functional tasks we proposed in the ATUE benchmark. According to the evolution process +shown in Figure 7, antibody evolution specificity distinct from that of proteins can be defined with +two main features: (i) ancestor germlines; (ii) the mutated amino acids of germlines. We implemented +statistical significance tests of (i) ancestor germlines subtype usage; (ii) the number of mutated amino +acids in antibodies against different labels of downstream tasks in ATUE to quantitatively assess the +"Task specificity". The analysis is now summarized in Table 7 In Appendix A.1. Generally, it is +clearly shown that ATUE benchmark comprises antibody tasks showing different scales of antibody +specificity for later on modeling analysis. Moreover, the which are used for statistical analysis of task +specificity and pre-training model objectives in our study. +Antigen Binding +In the Antigen binding dataset, both antibody binding and none antigen binding +sequences share the same germline subtype sequence (IGHV3.1) (Figure 8A) as well as the same +number of germline mutations 8B). Therefore, None of two antibody specific features show significant +distribution differences between data with different labels, demonstrating antigen binding is a task +with low antibody specificity. +15 + +Table 7: Task specificity. Summary of the statistical significance test of two antibody-specific features +for different tasks in the ATUE benchmark. +Task +Statistical significance (p-value) +General Specificity +Germline Usage Mutations Numbers +Antigen Binding +Nan +Nan +Low +Paratope Prediction +0.296 +0 +Medium +B cell classification +0 +0 +High +SARS antibody discovery +0 +0 +High +A +B +Figure 8: (A)IGHV gene segment usage distribution between binding and non-binding antibody. (B) +Germline mutation number distribution between binding antibody and non-binding antibodies. +Paratope Prediction +For the paratope prediction task, we first evaluate the germline subtype +distribution difference between sequences with different numbers of binding sites (Figure 9A). +Kruskal Wallis test showed a p-value of 0.296 suggesting germline subtype usage is not statistically +significant. Also, we find the binding sites can be significantly mapped with more germline mutations +than the non-binding sites, which is consistent with the knowledge of antibody specificity definition +(Figure 9B). One out of two antibody specific features show significant distribution differences +between data with different labels. Therefore, we define this task as a medium specificity task. +A +B +Figure 9: (A) Number of binding sites distribution between different IGHV gene segments. Com- +parison is performend using kruskal wallis test with p value 0.296. (B) Germline mutation number +distribution between binding and non-binding positions. Comparisons performed using t-tests show- +ing p value equals to 0. +16 + +1.0 +0.8 +e +0.6 +0.4 +0.2 +0.0 +0 +1 +Antigenbindingclasses10.4 +Mutations +10.2 +of +10.0 +Number +9.8 +9.6 +0 +1 +Antigen binding classesIGHV1 +IGHV2 +IGHV3 +IGHV4 +IGHV5 +IGHV610 +I sites +8 +Num of binding +6 +4 +2 +IGHV7 +Germline20.0 ++ +Num of Germline Mutations +17.5 +15.0 +12.5 ++ ++ ++ +10.0 ++ +7.5 +5.0 +2.5 +0.0 +nonbind +bind +Binding sitesB Cell Analysis +As shown in Figure 10, the distribution of the germline usage as well as the +number of germline mutations are significantly different between antibodies in B cells with different +developmental stages. This observation is highly consistent with previous studies Mroczek et al. +(2014); Ghraichy et al. (2021). Since both of the antibody specific feature show significant distribution +differences, this task is defined as a high-specificity task. +A +B +Figure 10: (A)IGHV gene segment usage distribution between different B cells. Comparison is +performend using chisquare test with p value 0. (B) Germline mutation number distribution between +different types of B cells. Comparisons performed using kruskal wallis test showing p value equals to +0. +SARS Antibody Discovery +Antibodies in SARS patients and healthy ones show a significant +difference in their germline subtype usage and the number of germline mutations (Figure 11). This +observation is highly consistent with previous studies showing SARS antibody convergent among +patients Galson et al. (2020). Since both of the antibody specific feature are highly significant, this +task is defined as a high-specificity task. +A +B +Figure 11: (A)IGHV gene segment usage distribution between antibody in SARS patients and health +ones. Comparison is performed using chisquare test with p value 0. (B) Germline mutation number +distribution between antibody in SARS patients and health ones. Comparisons performed using +kruskal wallis test showing p value equals to 0. +A.5 +TRAINING DETAILS +Antibody can be represented as A = {a1, a2, · · · , am} and the germline of individual antibody can +be represented as G = {g1, g2, · · · , gn}, where m and n are the lengths. Each token ai or gj in the +sequence is called a residue that belongs to the amino acid set A. A includes 20 common amino acids +17 + +1.0 +0.8 +e +P +0.4 +0.2 +0.0 +cell +IgD+ +IgD. +PC +immature +B cell classes25 +F mutations +S +20 +15 +of +nN +10 +5 +0 +cell +IgD. +cell +Cell +rer +in +B cell classesIGHJ1 +IGHJ2 +IGHJ3 +IGHJ4 +IGHJ5 +IGHJ6L +30 +25 +ions +mutati +20 +15 +of +w +nN +10 +5 +0 +B cell classes1.0 +0.8 +e +P 0.4 +0.2 +0.0 +. +B cell classesIGHV1 +IGHV2 +IGHV3 +IGHV4 +IGHV5 +IGHV6with a residue ‘X’ that indicates the residue is unknown (mostly in the germline). Typically, antibody +PLMs are trained with basic mask language modeling objective lMLM on the antibody sequences +S = A = {a1, · · · , am, }. +A.5.1 +EVOLUTION-AWARE PRETRAINING +In order to incorporate the evolutionary information into the pre-training, we pair the antibody +sequence A with its germline G and concatenate them into a long sequence with a special token +‘[SEP]’ as the delimiter: S = {s1, · · · , sm+n+1} = {a1, · · · , am, [SEP], g1, · · · , gn}. Thus, we +optimize MLM objective on the long sequence S: +lMLM = − 1 +|M| +� +i∈M +log p(si|S\M), +(1) +where M is the index set of masked tokens. It helps the model to learn the basic residue distribution +for antibody sequences. Besides, it can also capture the interaction between residues of the antibody +and its germline. +DTVMTQS……QYKHWPPYTFGRGTKLEIR +EIVMTQS……QYNNWPXXXXXXXXXXXXX +[SEP] +Transformer Encoder +[CLS] +Antibody +Germline +Ancestor Germline Prediction +(AGP) +1 1 0 0 0 … 1 1 +D T K H P Y … +Mutation Position Prediction +(MPP) +𝒀𝝐{𝟎, 𝟏} +mask mutation +substitute germline +(a) Pre-training with two biological evolution tasks. +DTVMTQS……QYKHWPPYTFGRGTKLEIR +EIVMTQS……QYNNWPXXXXXXXXXXXXX +[SEP] +Transformer Encoder +[CLS] +B cell +Classification +FWR +FWR +CDR3 +…… +FWR +FWR +CDR3 +…… +Antigen Binding +Sequence- +level +Individual- +level +Disease +Diagnostics +(b) Finetuning for three biological categories in ATUE. +Figure 12: EATLM. In Figure 12a, AGP randomly unpairs the germline sentence and predict the +ancestor relationship. MPP predicts the mutation position on the germline and the masked mutation +residue on the antibody. Based on the input, the three categories in ATUE can be divided into +sequence-level and individual-level (Figure 12b). For individual-level disease diagnostics, we score +each sequence in the individual profile and calculate the trimmed mean over all sequences to get the +individual score. +Ancestor Germline Prediction The ancestor relationship between the antibody and its germline +determines the shared biological functions obtained in the evolution. Antibody sequences with +similar residues evolved from different germline sequences may have different biological functions. +When stimulated by a foreign antigen, the common ancestor germline evolve to various antibody +sequences. Similar antibody sequences may have different germline sequences, which will affect their +biological functions. Thus, the aim of this task is to determine whether the antibody have evolutionary +relationship with the given germline. During training, we substitute the paired germline G with +random germline G′ = {g1, · · · , gn} in the batch via a probability p = 0.3. The new sequence is +denoted as S′ = {a1, · · · , am, [SEP], g′ +1, · · · , g′ +n} and the training loss can be described as: +la = − log p(y|S′), +(2) +where y ∈ {0, 1} indicate whether the noisy germline G′ is the ancestor of the antibody S. It can +help the model to distinguish the ancestor germline of the antibody by capturing the shared features. +Mutation Position Prediction The somatic hypermutations on the germline further give progeny +antibodies the specificity of binding with the specific antigen. In order to model this specificity, this +task focus on predicting the mutation positions and the residues mutated to. Specifically, for each +token gj in the germline G, the target is to predict a label yj ∈ {0, 1} to indicate whether this token +has been mutated. For the antibody sequence S, we mask the mutation position and predict these +tokens. The objective can be formalized as: +lm = − 1 +n +� +j∈{1,··· ,n} +log p(yj|S\M ′) − +1 +|M| +� +i∈M ′ +log p(ai|S\M ′). +(3) +18 + +maturation.Bind +Notbind +Binding. +sites. +0000.10110008 +HealthDisease:Here, M ′ is the ground-truth mutation position and we mask these tokens on the antibody sequence. +This task is more difficult than MLM which equally masks tokens in the L, because the tokens on the +mutation position of A get less information from the germline, compared with other shared residues +between the antibody and the germline. By optimize this objective, the model learn to capture the +specificity obtained from the somatic hypermutation in the evolutionary process. +A.5.2 +IMPLEMENTATION DETAILS +We use the base Transformer architecture (Vaswani et al., 2017) with 12 layers, 12 heads, and 768 +hidden states. The total parameters are 86M. We use Adam optimizer (Kingma & Ba, 2015) with +the maximum learning rate of 2e-4 and the warm-up step of 24,000. The maximum length is set +to 400 since most antibody sequences are shorter than 180. We first pre-train our model with the +MLM objective. During the pre-training, 15% tokens are randomly selected with 80% masked, 10% +replaced, and 10% kept. Then we conduct further pre-training on two antibody-related tasks with a +smaller learning rate of 1e-5. +For each task in ATUE, we finetune the model with supervised data. We follow the standard split of +Antigen Binding Prediction. For other tasks that do not provide a standard split, we conduct 10-cross +validation and report the average results. Since our pre-training model learns the representation of the +antibody sequence, we expand the CDR fragment to the full antibody via searching the biological +database for therapeutic antibody engineering tasks. For finetuning, we limit the max epochs to 30 +and use the Adam optimizer with a max learning rate of 3e-5. We use the mean representation of 12 +layers as the sequence representation. +Table 8: The pretraining details for different models. The ‘Germline’ is the prediction accuracy +of AGP, and the ‘Position’ and ‘Mutation’ are the accuracy of mutation positions and the mutated +residues respectively. +Model +Size +Loss +Step +Germline +Position +Mutation +AntiBERT +85M +0.437 +568000 +/ +/ +/ +AntiBERT w AGP +85M +0.558 +587000 +0.330 +/ +/ +AntiBERT w MPP +85M +1.744 +622000 +/ +0.965 +0.410 +ESM-1 FT +85M +0.2636 +136000 +/ +/ +/ +ESM-1b FT +650M +0.2704 +278000 +/ +/ +/ +EATLM w/o AGP & MPP +85M +0.143 +108000 +/ +/ +/ +EATLM w/o AGP +85M +1.744 +134000 +/ +1.000 +0.443 +EATLM w/o MPP +85M +0.001 +126000 +1.000 +/ +/ +EATLM +85M +1.856 +252500 +0.9960 +1.000 +0.443 +EATLM-large w/o AGP & MPP +650M +0.149 +193000 +/ +/ +/ +EATLM-large w/o AGP +650M +1.753 +267000 +/ +1.000 +0.438 +EATLM-large w/o MPP +650M +0.001 +200000 +1.000 +/ +/ +EATLM-large +650M +1.773 +384000 +0.996 +1.000 +0.431 +The model is trained with 108,000 steps and gets a 0.9606 token accuracy on the MLM task. It takes +further steps for AGP and MPP. The model quickly converges for AGP and gets a 0.99 accuracy on +the ancestor germline prediction because more than 80% residues are shared between the antibody +and its germline. For MPP, it can predict the mutation position with the accuracy of 1.00 and obtains +a 0.4430 accuracy in the mutation position. It means that the model can easily find the mutation +positions by the self-attention between the antibody and germline, but it is still difficult to predict +which residues this position will mutate to. We assume it is because the ancestor germline can undergo +different somatic hypermutations and get various progeny antibodies, resulting in different valid +mutations at the same position. We also compare this mutation accuracy with the model without MPP, +which is only trained with MLM on the concatenation of the antibody and its germline. With a high +prediction accuracy of 0.8889 on all positions, it achieves only a 0.0311 accuracy on the mutations. +It implies that the masking among all positions on the sequence can do accurate predictions of the +shared residues but hardly capture the mutation information. +19 + +We also conduct AGP and MPP to finetune the baseline model AntiBERT. The pre-training results +are shown in Table 8. We can find that without the concatenation of the antibody and its germline, it +is difficult to predict the ancestor relationship. It also underperforms than EATLM in MPP. +Negative sampling ratio +We have tried the ratio of 0.1/0.3/0.5/0.75 and found that this ratio has +little influence on performance and convergence speed. As we discuss in the appendix, the model can +quickly converge for AGP and get an accuracy of 0.99. +Finetuned Protein Language Models and Larger Architecture +We pre-train our method with a +larger architecture and compare it with ESM-1b, which also has 650M parameters. We also further +pre-trained the ESMs to transfer to the antibody field. After that, we evaluate them on the antigen +binding and paratope prediction task. The results are shown in Table 9. The result show that the larger +architecture does show advantage in terms of the performance improvement. For antigen binding, +ESM-1b has better performance than ESM-1. However, for paratope prediction, it performs worse. +In addition, for ESM, fine-tuning of the antibody dataset may cause the overfitting problem, leading +to the decrease in the performance of all three tasks. +Table 9: The performance of larger models and finetuned protein language models on three tasks. +The ESM-1 and ESM-1b models have 85M and 650M respectively. The EATLM-large have similar +architecture with ESM-1b. ‘FT’ indicates the model is further finetuned on our antibody pre-training +dataset. +Antigen Binding +Paratope +AUC +F1 +MCC +AUC +F1 +MCC +ESM-1 +0.917+0.001 0.854+0.002 0.689+0.002 0.886+0.009 0.669+0.024 0.547+0.026 +ESM-1 FT +0.914+0.001 0.858+0.001 0.695+0.003 0.883+0.009 0.657+0.031 0.534+0.030 +ESM-1b +0.924+0.002 0.860+0.002 0.707+0.003 0.873+0.009 0.655+0.042 0.524+0.036 +ESM-1b FT +0.916+0.003 0.844+0.003 0.686+0.004 0.869+0.010 0.660+0.021 0.527+0.015 +EATLM +0.922+0.052 0.862+0.021 0.699+0.027 0.887+0.008 0.698+0.017 0.575+0.024 +EATLM-large 0.921+0.004 0.854+0.004 0.677+0.010 0.887+0.007 0.685+0.015 0.561+0.020 +A.6 +DISCUSSION ABOUT OF EATLM +First, EATLM doesn’t use any 3D structure information during pre-training. As a special subgroup of +proteins, antibody structures provide much more information such as geometry than sequences. In the +future, recruiting structure information for antibody pre-training may be able to improve the results. +However, the data scale available for antibody structure is dramatically less than that of antibody +sequences. The largest dataset of antibody structures only contains thousands of 3D high-resolution +structures, while the number of antibody sequences is in billions. Using structure prediction methods +like AlphaFold may help to bridge the gap between sequences and structures. Second, EATLM +requires germline as input during downstream tasks, this will slow down the prediction speed. +A.7 +NEW SARS BINDER DISCOVERY +The main challenge for disease diagnosis is to distinguish the disease-related antibodies from millions +of antibody sequences in the individual profile, as stated in Section A.3. Here, with the help of a +sequence-level predictor, we can give each sequence a most likely label to help antibody discover, +whose accuracy has been verified by the excellent results on individual prediction, which may +accelerate the discovery of new antibody sequences. +SARS Sequence-level Predictor +As described in Section A.3, we first train a sequence-level +predictor for SARS-CoV-2. The results are shown in Table 10. Compared with Compared with Figure +4 in the main text, we find that good results in the sequence-level predictor do not necessarily mean +good results in the antibody discovery. It can be mainly affected by the noisy label of the sequence +level. +20 + +Table 10: Sequence-level predictor for SARS-CoV-2. +Sequence-level +SARS +AUC +F1 +MCC +No pretrain +0.894±0.029 0.801±0.045 0.637±0.078 +ESM-1 +0.903±0.061 0.799±0.032 0.648±0.093 +MSA-1b +0.914±0.025 0.810±0.050 0.671±0.080 +Ablang-H +0.915±0.027 0.817±0.038 0.668±0.068 +Ablang-L +0.893±0.035 0.801±0.054 0.637±0.099 +AntiBERT +0.916±0.026 0.810±0.033 0.661±0.060 +EATLM +0.904±0.027 0.808±0.035 0.643±0.069 +EATLM w/o AGP +0.904±0.029 0.808±0.039 0.644±0.078 +EATLM w/o MPP +0.901±0.026 0.808±0.035 0.648±0.069 +EATLM w/o MPP & AGP 0.901±0.026 0.807±0.034 0.645±0.067 +Figure out SARS Binders +As shown in Table 3 in the main body, we find 2 true SARS binders and +9 potential binders with the help of EATLM. Specifically, we first use our sequence-level predictor +to get a probability score for each sequence in the SARS dataset. Then we select the sequence with +high-ranked score (the probability > 0.5) and compare them with the public Cov-AbDab database +Raybould et al. (2021) 1, which contains data on published/patented antibodies known to bind to +SARS-CoV-2 (Raybould et al., 2021). Since the CDR3 fragment in the heavy chain is the most +relevant to the binding of antibody and antigen, we calculate the edit distance between the CDR3 +fragments in heavy chains (CDR-H3) with those of the known binder and use a threshold of 85% +similarity as the sequence identity. 85% Hamming distance for B cell antibody sequence clustering +(identify similar B cell antibody sequences responding to the same antigen/epitope) was previously +suggested in this paper (Gupta et al., 2017). This method then was widely used for B cell antibody +repertoire analysis in different studies (Montague et al., 2021; Wang et al., 2022). +SARS Binder Analysis +To provide a more intuitive analysis of the similarity between our predicted +antibody and true SARS-CoV-2 binders, we investigate the 3D structure of the true binding antibodies +and the mutation site of our predicted sequence on the corresponding structure. High resolution +structure of true binding antibody #3 in Table 3 with SARS-CoV-2 are shown in Figure 13 (PDB +code: 7N62). The interaction interface between the antibodies and SARS-CoV-2 spike/RBD are +shown in Figure 3 in main body. CDR-H3 were shown in orange. Only one single atom highlighted +in red is different between predicted binder and true binder. Obviously, these different residues don’t +localize to direct binding site and CDR-H3 founding core, suggesting the sequence difference likely +will not affect antibody virus interaction. Furthermore, we found the epitopes of the 11 identified +SARS-CoV-2 antibodies cover a wide range of different structures from traditional RBD domain to +novel non-RBD epitopes like S2 and NTD as shown in Table 3. This result shows our method enables +diverse-epitope antibody discovery. +Table 11: SARS-CoV-2 antibody hit rate with different probability thresholds. ‘Total’ is the total +number of sequences whose probabilities are higher than the threshold. ‘Hit’ is the number of +sequences that meet the similarity requirements with existing binders. +Threshold +Total +Hit +Hit rate (%) +0.5 +13253 +66 +0.498 +0.7 +10227 +54 +0.528 +0.8 +9338 +49 +0.525 +0.9 +8178 +47 +0.562 +Probability Threshold Sensitivity +In order to investigate the influence of the threshold used to +determine the potential binders, we try different thresholds in Table 11. Here, the probability threshold +1http://opig.stats.ox.ac.uk/webapps/covabdab/ +21 + +Figure 13: High resolution structure of mutation in the predicted binder (AKDQDDAYYYYYYMDV) +with the existing binding antibody (AKDQDDGYYYYYYMDV). +means that if the sequence predictor give a probability higher than the threshold for one sequence, it +will be viewed as a potential binder. If the predicted binder have a sequence similarity higher than +85% with the existing binders in Cov-AbDab, we view it as one hit. As the threshold score increases, +the hit rate corresponding increases from 0.528% to 0.562%, indicating that our model may enable +priority selection of SARS-CoV-2 antibodies and reduce experimental costs. +Sequence Similarity Sensitivity +In previous work, two antibodies with the CDR-H3 similarity +over 85% can be viewed as similar and have a high probability to share the same functionality. And +here we also check the influence on the binder matching of different thresholds of the similarity. The +results are shown in 14. Here, we fix the probability threshold as 0.5. As we can see, the baselines +have similar trends in all threshold. If we relax the threshold, there will be more matching sequences. +However, the predictors will have less advantage over the random order, which indicates that the +ranking is less important. +The Potential of New Binder Discovery +During the training of our sequence-level predictor, we +have no reliable ground-truth labels, which means that the model has never known which sequences +can bind to SARS in the real-world scenario. However, the model can learn from the noisy data and +rank the real SARS binders with high probabilities. Sequence identity of 1 means that the CDR-H3 +fragment can be directly found in the Cov-AbDab database, which implies that the sequences have +been verified by wet laboratory testing. The other sequences with an identity over 90% are thought +to have a similar binding performance to existing binders, indicating that they are promising SARS +binders that can help the discovery of therapeutic antibodies for SARS-CoV-2. +0 +2000 4000 6000 8000 10000120001400016000 +0 +50 +100 +150 +200 +EATLM +Ablang-H +Ablang-L +ESM-1b +Transformer +AntiBERT +Expected +(a) Threshold=0.8 +0 +2000 4000 6000 8000 10000120001400016000 +0 +10 +20 +30 +40 +EATLM +Ablang-H +Ablang-L +ESM-1b +Transformer +AntiBERT +Expected +(b) Threshold=0.85 +0 +2000 4000 6000 8000 10000120001400016000 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +EATLM +Ablang-H +Ablang-L +ESM-1b +Transformer +AntiBERT +Expected +(c) Threshold=0.9 +Figure 14: The cumulative count of matching sequences number. The dashed line is the expected +results for a random order. The x-axis is the sequence number and the y-axis is the cumulative +matched sequence number. +22 + +SARS-CoV-2 spike +G2Amut +CDRH3 +No.3 +Pred binder: +AKDQDDAYYYYYYMDV +True binder(7N62): AKDQDDGYYYYYYMDVA.8 +EXTENTED STUDY FOR DISEASE DIAGNOSIS +It would be interesting to see whether our sequence classifier can be used for other applications, +such as disease diagnosis. Each human is estimated to maintain about 108 − 1010 distinct antibody +sequences, constructing an informative encyclopedia recording the past and present health and disease. +Interpreting the pattern of the sequences has already proved useful in disease diagnosis and allows us +to assess many infectious diseases without expensive laboratory testing. However, it is difficult to +distinguish which antibody sequence from the numerous sequences is responsible for the recognition +of the specific antigen, which hinders the discovery of the antibody for diseases (Zaslavsky et al., +2022; Lu et al., 2018; Greiff et al., 2020). +Benefiting from the recent high-throughput sequencing, we can obtain millions of antibody sequences +from the individual human. At the same time, we can get a disease label that indicates whether the +human is infected by the disease. The main challenge is that we can hardly get the sequence-level +disease label that indicates whether the antibody sequence is related to the disease. Thus, we follow +the practice of Roskin et al. (2020) to use the individual label as the rough sequence label and train a +sequence-level predictor. Then we use this predictor to predict sequences of the individual profile and +make the trimmed mean score as the individual score. +We use the same data processing as Antibody Discovery stated in Section A.3. +For +health/SARS/HIV/Ebola/Allergy/SLE/MS, we set the ‘Disease’ field to ‘None’, ‘Ebola’, ‘Allgery’, +‘SLE’,‘MS’. Then we obtain 87/133/51/14/12/8/8 patient profiles for each type. We also do 10-cross +validation and select sequences with high redundancy. +Disease Classification +We use all these diseases profile to build the Q7 classification task for +disease diagnosis. Previous biological studies mainly use this multi-classification task for disease +diagnosis Zaslavsky et al. (2022); Wang et al. (2022), highlighting the discriminatory power among +different diseases are important for disease diagnosis. The results are shown in Table 12. We found +both PPLM an PALM show comparable results as the random initialized model, suggesting the +finetuning part play more important role and pretrained language model cannot help this task. +Table 12: The Q7 disease classification task. ‘ACC’ is the accuracy rate. +ACC +No pretrain +0.754±0.023 +ESM-1 +0.747±0.016 +ESM-1 FT +0.762±0.024 +MSA-1b +0.746±0.019 +Ablang-H +0.704±0.033 +Ablang-L +0.702±0.040 +AntiBERT +0.750±0.016 +EATLM +0.756±0.020 +EATLM w/o AGP +0.754±0.020 +EATLM w/o MPP +0.755±0.020 +EATLM w/o AGP & MPP 0.756±0.021 +Sequence-level Predictor for Various Disease +As before, we train a sequence-level predictor for +each disease. The results are shown in Table 13. Compared with Table 4 in the main text, we +find that good results in the sequence-level predictor do not necessarily mean good results in the +individual-level predictor. It is mainly due to the trimmed mean we use to get individual-level results, +which is a central estimate that is robust to noise labels. Overall, our model has comparable results to +other models in terms of sequence prediction with noisy labels, and has better results for individual +diagnosis. +Individual-level Predictor for Various Disease +It is observed our evolution-aware EATLM per- +forms the best in the individual-level classifier to determine whether the patient suffering from SARS. +Besides, PALMs significantly outperform PPLMs. The results is shown in Table 14. +23 + +Table 13: Sequence-level predictor for disease diagnosis. +Sequence-level +SARS +HIV +AUC +F1 +MCC +AUC +F1 +MCC +No Pretrain +0.894±0.029 0.801±0.045 0.637±0.078 0.893±0.081 0.622±0.223 0.563±0.209 +ESM-1 +0.903±0.061 0.799±0.032 0.648±0.093 0.927±0.039 0.739±0.082 0.701±0.080 +MSA-1b +0.914±0.025 0.810±0.050 0.671±0.080 0.903±0.064 0.680±0.103 0.617±0.102 +Ablang-H +0.915±0.027 0.817±0.038 0.668±0.068 0.895±0.058 0.684±0.087 0.604±0.108 +Ablang-L +0.893±0.035 0.801±0.054 0.637±0.099 0.881±0.072 0.668±0.111 0.584±0.131 +AntiBERT +0.916±0.026 0.810±0.033 0.661±0.060 0.921±0.046 0.742±0.088 0.689±0.101 +EATLM +0.904±0.027 0.808±0.035 0.643±0.069 0.930±0.044 0.744±0.079 0.677±0.105 +EATLM w/o AGP +0.904±0.029 0.808±0.039 0.644±0.078 0.933±0.041 0.747±0.077 0.679±0.103 +EATLM w/o MPP +0.901±0.026 0.808±0.035 0.648±0.069 0.932±0.037 0.749±0.072 0.682±0.096 +EATLM w/o MPP & AGP 0.901±0.026 0.807±0.034 0.645±0.067 0.930±0.045 0.749±0.074 0.678±0.098 +Ebola +Allergy +AUC +F1 +MCC +AUC +F1 +MCC +No Pretrain +0.967±0.031 0.796±0.115 0.787±0.113 0.994±0.006 0.976±0.017 0.958±0.019 +ESM-1 +0.970±0.035 0.845±0.098 0.836±0.097 0.994±0.006 0.988±0.006 0.975±0.011 +MSA-1b +0.970±0.026 0.803±0.103 0.799±0.096 0.966±0.019 0.918±0.037 0.835±0.093 +Ablang-H +0.951±0.024 0.700±0.087 0.691±0.079 0.821±0.093 0.734±0.091 0.459±0.190 +Ablang-L +0.945±0.027 0.719±0.110 0.710±0.100 0.782±0.136 0.735±0.104 0.381±0.262 +AntiBERT +0.959±0.030 0.804±0.110 0.795±0.106 0.993±0.008 0.984±0.009 0.970±0.014 +EATLM +0.969±0.036 0.848±0.096 0.841±0.094 0.995±0.006 0.988±0.005 0.975±0.011 +EATLM w/o AGP +0.967±0.032 0.843±0.097 0.835±0.093 0.996±0.004 0.983±0.006 0.968±0.007 +EATLM w/o MPP +0.969±0.037 0.840±0.094 0.828±0.098 0.996±0.004 0.980±0.005 0.961±0.011 +EATLM w/o MPP & AGP 0.970±0.025 0.821±0.098 0.811±0.091 0.996±0.002 0.981±0.007 0.963±0.009 +SLE +MS +AUC +F1 +MCC +AUC +F1 +MCC +No Pretrain +0.994±0.004 0.982±0.009 0.960±0.012 0.841±0.182 0.847±0.129 0.617±0.378 +ESM-1 +0.992±0.005 0.979±0.014 0.954±0.021 0.879±0.127 0.845±0.123 0.621±0.358 +MSA-1b +0.998±0.001 0.988±0.007 0.973±0.011 0.853±0.150 0.836±0.135 0.594±0.392 +Ablang-H +0.994±0.003 0.964±0.022 0.931±0.021 0.836±0.184 0.828±0.108 0.562±0.353 +Ablang-L +0.995±0.002 0.977±0.017 0.956±0.021 0.846±0.175 0.852±0.115 0.633±0.351 +AntiBERT +0.994±0.009 0.952±0.068 0.919±0.096 0.893±0.095 0.854±0.112 0.701±0.214 +EATLM +0.990±0.008 0.970±0.018 0.939±0.018 0.847±0.135 0.798±0.156 0.572±0.294 +EATLM w/o AGP +0.990±0.007 0.962±0.031 0.929±0.040 0.846±0.132 0.799±0.162 0.584±0.284 +EATLM w/o MPP +0.981±0.021 0.940±0.063 0.891±0.086 0.809±0.185 0.807±0.152 0.535±0.381 +EATLM w/o MPP & AGP 0.988±0.010 0.952±0.051 0.915±0.070 0.827±0.159 0.798±0.147 0.547±0.311 +24 + +Table 14: Individual-level predictor for disease diagnosis. +Pateint-level +SARS +HIV +AUC +F1 +MCC +AUC +F1 +MCC +No pretrain +0.975±0.033 0.962±0.024 0.902±0.062 0.920±0.092 0.815±0.117 0.776±0.122 +ESM-1 +0.979±0.017 0.933±0.032 0.824±0.090 0.967±0.056 0.883±0.107 0.849±0.135 +MSA-1b +0.989±0.012 0.954±0.033 0.887±0.078 0.962±0.039 0.833±0.068 0.789±0.072 +Ablang-H +0.988±0.013 0.968±0.024 0.920±0.063 0.960±0.040 0.788±0.134 0.744±0.137 +Ablang-L +0.990±0.015 0.938±0.039 0.843±0.091 0.931±0.069 0.798±0.131 0.742±0.170 +AntiBERT +0.983±0.017 0.976±0.022 0.938±0.056 0.969±0.053 0.878±0.112 0.850±0.134 +EATLM +0.977±0.028 0.983±0.017 0.955±0.045 0.978±0.054 0.914±0.081 0.884±0.106 +EATLM w/o AGP +0.983±0.019 0.973±0.025 0.929±0.066 0.978±0.054 0.903±0.076 0.869±0.099 +EATLM w/o MPP +0.987±0.015 0.969±0.024 0.921±0.061 0.976±0.053 0.906±0.092 0.880±0.112 +EATLM w/o MPP & AGP 0.986±0.014 0.969±0.023 0.920±0.061 0.973±0.052 0.919±0.096 0.896±0.117 +Ebola +Allergy +AUC +F1 +MCC +AUC +F1 +MCC +No Pretrain +0.994±0.017 0.933±0.133 0.934±0.132 1.000±0.000 1.000±0.000 1.000±0.000 +ESM-1 +1.000±0.000 0.933±0.133 0.934±0.132 1.000±0.000 1.000±0.000 1.000±0.000 +MSA-1b +1.000±0.000 0.933±0.133 0.934±0.132 1.000±0.000 1.000±0.000 1.000±0.000 +Ablang-H +0.944±0.167 0.767±0.213 0.750±0.254 1.000±0.000 0.960±0.080 0.916±0.169 +Ablang-L +0.961±0.100 0.731±0.205 0.703±0.277 1.000±0.000 0.960±0.080 0.916±0.169 +AntiBERT +0.978±0.067 0.933±0.133 0.934±0.132 1.000±0.000 1.000±0.000 1.000±0.000 +EATLM +1.000±0.000 0.947±0.111 0.944±0.114 1.000±0.000 1.000±0.000 1.000±0.000 +EATLM w/o AGP +1.000±0.000 0.933±0.133 0.934±0.132 1.000±0.000 1.000±0.000 1.000±0.000 +EATLM w/o MPP +1.000±0.000 0.933±0.133 0.934±0.132 1.000±0.000 1.000±0.000 1.000±0.000 +EATLM w/o MPP & AGP 0.994±0.017 0.933±0.133 0.934±0.132 1.000±0.000 1.000±0.000 1.000±0.000 +SLE +MS +AUC +F1 +MCC +AUC +F1 +MCC +No Pretrain +1.000±0.000 1.000±0.000 1.000±0.000 0.700±0.200 0.900±0.137 0.893±0.400 +ESM-1 +1.000±0.000 1.000±0.000 1.000±0.000 0.900±0.200 0.933±0.133 0.900±0.200 +MSA-1b +1.000±0.000 1.000±0.000 1.000±0.000 0.700±0.400 0.893±0.137 0.700±0.400 +Ablang-H +1.000±0.000 1.000±0.000 1.000±0.000 0.800±0.245 0.893±0.137 0.700±0.400 +Ablang-L +1.000±0.000 1.000±0.000 1.000±0.000 1.000±0.000 0.960±0.080 0.800±0.400 +AntiBERT +1.000±0.000 1.000±0.000 1.000±0.000 1.000±0.000 1.000±0.000 1.000±0.000 +EATLM +1.000±0.000 1.000±0.000 1.000±0.000 1.000±0.000 0.867±0.163 0.800±0.245 +EATLM w/o AGP +1.000±0.000 1.000±0.000 1.000±0.000 1.000±0.000 0.867±0.163 0.800±0.245 +EATLM w/o MPP +1.000±0.000 1.000±0.000 1.000±0.000 1.000±0.000 0.827±0.150 0.600±0.374 +EATLM w/o MPP & AGP 1.000±0.000 1.000±0.000 1.000±0.000 1.000±0.000 0.827±0.150 0.800±0.374 +25 + diff --git a/vNFLT4oBgHgl3EQfjy8P/content/tmp_files/load_file.txt b/vNFLT4oBgHgl3EQfjy8P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..09703a2ec6955b8823f7d67bb3de73033b39756d --- /dev/null +++ b/vNFLT4oBgHgl3EQfjy8P/content/tmp_files/load_file.txt @@ -0,0 +1,2069 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf,len=2068 +page_content='ON PRE-TRAINED LANGUAGE MODELS FOR ANTI- BODY Danqing Wang∗ Department of Computer Science University of California, Santa Barbara danqingwang@ucsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='edu Fei Ye ByteDance AI Lab ByteDance yefei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='joyce@bytedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='com Zhou Hao∗ Insititute for AI Industry Research Tsinghua University zhouhao@air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='cn ABSTRACT Antibodies are vital proteins offering robust protection for the human body from pathogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The development of general protein and antibody-specific pre-trained language models both facilitate antibody prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, few studies comprehensively explore the representation capability of distinct pre-trained lan- guage models on different antibody problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Here, to investigate the problem, we aim to answer the following key questions: (1) How do pre-trained language models perform in antibody tasks with different specificity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2) How many ben- efits will the model gain if we introduce the specific biological mechanism to the pre-training process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (3) Do the learned antibody pre-trained representations make sense in real-world antibody problems, like drug discovery and immune process understanding?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Previously, no benchmark available largely hindered the study to answer these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To facilitate the investigation, we provide an AnTibody Understanding Evaluation (ATUE) benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We comprehensively evaluate the performance of protein pre-trained language models by empirical study along with conclusions and new insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Our ATUE and code is released at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='com/dqwang122/EATLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 1 INTRODUCTION Antibodies are special type of proteins extensively used as diagnostic and therapeutic tools against diverse diseases, such as SARS-CoV-2 (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Deciphering the information stored in antibody sequences is highly important with regard to real-world therapeutic antibody development and immune understanding (Greiff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Yermanos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Fortunately, the recent progress of general Pre-trained Protein Language Models (PPLM) and specific Pre-trained Antibody Language Models (PALM) provide new prospects for antibody-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For example, many studies have shown that the representations learned by PPLMs are promising to transfer to antibody tasks (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Zaslavsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For PALMs, they have been shown to improve model performance in antibody paratope predictions (Leem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Despite the current success, few studies comprehensively investigate the representation capability of different pre-trained language models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' general PPLM and specific PALM) on distinct antibody tasks, which limits our ability to design better architectures that can help antibody discovery and modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For example, in Figure 1, we compare the performance of the pre-trained protein language model ESM (Rives et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021), the pre-trained antibody language model AntiBERT (Leem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021), and the model trained from scratch on three antibody tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' These three tasks range from low to high in terms of antibody specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Here, specificity refers to the antibody’s evolution, for example, the antibody evolves to obtain the ability to bind antigen (The definition is discussed in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1 and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ∗Work was done in ByteDance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='12112v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='CL] 28 Jan 2023 Performance Increase (%) Antibody Specificty Relatedness Low Medium High 0 10 ‑10 No Pretrain ESM‑1 AntiBERT EATLM Figure 1: Performance of pre-trained language models on tasks with different specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ESM- 1 belongs to PPLMs and AntiBERT belongs to PALMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' EATLM is the method with a specific antibody mechanism introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It can be observed that although ESM performs well in task that has low relevance with antibod- ies, the performance dramatically degrades in tasks of higher relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Besides, AntiBERT does not show clear advantages over the non- pretrain model in the high-specificity task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results point out the weaknesses of current pre- training language models for antibody studies: Direct adaptation of general PPLM representa- tions may damage the performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Pre-training strategies for PALMs cannot fit the specific an- tibody biological function well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' All of these call for a comprehensive guideline of model de- signing for different antibody tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Mainly, we focus on answering the following questions: (I) How do pre-trained language models per- form in antibody tasks with different speci- ficity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Addressing of the question is mainly hindered by two challenges: the lack of a reliable antibody-specific benchmark for performance evaluation and a comprehensive study of current PPLMs and PALMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (II) How many additional benefits will the model gain if the biological mechanism is introduced to the pre-training process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibody-specific evolution has been em- ployed by many computational biology studies and shows promising results in antibody-related tasks, such as disease diagnosis and therapeutic antibody development (Yermanos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Miho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then, it is interesting to know whether antibody representation learning can benefit from the incorporation of antibody-specific evolution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (III) Does the learned antibody pre-trained representations make sense in real-world antibody problems, like drug discovery and immune process understanding?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Since antibody is so important that is extensively used for drug development in the real world, it is interesting to know whether pre-trained representation can indeed help biologists to understand antibody functions or discover drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To investigate these questions, we first propose antibody study benchmark AnTibody Understanding Evaluation (ATUE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This is the first antibody benchmark with four real-world supervised tasks covering therapeutic antibody engineering, B cell analysis, and antibody discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To evaluate models on different aspects of antibody biological functions, these tasks are designed to have specificity ranging from low to high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Based on ATUE, we perform comprehensive empirical studies to investigate the representation ability of distinct pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We also explore the influence of involving specific biological mechanisms in antibody pre-training by introducing two simple objectives to tailor masked language modeling for evolution: (1) Ancestor germline prediction guides the model to discriminate the evolutionary relationship between antibody and ancestral sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2) Mutation position prediction mimics hypermutation during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The method is used as an analytic tool to investigate the representation ability of antibody evolution- tailored language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Finally, we take a close look at the SARS-CoV-2 antibody discovery to investigate the pre-trained representation under a real-world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Our contributions can be divided three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For facilitating antibody application studies, we develop the first comprehensive antibody bench- mark to benefit the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We also introduce two objectives tailored for antibody evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For providing guidelines for a better antibody representation, we have the following key observa- tions: (i) PPLMs perform well on antibody tasks that have a high relationship with structure, such as the binding with antigen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, they perform worst in tasks with high antibody specificity, indicating representation benefiting protein structure prediction is harmful to antibody-specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (ii) In most cases, PALMs perform as well as or even better than PPLMs with less pre- training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (iii) PALMs can be improved by incorporating the evolution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, the evolution information from MSAs does not always benefit the antibody tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (iv) The intro- duction of two antibody biological mechanisms facilitates PALMs with more antibody-specific features and improves model performance in the task with high antibody specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This is the first attempt showing how antibody specific evolutionary information can be incorporated in pre-training language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 2 For accelerating real-world antibody discovery, we identified 11 potential SARS-CoV-2 binders whose sequences are highly identical to existing therapeutic antibodies binding with the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 2 RELATED WORK Our work focuses on researching the effectiveness of protein and antibody pre-trained language models for antibody-specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Below we review the representative existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We list the details in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Pretrained Protein Language Models (PPLMs) There is an increasing interest in exploring large- scale language models using protein sequences (Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Madani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Meier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' These models have been shown to achieve state-of-art capacity in predicting protein structure and function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ProtTrans (Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) and ESM-1b (Rives et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) take individual protein sequences as input and adopt Transformer language models for pre-training, demonstrating that self-supervision is a promising paradigm for protein secondary structure, contact, homology predictions, and function prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To extract evolutionary information from protein sequences, Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2021) proposed the MSA-transformer/MSA-1b model utilizing multiple sequence alignment (MSA) instead of a single query sequence as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This model is superior to ESM-1b for structure prediction, demonstrating evolution information can benefit protein representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Despite the progress in the field, few studies reported protein PLM transfer learning results on antibody tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Pretrained Antibody Language Models (PALMs) Encouraged by the success of PLMs in protein representation learning, series work seeks to learn antibody representations based on sequences of antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (Leem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Ruffolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Olsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Prihoda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' AntiBERTy (Ruffolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) proposed the first antibody-specific language model, exploring a Transformer trained on 558M natural antibody sequences in the OAS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The case study in the work showed the representation obtained from the PLM is useful for antibody sequence clustering into trajectories resembling affinity maturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Olsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2022b) train two language models for antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A heavy chain version Ablang-H is trained on 14M sequences and a light chain version Ablang-L on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='19M sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The study reported transfer learning results on restoring missing residues of antibody sequences, which is a task similar to pre-training objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' AntiBERTa (Leem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) train the antibody language model on OAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Although this paper claims finetuning AntiBERTa for paratope position prediction can achieve state-of-the-art performance, the experimental results lack standard deviations, making it unclear how significant the results obtained are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Recently, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2022) proposed a antibody-specific language model and explored its performance in SARS-CoV-2 antigen binding, showing context-dependent representations of antibody sequences benefit binding prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Table 1: Pre-training language models for protein and antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Evolution denotes whether evolutionary-related sequences are used during the pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' MLM is masked language modeling pretraining objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' HC, antibody heavy chain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' LC, antibody light chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Model Category Dataset Evolution Objective Antibody Tasks ESM-1 (Rives et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) PPLM UniRef50 (27M) × MLM MSA-1b (Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) PPLM UniRef50 (26M MSAs) ✓ MLM Ablang-H (Olsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022b) PALM OAS (14M HC) × MLM Reconstruction Ablang-L (Olsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022b) PALM OAS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='19M LC) × MLM Reconstruction AntiBERTa (Leem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) PALM OAS (72M) × MLM Paratope Prediction EATLM PALM OAS (20M) ✓ MLM, AGP & MPP ATUE 3 FRAMEWORK In this section, we first give a brief introduction to the antibody and its specific evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then we propose the first antibody-specific benchmark (ATUE) composed of four tasks with different specificities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Finally, we implement several PPLMs and PALMs baselines and design an evolution- aware PALM to incorporate the biological mechanism into the pre-training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1 BACKGROUND Antibody Antibodies are vital proteins generated by the immune system to remove harmful foreign pathogens in the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' they can specifically bind to antigens on the pathogen and recognize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibodies are composed of two identical heavy chains and two identical light chains and form a large Y-shaped structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Two tips on it contain highly variable loops, called Complementarity Determining Regions (CDR), which function for antigen binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibody Specific Evolution Notably, the antibody evolution process is significantly different from that of proteins, providing a good opportunity for us to investigate the impact of general PPLMs on specific subdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To perform its protective function, the antibody sequence undergoes evolution selection to search for optimal patterns that can specifically recognize pathogens (Honjo & Habu, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Deciphering the information stored in antibody sequences may benefit our understanding of disease and accelerate therapeutic antibody development (Greiff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Yermanos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' During evolution, the random recombination of V/D/J-gene segments provides the initial diversity for the ancestor sequence (germline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Upon exposure to a pathogen, this sequence undergoes frequent sequence mutations to search for progeny antibody sequences with optimal binding specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In other words, gene recombination provides millions of germlines in the human body, and the germlines further mutate into a huge number of progeny antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Thus, the ancestor relationship between an antibody and its corresponding germline as well as the mutation it undergoes together determine the unique biological functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In brief, the evolutionary relationships between antibodies arise to gain new functions such as antigen binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It is significantly different from that of proteins, which are to certain functions across different organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We further illustrate this process in Figure 7 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Unsupervised Antibody Corpus To obtain the evolutionary information of antibody sequences, we utilize Observed Antibody Space (OAS), a database containing more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 billion natural antibody sequences (Kovaltsuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Olsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022a) The antibody sequences in the database have been precisely annotated with evolutionary and structural information, including the paired germline and CDR3 for each antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To pair the antibody with its germline used in the pretraining task, we used the annotated sequences provided in the OAS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Further information on data processing can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2 ANTIBODY UNDERSTANDING EVALUATION (ATUE) We provide four biologically relevant downstream prediction tasks to serve as antibody benchmarks, covering four major application aspects for antibodies in the real world: therapeutic antibody engineering, disease diagnostics, antibody discovery, and B cell maturation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Notably, the antibody evolution relatedness of these tasks ranges from low to high, offering scaled tasks with subdomain specificity for pre-trained language model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Detailed information is listed in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' All data are publicly open and used under the right license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For each task, we focus on the following aspects and leave the details in Appendix: [Definition] The formal definition of the task and the understanding ability required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Impact] The importance of the task in the biological area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Details in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Dataset] The data source and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Details of data processing in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2 [Specificity] Antibody’s specific evolution characteristics that is different from general pro- teins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The definition of antibody specific evolution is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Quantitative analysis the scale of antibody specificity for every task is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We use several classification metrics to evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Accuracy (ACC) calculates the ratio of correct predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Matthews Correlation Coefficient (MCC) is the coefficient between true and predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' F1 is the average weighted score of precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' AUC is the area under the ROC curve, which shows the performance at all classification thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antigen Binding Prediction is a binary sequence classification task to determine whether the CDR region of the antibody can bind to the specific antigen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Impact] A better understanding of the binding affinity between antibody and antigen can accelerate the affinity optimization of therapeutic antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Dataset] We collect the antigen binding data from (Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) and follow the training/validation/test split of 15,128/3,242/3,242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The paratope data is collected from 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibody discovery Q2 classification C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' B cell classification Q6 classification maturation SARS‑CoV‑2 virus 000010110100 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antigen Binding Q2 classification .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Bind B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Paratope Prediction Sequence labeling Task Specificity High Low Figure 2: Antibody prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The specificity of tasks ranges from low to high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (Liberis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018) with 1,662 CDR segments on 277 antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Specificity] Low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' All the antibodies sequence in the dataset are derived from a single germline sequence indicating the task is not antibody-specific evolution-related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Paratope Prediction is to identify binding positions on the antibody sequence, which is a sequence labeling task to predict a 0/1 label for each residue of CDR fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Impact] The exploration of paratope (binding positions between antibody and antigen) can help to understand the binding mechanisms of therapeutic antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Dataset] The paratope data is collected from (Liberis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018) with 1,662 CDR segments on 277 antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Specificity] This task is medium specificity related because only partial antibodies from the database are derived from evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' B Cell Maturation Analysis It is a 6-category classification task to distinguish the maturation stage of B cell antibody sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Each sequence belongs to one of {immature, transitional, mature, plasmacytes, memory IgD+, memory IgD-}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It requires the model to learn a representation sensitive to different maturation states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Impact] It benefits the understanding of the mechanism during immune evolution, which is a critical biological process in the immune system affecting the function and antigen specificity of antibodies (Ghraichy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Meffre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Dataset] We collect 88,094 sequences from (Mroczek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Specificity] High.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibody evolution is highly coupled with B cell maturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (Meffre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2000) Antibody Discovery The task is a binary sequence classification task to distinguish which antibody is directly responsible for SARS-CoV-2 binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The task is highly challenging from two aspects: (1) Less than 1% of antibodies from SARS-CoV-2 patients are directly responsible for virus binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2) It is hard to get a reliable sequence-level classifier using unreliable and noisy individual-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Impact] Antibody discovery from B cell repertoire has been widely recognized as a important approach to accelerate antibody discovery for diverse antigens (Weiner, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Pedrioli & Oxenius, 2021), and achieved great success for SARS-CoV-2 antibody discovery (Kovaltsuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Shiakolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Dataset] We collected antibody sequences from 133 SARS-CoV-2 patients and 87 health persons from OAS and followed the processing pipeline of (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Inspired Zaslavsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2022), we match the high-ranked sequences with the sequences in the CoV- AbDab (Raybould et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) database, which have been proved to bind SARS-CoV-2 using wet-lab experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' [Specificity] High.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It is widely reported antibodies derived from the same disease such as SARS-CoV-2 share strong convergent germline signals (Galson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3 EXPERIMENT SETUP Based on the antibody benchmark ATUE, we evaluate the performance of current pertaining language models in different specificity tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Furthermore, to investigate the benefit of introducing the 5 XXbiological mechanism, we incorporate evolution information as the extra pretraining objectives for PALMs and propose EATLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The detailed description can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 Current Pre-trained language models Existing antibody and protein language models are summa- rized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Since the code and pre-training data of AntiBERTa are not released, we train a BERT model named AntiBERT on the full OAS database following the same setting as the original study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' MSA-1b (Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) takes protein-specific evolutionary sequences (Multiple Sequence Alignment, MSA) as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Because it is hard to align sequences between antibodies due to the diversity of CDR3, we take the germline and create pseudo-MSAs with depth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We add a linear layer on top of the language models and finetune the whole model on the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Evolution-aware antibody pretraining method To incorporate the biological mechanism into the pre-training, we propose a model with evolution information: Antibody EvoluTion-aware pretraining Language Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The antibody can be represented as A and the germline of the individual antibody can be represented as G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Typically, PALMs are trained with basic masked language modeling (MLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Based on it, we design another two pre-training objectives to simulate the biological mechanism of antibody evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The evolutionary relationship between the antibody and its germline includes two folds: (i) Whether the antibody and the germline have an evolutionary relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (ii) How to mutate residues from the germline to get the specific antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Two evolution-related objectives are introduced to solve the above questions: ancestor germline prediction (AGP) and mutation position prediction (MPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For ancestor germline prediction, we substitute the paired germline G with random germline G′ in the batch via a probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The model is made to distinguish the ancestor germline of the antibody by capturing the shared features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To predict mutation position, for each token in the germline G, the objective is to predict a 0/1 label for each token to indicate whether this token has been mutated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For the antibody sequence S, we mask the mutation position and predict these tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The technical details are described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Hyper-parameters We use the base Transformer architecture (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2017) with 12 layers, 12 heads, and 768 hidden states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For each task in ATUE, we finetune the model with supervised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We follow the standard split of Antigen Binding Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For other tasks that do not provide a standard split, we use a 10-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Since our pre-training model learns the representation of the antibody sequence, we expand the CDR fragment to the full antibody by searching the biological database for therapeutic antibody engineering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We also use the same Transformer architecture to train from scratch for each downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This model is indicated as non-pretrain since it is not pre-trained on a protein/antibody database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Reproduction We conduct 10-fold validation on paratope prediction, B cell maturation analysis, and antibody discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For antigen binding prediction, we conduct three repetitive experiments with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We report the average results and the standard derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The benchmark and code will be released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 4 RESULTS AND ANALYSIS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1 SUPERVISED ANTIBODY PREDICTION Antigen binding Here, we evaluate the performance PLMs models for antibody binding and paratope prediction, which are less antibody specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results are shown in Table 2 It is observed that PPLMs and PALMs achieve comparable performance on this task, indicating PALMs can learn similar general protein representation as PPLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Among different PALMs, Ablang-H outperforms Ablang-L and AntiBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It indicates that separate training for heavy and light chain sequences is helpful for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The introduction of AGP and MPP provides a little improvement over AUC and F1 metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Paratope prediction As the results shown in Table 2, for paratope prediction, both PPLMs and PALMs can significantly boost the prediction accuracy over the model with pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, PALM doesn’t show much significant advantage over PPLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' EATLM achieves the best performance, especially on F1 and MCC, while other pertaining models have a high recall and a low precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It indicates that those models tend to predict more residues as the binding site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, with the introduction of mutation residue prediction, EATLM can focus on the mutated positions that are 6 Table 2: Performance of PPLMs and PALMs on antibody tasks with increasing specificity.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8 Figure 3: B cell evolution category prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For each pij in i−th row and j-th column, it means the frequency for the model to predict the antibody in i category to j category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The number is normalized by row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' B Cell Analysis In this task, we examine the repre- sentation capacity of different pre-trained language models for distinguishing different B cell mature states during evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In general, we find PPLMs can hardly distin- guish the minor differences between B cell sequences by showing compromising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Both the perfor- mance of ESM-1 and MSA-1b is significantly worse than randomly initialized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' MSA-1b performs the worst in all pre-trained language models, indi- cating representation that can perform well in pro- tein structure prediction will be harmful to antibody- specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Conversely, all PALMs show promis- ing results for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The reason may lie in that the general protein has little relationship with the special antibody mature process and can hardly capture this feature during the protein pretraining process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' EATLM significantly outperforms the other PALMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This is because by explicitly modeling the biological mechanism, our model can effectively capture the evolution feature and better distinguish between B cells at different stages of maturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We conduct further analysis to figure out whether our EATLM successfully captures sequence char- acteristics during the evolutionary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We explore the probabilities of predicting antibodies in class i to class j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results shown in Figure 3 reveal EATLM can easily classify the immature B cell with an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It is consistent with the biological study that CDR3 sequence length in immature B cells is significantly shorter than that of the other mature B cells (Ghraichy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' From the diagonal, we can figure out that our model tends to mistake the B cell sequences with their previous or post-evolutionary stage, consistent with the biological process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibody Discovery Here, we examine PPLMs and PALMS for their capacity in benefiting real- world problems and enabling the discovery of antigen-specific antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Following the methods in this study Zaslavsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2022), we considered two steps for the SARS-CoV-2 specific antibody discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' First, we generate a sequence classifier to distinguish SARS-CoV-2 antibodies using noisy individual-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then, the high-ranked sequences are matched with the sequences with true binding sequences in the CoV-AbDab (Raybould et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021) database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We use the 90% sequence identity as the threshold to determine whether the antibody is similar to existing SARS binders in the CoV-AbDab database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A high similarity indicates a high probability of having the same biological functionality as the existing binders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The details can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 7 No Pretrain EATLM AntiBERT Ablang_heavy Ablang_light ESM_1 Expected 1 2 Cumulative sum of matched antibodies Number of antibodies Figure 4: The cumulative sum of matched sequences number in the order of the pre- dicted probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' EATLM outperforms other PALMs and PPLMs for finding SARS-CoV-2 binder faster highlighted in 1⃝ and finding all antibodies faster highlighted in 2⃝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In Figure 4, we plot the cumulative sum of matched sequences number in the order of the predicted prob- ability of the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We can find that the se- quences predicted with high probability by PALMs match with the existing binders better than PPLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It means that PALMs can figure out the potential binders more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Besides, EATLM signif- icantly outperforms other PALMs as the red line shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In the early stage, this method is the quickest way to find potential binders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then it loses to Ablang- H but finally overtakes again and converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It means that EATLM is the first one to figure out all potential binders in this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Furthermore, we list several potential binders found by EATLM in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Without supervised labels, EATLM gives a high probability of 2 SARS-CoV-2 existing binding antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Besides, EATLM proposes 9 potential sequences with high CDR-H3 sequence identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Further analysis of the results reveals that EATLM enables diverse-epitope antibody discovery and priority selection, demonstrating that EATLM benefits therapeutic antibody discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The detailed explanations can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To examine whether the antibody sequences with 90% sequence identity can indeed bind the same target, we investigate the 3D structure of the true binding antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Table 4 shows only one signle atom difference between the predicted binder and the existing binder, suggesting the predicted binders are highly possible to interact with SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Table 3: The CDR3-H3 region of high-ranked sequences to bind to SARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We show the CDR3 fragment of the heavy chain in the antibody sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ‘Identity’ is the similarity between the predicted binder and the true binder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The epitope of the true binders is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The origin of the majority of the true binder sequences is B cells from patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The different amino acids between the predicted binder and the existing binder are highlighted in red No Predicted Binder Existing Binder Epitope Identity 1 AREGIVGATTGFDY AREGIVGATTGFDY spike 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='000 2 ARDLGGYFDY ARDLGGYFDY RBD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='000 3 AKDQDDAYYYYYYMDV AKDQDDGYYYYYYMDV NTD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='938 4 ASYYYDSSGYHYGMDV ASYYYDSSGYYYGMDV RBD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='938 5 ARRGLGLYYYGMDV ARRGDGLYYYGMDV S2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='929 6 ARAFRGSYYYGMDV ARATRGSYYYGMDV S2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='929 7 ARLSGSSWYFDY ARLSGSSWDFDY spike 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='917 8 ARLGSSSWYFDY ARVGSSSWYFDY spike 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='917 9 ARGWLRGYFDL ARRGWLRGYFDL RBD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='909 10 ARDWGELYFDY ARDWGEYYFDY RBD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='909 11 ARDLGGVFDY ARDLGGYFDY RBD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='900 Table 4: 3D structure of the true SARS-CoV- 2 binding antibody No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' G2A highlights the single atom difference in No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3, indicating the predicted binder is highly likely to bind the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' G2A PDB:7N62 SARS‑CoV‑2 NTD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2 HOW DOES EVOLUTION PRETRAINING TASK INFLUENCE THE REPRESENTATION?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To understand why EATLM shows better performance on antibody tasks, we analyze the pre-trained representations to evaluate the effectiveness of the evolution-aware pretraining strategies from two aspects: (1) Does the pre-trained antibody representation reflect its ancestor relationship?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2) Is the specificity of antibodies captured by the evolution objective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Ancestor Gerlime Visualization We perform UMAP visualization analyses in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' First, we observe that antibodies evolved from the same germline are nicely clustered together (Figure 5a and 5b), indicating the learned embedding is encoded with germline information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Besides, sequences with similar scales of evolutionary distance tend to cluster together, and a clear gradation of evolutionary 8 distance can be observed in Figure 5c and 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The visualization provides a sanity check for the ability of EATLM to extract the sequence information of antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Accuracy of Mutation Position Based on the specific evolution process described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1, we can find the mutation during the evolution process bring specificity to the antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Thus, we explore the model’s ability to predict mutated residue from the masked token, which can reflect the specificity feature the model captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We find that although AntiBERT can predict with an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8889 on all positions, it fails on mutation positions with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0311 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In contrast, EATLM achieves an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='443 on mutation position, which indicates that the model captures the specificity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Note that during the MPP training, we mask the mutation position on antibody sequences, which are different from its germline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Thus, the model cannot get the mutated residue from the germline directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The only way is to learn the underlying mutation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The full results are shown in Table 12a in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 25 20 15 10 5 0 5 10 20 10 0 10 20 30 0 1 2 3 4 5 6 (a) Health germlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 15 10 5 0 5 10 10 5 0 5 10 15 0 1 2 3 › (b) Patient germlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 10 5 0 5 10 15 20 15 10 5 0 5 10 15 20 0 5 10 15 20 25 30 (c) Health distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 20 10 0 10 20 15 10 5 0 5 10 15 0 5 10 15 20 25 30 (d) Patient distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Figure 5: UMAP Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (a-b) for sequences with different germlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' One color indicates one germline with its descendant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (c-d) for evolutionary phylogeny distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The shade of color indicates its distance from the ancestor germline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3 KEY OBSERVATIONS The performance of pre-trained language models highly depends on the task specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For tasks with low antibody-specificity relatedness, PPLMs show comparable performance with PALMs, indicating that transferring the general protein representations from PPLMs is an effective and time-saving way in these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' PALMs show their advantage and outperform PPLMs on medium specificity relatedness tasks, like paratope prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For tasks with high specificity, it is observed the performance of PPLMs dramatically decreased, suggesting general pre-trained protein models are not sufficient in antibody-specific representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Interestingly, the performance of MSA-1b is 20% less than the model without pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This observation is consistent with the biological study that the mechanism of antibody evolution is significantly different from that of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Therefore, the incorporation of protein evolution information is not always good for antibody tasks, especially the tasks that require antibody evolution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Performance Increase (%) Task Specificity Low Medium High 0 10 ‑10 ‑20 20 ‑30 No Pretrain ESM‑1 AntiBERT EATLM MSA‑1b Ablang‑H Ablang‑L Figure 6: Performance summary of vari- ous pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Incorporation of biological evolution mechanism into PALM generally benefits antibody prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Evolution-related training objectives help detect mutation positions on antibodies, which is a distinguishing feature from germline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Interestingly, the performance increase of EATLM compared with other PALMs is correlated with task specificity relatedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The ablation study showed the removal of the evolution-related pretraining objectives re- sults in performance decreasing, confirming that evolution- related objectives benefit the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Future studies to have more in-depth research in this direction could be promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibody pre-trained representations are helpful for real-world drug discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Using the language model, we predict the probability for each antibody of being a binder with SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Without accurate sequence-level labels, we successfully identify 11 potential antibody binders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 9 5 CONCLUSIONS AND LIMITATIONS In this paper, we have presented comprehensive empirical studies to characterize the impact of pre-trained protein and antibody language models for antibody prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Four important antibody tasks from four biological categories with evolution relatedness ranging from high to low are provided ATUE to facilitate the antibody and ML field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, there are some limitations to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' First, for the ATUE, the task diversity is highly limited because of data scarcity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' With the data increase in this field, we expect we can update our benchmark with more disease diversity and data size in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Also, we do not test any 3D structure information during antibody pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' As a special subgroup of proteins, antibody structures provide much more information such as geometry than sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In the future, recruiting structure information for antibody pre-training may be able to improve the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' REFERENCES Yunlong Cao, Bin Su, Xianghua Guo, Wenjie Sun, Yongqiang Deng, Linlin Bao, Qinyu Zhu, Xu Zhang, Yinghui Zheng, Chenyang Geng, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Potent neutralizing antibodies against sars-cov-2 identified by 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Disease diagnostics using machine learning of immune receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' bioRxiv, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Feng Zhu, Thomas Althaus, Chee Wah Tan, Alizée Costantini, Wan Ni Chia, Nguyen Van Vinh Chau, Giada Mattiuzzo, Nicola J Rose, Eric Voiglio, Lin-Fa Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Who international standard for sars-cov-2 antibodies to determine markers of protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The Lancet Microbe, 3(2):e81–e82, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 12 A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1 ANTIBODY SPECIFIC EVOLUTION Antibodies, composed of two identical heavy chains and two identical light chains, form a large Y-shaped structure, where the two tips are responsible for pathogens binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibody evolution , described by sequence-sequence relationships between ancestor and progeny antibodies, reflects antibodies’ key antigen-binding function (Honjo & Habu, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' During antibody evolution (Figure 7), the initial diversity is encoded into the ancestor sequence through randomly recombination of V-, D- and J-gene segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Upon exposure to a pathogen, the sequence undergoes frequent sequence mutations to search for progeny sequences with optimal binding specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Sequence evolution analysis has been employed by many computational biology studies and shows promising results in antibody-related tasks, such as disease diagnosis and therapeutic antibody development (Yermanos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Miho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Importantly, antibody evolution is significantly different from that of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibodies only contain hundreds of thousands ancestor sequences so-called germline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To bind dozens of millions of diverse antigens, antibodies need to mutate from the ancestor sequences to gain new functions (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Therefore, the non-conserved amino acids (mutated ones) plays important roles for structure and function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' On the contrary, the conserved amino acids (not mutated) in proteins determine structure and function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' During protein evolution, evolutionary pressure to maintain protein structure and functions leads to the conservation or co-evolution of residues located in structural folding core for binding interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Diverse methods have been developed to extract the co-evolution information from conserved amino acids sequences for structure and function prediction, such as AlphaFold (Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In brief (Figure 7), antibody evolution specificity distinct from that of proteins can be defined with two main features: (i) ancestor germlines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (ii) the mutated amino acids of germlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' V(D)J recombination Antigen B Antigen C Antigen A Protein evolution GAMQ VKQA KGLE SSITSGTNN R W PG WV IYY ASMN IRTT RSIE GPISAGSQQ K W PG WV IYY GAMQ IRAT RGID GSVSTPSGQ R W PG WV IYY GGMP AKSG KAVD GPLSGPSYA K W PG WV IYY ATMQ AQTP KSLD QSLGGPSGA Q W PG WV IYY ATNN VNQP KGLD PSITSGSGN R W PG WV IYY Protein B Protein A GAMQWVKQAPGKGLEWVSSITSGSNNIYYR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' FAE VPM YP F KLVPM QL F EKLVP W L FAEK LV PM FAEKLVPM Antibody evolution ancestor germline mutation ( ) non mutated function mutation ) (mutated ancestor functions Figure 7: Evolution specificity of B cell antibodies comparing with general proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The evolutionary sequence relationships are highlighted using blue dash lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2 DATA PROCESSING DETAILS Pairing Antibody with Germline For germline annotation in the pre-training task, we used the annotated germline sequences provided in the OAS database (Kovaltsuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For downstream benchmarks tasks like B-cell classification, therapeutic antibody engineering, and disease diagnosis, we completely followed the methods shown in the OAS database paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' IgBLAST, an immunoin- formatic benchmarking tool for the analysis of B-cell antibody repertoires was used for germline annotation (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The antibody nucleotide-containing FASTA file was aligned to germline 13 and translated to amino acids using IgBLASTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The antibody amino-acid sequence was aligned using IgBLASTp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The germline databases for human patients used ImMunoGeneTics (IMGT) germline sequences derived from IMGT (Lefranc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For each antibody, usually, multiple germline sequences can be obtained and only the single sequence showing the highest confidence score for the alignment was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Pre-training Data Processing We downloaded OAS Oct 2021 version from its website and re- moved duplicate sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To avoid data leakage, we cluster sequences based on the CDR3 sequence and filter each cluster by 70% identity over the whole sequence using Linclust (Steinegger & Söding, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then, we shuffle the dataset and split it into 100k-size chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The last chunk is used as the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The dataset size is 20,245,249 and 45,249 are used for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Table 5: ATUE benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The five downstream tasks are divided into three biological categories and each category focuses on different input levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ‘Q6’ indicates the classification task have 6 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For disease diagnosis, the size indicate the number of profiles, where each profile contains thousands to millions of sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Specificity Task Name Input Formalization Size Low Antigen Binding Prediction CDR fragment Q2 Cls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 21,612 seqs Medium Paratope Prediction CDR residue Q2 Labeling 1,662 seqs, 21,342 positions High SARS Antibody Classification Sequence Q2 Cls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 22,000 seqs High B Cell Classification Sequence Q6 Cls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 88,094 seqs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3 ATUE DETAILS We summarize the tasks used in ATUE in Table 5 and discuss each task in detail in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antigen Binding Accurate antigen-binding prediction approaches could allow significantly more efficient antibody discovery with higher affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Machine learning methods have already achieved some success in antibody binding capacity optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We collect the antigen-binding data from Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2021) and follow the training/validation/test split of 15,128/3,242/3,242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The original dataset only has CDR3 fragments, and we extend them to the full antibody sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For cross- validation, we split the dataset by antibody sequences to ensure that no antibody sequences overlap between 90% training and 10% validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Paratope Prediction Paratope is the antibody residues involved in antigen binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The ability to accurately map the paratope can provide detailed knowledge about the binding mechanism and accelerate antibody discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 1D sequence-based deep learning methods have been employed for paratope prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The paratope data is collected from Liberis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2018) with 1,662 CDR segments on 277 antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Each antibody contains three CDR fragments (CDR1, CDR2 and CDR3) in the heavy chain and three CDR fragments in the light chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We also search the full sequence for each antibody and use the whole sequence as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For cross-validation, we split the dataset by antibody sequences to ensure that no antibody sequences overlap between 90% training and 10% validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Table 6: The statistics of B cell classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Type Size Immature b cell 14,145 Transitional b cell 13,197 Mature b cell 16,139 Plasmacytes PC 22,236 Memory IgD- 8,437 Memory IgD+ 13,940 14 B Cell Analysis We formulate a 6-category classification task for B cell maturation analysis, which includes {immature, transitional, mature, memory IgD+, memory IgD-, plasmacytes,}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The analysis of B cell maturation plays an important role in understanding the mechanisms underlying B cell responses in immune system Ghraichy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Meffre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The order of B cell type follows the evolutionary process in the immune system, from immature state to transitional state and finally becomes a memory B cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Both memory IgD- and IgD+ belong to memory B cells with different isotypes, and they have a high affinity to foreign antigens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Among the other categories, the Plasmacytes PC sequences also have some affinity ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It is widely reported that changes in antibody sequence patterns correlate with B-cell maturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Therefore, we use this task to evaluate the representation learning capacity of the language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We collect 88,094 sequences from Mroczek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' They extracted from the peripheral blood of healthy adults and got six types of B cells with different maturity and antibody sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The distribution of various types of B cells in the dataset is shown in Table 6 Antibody Discovery Antibody discovery from B cell repertoire has been widely recognized as a novel trend to improve the efficiency of antibody discovery for diverse antigens (Weiner, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Pedri- oli & Oxenius, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, previous studies highly rely on expensive wet-lab experiments (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Shiakolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Deep learning based method have shown potential capacity in help antibody discovery by reduce cost and increase efficiency (Widrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Here, we ask whether pretrained models can benefit real-world problems and enable fast-track neutralization SARS-CoV-2 antibody discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In the first step, we develop a sequence classifier to distinguish which antibody sequence from the numerous sequences is responsible for the recognition of the SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This task is highly challenging since we can hardly get the sequence-level disease label that indicates whether the antibody sequence is related to the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Thus, we follow the practice of Roskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Zaslavsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2022) to use the individual label as the rough sequence label and train a sequence- level predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then, with the help of a sequence-level predictor, we can give each sequence a most likely label to help antibody discover, whose accuracy has been verified by the excellent results on individual prediction, which may accelerate the discovery of new antibody sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We follow the condition of Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2021) to filter SARS-CoV-2 antibody data from the OAS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The basic condition is ‘Chain = heavy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Isotype = IGHG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' BSource = PBMC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Species = human;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Vaccine = None’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We further add the condition of ‘Unique Sequences >= 10000’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For health/SARS we set the ‘Disease’ field to ‘None’, ‘SARS-CoV-2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then we obtain 87/133 patient profiles for each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' To make a balanced dataset, we limit the size of the health profile and mix up the healthy ones and the ones with the SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For cross-validation, we randomly split the dataset by profiles 10 times: 90% for training and 10% for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We further select sequences with top100 redundancy to make the positive labels more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='4 QUANTITATIVE ANALYSIS OF ATUE TASK SPECIFICITY It is important to include statistical significance tests relative to the antibody-specific features in antibody functional tasks we proposed in the ATUE benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' According to the evolution process shown in Figure 7, antibody evolution specificity distinct from that of proteins can be defined with two main features: (i) ancestor germlines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (ii) the mutated amino acids of germlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We implemented statistical significance tests of (i) ancestor germlines subtype usage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (ii) the number of mutated amino acids in antibodies against different labels of downstream tasks in ATUE to quantitatively assess the "Task specificity".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The analysis is now summarized in Table 7 In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Generally, it is clearly shown that ATUE benchmark comprises antibody tasks showing different scales of antibody specificity for later on modeling analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Moreover, the which are used for statistical analysis of task specificity and pre-training model objectives in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antigen Binding In the Antigen binding dataset, both antibody binding and none antigen binding sequences share the same germline subtype sequence (IGHV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1) (Figure 8A) as well as the same number of germline mutations 8B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Therefore, None of two antibody specific features show significant distribution differences between data with different labels, demonstrating antigen binding is a task with low antibody specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 15 Table 7: Task specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Summary of the statistical significance test of two antibody-specific features for different tasks in the ATUE benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Task Statistical significance (p-value) General Specificity Germline Usage Mutations Numbers Antigen Binding Nan Nan Low Paratope Prediction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='296 0 Medium B cell classification 0 0 High SARS antibody discovery 0 0 High A B Figure 8: (A)IGHV gene segment usage distribution between binding and non-binding antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (B) Germline mutation number distribution between binding antibody and non-binding antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Paratope Prediction For the paratope prediction task, we first evaluate the germline subtype distribution difference between sequences with different numbers of binding sites (Figure 9A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Kruskal Wallis test showed a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='296 suggesting germline subtype usage is not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Also, we find the binding sites can be significantly mapped with more germline mutations than the non-binding sites, which is consistent with the knowledge of antibody specificity definition (Figure 9B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' One out of two antibody specific features show significant distribution differences between data with different labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Therefore, we define this task as a medium specificity task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A B Figure 9: (A) Number of binding sites distribution between different IGHV gene segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Com- parison is performend using kruskal wallis test with p value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (B) Germline mutation number distribution between binding and non-binding positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Comparisons performed using t-tests show- ing p value equals to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 0 1 Antigenbindingclasses10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='4 Mutations 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2 of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 Number 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='6 0 1 Antigen binding classesIGHV1 IGHV2 IGHV3 IGHV4 IGHV5 IGHV610 I sites 8 Num of binding 6 4 2 IGHV7 Germline20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 + Num of Germline Mutations 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 + + + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 nonbind bind Binding sitesB Cell Analysis As shown in Figure 10, the distribution of the germline usage as well as the number of germline mutations are significantly different between antibodies in B cells with different developmental stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This observation is highly consistent with previous studies Mroczek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Ghraichy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Since both of the antibody specific feature show significant distribution differences, this task is defined as a high-specificity task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A B Figure 10: (A)IGHV gene segment usage distribution between different B cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Comparison is performend using chisquare test with p value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (B) Germline mutation number distribution between different types of B cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Comparisons performed using kruskal wallis test showing p value equals to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' SARS Antibody Discovery Antibodies in SARS patients and healthy ones show a significant difference in their germline subtype usage and the number of germline mutations (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This observation is highly consistent with previous studies showing SARS antibody convergent among patients Galson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Since both of the antibody specific feature are highly significant, this task is defined as a high-specificity task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A B Figure 11: (A)IGHV gene segment usage distribution between antibody in SARS patients and health ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Comparison is performed using chisquare test with p value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (B) Germline mutation number distribution between antibody in SARS patients and health ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Comparisons performed using kruskal wallis test showing p value equals to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 TRAINING DETAILS Antibody can be represented as A = {a1, a2, · · · , am} and the germline of individual antibody can be represented as G = {g1, g2, · · · , gn}, where m and n are the lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Each token ai or gj in the sequence is called a residue that belongs to the amino acid set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A includes 20 common amino acids 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8 e P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 cell IgD+ IgD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' PC immature B cell classes25 F mutations S 20 15 of nN 10 5 0 cell IgD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' cell Cell rer in B cell classesIGHJ1 IGHJ2 IGHJ3 IGHJ4 IGHJ5 IGHJ6L 30 25 ions mutati 20 15 of w nN 10 5 0 B cell classes1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8 e P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' B cell classesIGHV1 IGHV2 IGHV3 IGHV4 IGHV5 IGHV6with a residue ‘X’ that indicates the residue is unknown (mostly in the germline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Typically, antibody PLMs are trained with basic mask language modeling objective lMLM on the antibody sequences S = A = {a1, · · · , am, }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1 EVOLUTION-AWARE PRETRAINING In order to incorporate the evolutionary information into the pre-training, we pair the antibody sequence A with its germline G and concatenate them into a long sequence with a special token ‘[SEP]’ as the delimiter: S = {s1, · · · , sm+n+1} = {a1, · · · , am, [SEP], g1, · · · , gn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Thus, we optimize MLM objective on the long sequence S: lMLM = − 1 |M| � i∈M log p(si|S\\M), (1) where M is the index set of masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It helps the model to learn the basic residue distribution for antibody sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Besides, it can also capture the interaction between residues of the antibody and its germline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' DTVMTQS……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='QYKHWPPYTFGRGTKLEIR EIVMTQS……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='QYNNWPXXXXXXXXXXXXX [SEP] Transformer Encoder [CLS] Antibody Germline Ancestor Germline Prediction (AGP) 1 1 0 0 0 … 1 1 D T K H P Y … Mutation Position Prediction (MPP) 𝒀𝝐{𝟎, 𝟏} mask mutation substitute germline (a) Pre-training with two biological evolution tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' DTVMTQS……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='QYKHWPPYTFGRGTKLEIR EIVMTQS……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='QYNNWPXXXXXXXXXXXXX [SEP] Transformer Encoder [CLS] B cell Classification FWR FWR CDR3 ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' FWR FWR CDR3 ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antigen Binding Sequence- level Individual- level Disease Diagnostics (b) Finetuning for three biological categories in ATUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Figure 12: EATLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In Figure 12a, AGP randomly unpairs the germline sentence and predict the ancestor relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' MPP predicts the mutation position on the germline and the masked mutation residue on the antibody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Based on the input, the three categories in ATUE can be divided into sequence-level and individual-level (Figure 12b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For individual-level disease diagnostics, we score each sequence in the individual profile and calculate the trimmed mean over all sequences to get the individual score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Ancestor Germline Prediction The ancestor relationship between the antibody and its germline determines the shared biological functions obtained in the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antibody sequences with similar residues evolved from different germline sequences may have different biological functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' When stimulated by a foreign antigen, the common ancestor germline evolve to various antibody sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Similar antibody sequences may have different germline sequences, which will affect their biological functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Thus, the aim of this task is to determine whether the antibody have evolutionary relationship with the given germline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' During training, we substitute the paired germline G with random germline G′ = {g1, · · · , gn} in the batch via a probability p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The new sequence is denoted as S′ = {a1, · · · , am, [SEP], g′ 1, · · · , g′ n} and the training loss can be described as: la = − log p(y|S′), (2) where y ∈ {0, 1} indicate whether the noisy germline G′ is the ancestor of the antibody S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It can help the model to distinguish the ancestor germline of the antibody by capturing the shared features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Mutation Position Prediction The somatic hypermutations on the germline further give progeny antibodies the specificity of binding with the specific antigen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In order to model this specificity, this task focus on predicting the mutation positions and the residues mutated to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Specifically, for each token gj in the germline G, the target is to predict a label yj ∈ {0, 1} to indicate whether this token has been mutated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For the antibody sequence S, we mask the mutation position and predict these tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The objective can be formalized as: lm = − 1 n � j∈{1,··· ,n} log p(yj|S\\M ′) − 1 |M| � i∈M ′ log p(ai|S\\M ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (3) 18 maturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='Bind Notbind Binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='10110008 HealthDisease:Here, M ′ is the ground-truth mutation position and we mask these tokens on the antibody sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This task is more difficult than MLM which equally masks tokens in the L, because the tokens on the mutation position of A get less information from the germline, compared with other shared residues between the antibody and the germline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' By optimize this objective, the model learn to capture the specificity obtained from the somatic hypermutation in the evolutionary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2 IMPLEMENTATION DETAILS We use the base Transformer architecture (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2017) with 12 layers, 12 heads, and 768 hidden states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The total parameters are 86M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We use Adam optimizer (Kingma & Ba, 2015) with the maximum learning rate of 2e-4 and the warm-up step of 24,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The maximum length is set to 400 since most antibody sequences are shorter than 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We first pre-train our model with the MLM objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' During the pre-training, 15% tokens are randomly selected with 80% masked, 10% replaced, and 10% kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then we conduct further pre-training on two antibody-related tasks with a smaller learning rate of 1e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For each task in ATUE, we finetune the model with supervised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We follow the standard split of Antigen Binding Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For other tasks that do not provide a standard split, we conduct 10-cross validation and report the average results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Since our pre-training model learns the representation of the antibody sequence, we expand the CDR fragment to the full antibody via searching the biological database for therapeutic antibody engineering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For finetuning, we limit the max epochs to 30 and use the Adam optimizer with a max learning rate of 3e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We use the mean representation of 12 layers as the sequence representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Table 8: The pretraining details for different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The ‘Germline’ is the prediction accuracy of AGP, and the ‘Position’ and ‘Mutation’ are the accuracy of mutation positions and the mutated residues respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Model Size Loss Step Germline Position Mutation AntiBERT 85M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='437 568000 / / / AntiBERT w AGP 85M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='558 587000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='330 / / AntiBERT w MPP 85M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='744 622000 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='410 ESM-1 FT 85M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2636 136000 / / / ESM-1b FT 650M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='2704 278000 / / / EATLM w/o AGP & MPP 85M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='143 108000 / / / EATLM w/o AGP 85M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='744 134000 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='443 EATLM w/o MPP 85M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='001 126000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='000 / / EATLM 85M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='856 252500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='9960 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='443 EATLM-large w/o AGP & MPP 650M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='149 193000 / / / EATLM-large w/o AGP 650M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='753 267000 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='438 EATLM-large w/o MPP 650M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='001 200000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='000 / / EATLM-large 650M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='773 384000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='996 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='431 The model is trained with 108,000 steps and gets a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='9606 token accuracy on the MLM task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It takes further steps for AGP and MPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The model quickly converges for AGP and gets a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='99 accuracy on the ancestor germline prediction because more than 80% residues are shared between the antibody and its germline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For MPP, it can predict the mutation position with the accuracy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='00 and obtains a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='4430 accuracy in the mutation position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It means that the model can easily find the mutation positions by the self-attention between the antibody and germline, but it is still difficult to predict which residues this position will mutate to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We assume it is because the ancestor germline can undergo different somatic hypermutations and get various progeny antibodies, resulting in different valid mutations at the same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We also compare this mutation accuracy with the model without MPP, which is only trained with MLM on the concatenation of the antibody and its germline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' With a high prediction accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8889 on all positions, it achieves only a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0311 accuracy on the mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It implies that the masking among all positions on the sequence can do accurate predictions of the shared residues but hardly capture the mutation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 19 We also conduct AGP and MPP to finetune the baseline model AntiBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The pre-training results are shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We can find that without the concatenation of the antibody and its germline, it is difficult to predict the ancestor relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It also underperforms than EATLM in MPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Negative sampling ratio We have tried the ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='75 and found that this ratio has little influence on performance and convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' As we discuss in the appendix, the model can quickly converge for AGP and get an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Finetuned Protein Language Models and Larger Architecture We pre-train our method with a larger architecture and compare it with ESM-1b, which also has 650M parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We also further pre-trained the ESMs to transfer to the antibody field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' After that, we evaluate them on the antigen binding and paratope prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results are shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The result show that the larger architecture does show advantage in terms of the performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For antigen binding, ESM-1b has better performance than ESM-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, for paratope prediction, it performs worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In addition, for ESM, fine-tuning of the antibody dataset may cause the overfitting problem, leading to the decrease in the performance of all three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Table 9: The performance of larger models and finetuned protein language models on three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The ESM-1 and ESM-1b models have 85M and 650M respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The EATLM-large have similar architecture with ESM-1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ‘FT’ indicates the model is further finetuned on our antibody pre-training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Antigen Binding Paratope AUC F1 MCC AUC F1 MCC ESM-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='917+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='854+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='689+0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='887+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='698+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='575+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='024 EATLM-large 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='921+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='854+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='677+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='887+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='685+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='561+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='020 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='6 DISCUSSION ABOUT OF EATLM First, EATLM doesn’t use any 3D structure information during pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' As a special subgroup of proteins, antibody structures provide much more information such as geometry than sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' In the future, recruiting structure information for antibody pre-training may be able to improve the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, the data scale available for antibody structure is dramatically less than that of antibody sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The largest dataset of antibody structures only contains thousands of 3D high-resolution structures, while the number of antibody sequences is in billions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Using structure prediction methods like AlphaFold may help to bridge the gap between sequences and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Second, EATLM requires germline as input during downstream tasks, this will slow down the prediction speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='7 NEW SARS BINDER DISCOVERY The main challenge for disease diagnosis is to distinguish the disease-related antibodies from millions of antibody sequences in the individual profile, as stated in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Here, with the help of a sequence-level predictor, we can give each sequence a most likely label to help antibody discover, whose accuracy has been verified by the excellent results on individual prediction, which may accelerate the discovery of new antibody sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' SARS Sequence-level Predictor As described in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3, we first train a sequence-level predictor for SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results are shown in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Compared with Compared with Figure 4 in the main text, we find that good results in the sequence-level predictor do not necessarily mean good results in the antibody discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It can be mainly affected by the noisy label of the sequence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 20 Table 10: Sequence-level predictor for SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Sequence-level SARS AUC F1 MCC No pretrain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='894±0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='099 AntiBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='916±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='810±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='661±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='060 EATLM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='904±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='808±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='643±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='069 EATLM w/o AGP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='904±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='808±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='644±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='078 EATLM w/o MPP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='901±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='808±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='648±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='069 EATLM w/o MPP & AGP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='901±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='807±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='645±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='067 Figure out SARS Binders As shown in Table 3 in the main body, we find 2 true SARS binders and 9 potential binders with the help of EATLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Specifically, we first use our sequence-level predictor to get a probability score for each sequence in the SARS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then we select the sequence with high-ranked score (the probability > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5) and compare them with the public Cov-AbDab database Raybould et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2021) 1, which contains data on published/patented antibodies known to bind to SARS-CoV-2 (Raybould et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Since the CDR3 fragment in the heavy chain is the most relevant to the binding of antibody and antigen, we calculate the edit distance between the CDR3 fragments in heavy chains (CDR-H3) with those of the known binder and use a threshold of 85% similarity as the sequence identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 85% Hamming distance for B cell antibody sequence clustering (identify similar B cell antibody sequences responding to the same antigen/epitope) was previously suggested in this paper (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This method then was widely used for B cell antibody repertoire analysis in different studies (Montague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' SARS Binder Analysis To provide a more intuitive analysis of the similarity between our predicted antibody and true SARS-CoV-2 binders, we investigate the 3D structure of the true binding antibodies and the mutation site of our predicted sequence on the corresponding structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' High resolution structure of true binding antibody #3 in Table 3 with SARS-CoV-2 are shown in Figure 13 (PDB code: 7N62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The interaction interface between the antibodies and SARS-CoV-2 spike/RBD are shown in Figure 3 in main body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' CDR-H3 were shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Only one single atom highlighted in red is different between predicted binder and true binder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Obviously, these different residues don’t localize to direct binding site and CDR-H3 founding core, suggesting the sequence difference likely will not affect antibody virus interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Furthermore, we found the epitopes of the 11 identified SARS-CoV-2 antibodies cover a wide range of different structures from traditional RBD domain to novel non-RBD epitopes like S2 and NTD as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' This result shows our method enables diverse-epitope antibody discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Table 11: SARS-CoV-2 antibody hit rate with different probability thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ‘Total’ is the total number of sequences whose probabilities are higher than the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ‘Hit’ is the number of sequences that meet the similarity requirements with existing binders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Threshold Total Hit Hit rate (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 13253 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='7 10227 54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8 9338 49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='9 8178 47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='562 Probability Threshold Sensitivity In order to investigate the influence of the threshold used to determine the potential binders, we try different thresholds in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Here, the probability threshold 1http://opig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='uk/webapps/covabdab/ 21 Figure 13: High resolution structure of mutation in the predicted binder (AKDQDDAYYYYYYMDV) with the existing binding antibody (AKDQDDGYYYYYYMDV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' means that if the sequence predictor give a probability higher than the threshold for one sequence, it will be viewed as a potential binder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' If the predicted binder have a sequence similarity higher than 85% with the existing binders in Cov-AbDab, we view it as one hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' As the threshold score increases, the hit rate corresponding increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='528% to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='562%, indicating that our model may enable priority selection of SARS-CoV-2 antibodies and reduce experimental costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Sequence Similarity Sensitivity In previous work, two antibodies with the CDR-H3 similarity over 85% can be viewed as similar and have a high probability to share the same functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' And here we also check the influence on the binder matching of different thresholds of the similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results are shown in 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Here, we fix the probability threshold as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' As we can see, the baselines have similar trends in all threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' If we relax the threshold, there will be more matching sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, the predictors will have less advantage over the random order, which indicates that the ranking is less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The Potential of New Binder Discovery During the training of our sequence-level predictor, we have no reliable ground-truth labels, which means that the model has never known which sequences can bind to SARS in the real-world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, the model can learn from the noisy data and rank the real SARS binders with high probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Sequence identity of 1 means that the CDR-H3 fragment can be directly found in the Cov-AbDab database, which implies that the sequences have been verified by wet laboratory testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The other sequences with an identity over 90% are thought to have a similar binding performance to existing binders, indicating that they are promising SARS binders that can help the discovery of therapeutic antibodies for SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 0 2000 4000 6000 8000 10000120001400016000 0 50 100 150 200 EATLM Ablang-H Ablang-L ESM-1b Transformer AntiBERT Expected (a) Threshold=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8 0 2000 4000 6000 8000 10000120001400016000 0 10 20 30 40 EATLM Ablang-H Ablang-L ESM-1b Transformer AntiBERT Expected (b) Threshold=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='85 0 2000 4000 6000 8000 10000120001400016000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='5 EATLM Ablang-H Ablang-L ESM-1b Transformer AntiBERT Expected (c) Threshold=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='9 Figure 14: The cumulative count of matching sequences number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The dashed line is the expected results for a random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The x-axis is the sequence number and the y-axis is the cumulative matched sequence number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 22 SARS-CoV-2 spike G2Amut CDRH3 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3 Pred binder: AKDQDDAYYYYYYMDV True binder(7N62): AKDQDDGYYYYYYMDVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='8 EXTENTED STUDY FOR DISEASE DIAGNOSIS It would be interesting to see whether our sequence classifier can be used for other applications, such as disease diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Each human is estimated to maintain about 108 − 1010 distinct antibody sequences, constructing an informative encyclopedia recording the past and present health and disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Interpreting the pattern of the sequences has already proved useful in disease diagnosis and allows us to assess many infectious diseases without expensive laboratory testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' However, it is difficult to distinguish which antibody sequence from the numerous sequences is responsible for the recognition of the specific antigen, which hinders the discovery of the antibody for diseases (Zaslavsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Greiff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Benefiting from the recent high-throughput sequencing, we can obtain millions of antibody sequences from the individual human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' At the same time, we can get a disease label that indicates whether the human is infected by the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The main challenge is that we can hardly get the sequence-level disease label that indicates whether the antibody sequence is related to the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Thus, we follow the practice of Roskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2020) to use the individual label as the rough sequence label and train a sequence-level predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then we use this predictor to predict sequences of the individual profile and make the trimmed mean score as the individual score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We use the same data processing as Antibody Discovery stated in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' For health/SARS/HIV/Ebola/Allergy/SLE/MS, we set the ‘Disease’ field to ‘None’, ‘Ebola’, ‘Allgery’, ‘SLE’,‘MS’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Then we obtain 87/133/51/14/12/8/8 patient profiles for each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We also do 10-cross validation and select sequences with high redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Disease Classification We use all these diseases profile to build the Q7 classification task for disease diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Previous biological studies mainly use this multi-classification task for disease diagnosis Zaslavsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' (2022), highlighting the discriminatory power among different diseases are important for disease diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results are shown in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' We found both PPLM an PALM show comparable results as the random initialized model, suggesting the finetuning part play more important role and pretrained language model cannot help this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Table 12: The Q7 disease classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ‘ACC’ is the accuracy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' ACC No pretrain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='754±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='023 ESM-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='747±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='016 ESM-1 FT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='762±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='024 MSA-1b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='746±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='019 Ablang-H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='704±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='033 Ablang-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='702±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='040 AntiBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='750±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='016 EATLM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='756±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='020 EATLM w/o AGP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='754±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='020 EATLM w/o MPP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='755±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='020 EATLM w/o AGP & MPP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='756±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='021 Sequence-level Predictor for Various Disease As before, we train a sequence-level predictor for each disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results are shown in Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Compared with Table 4 in the main text, we find that good results in the sequence-level predictor do not necessarily mean good results in the individual-level predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' It is mainly due to the trimmed mean we use to get individual-level results, which is a central estimate that is robust to noise labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Overall, our model has comparable results to other models in terms of sequence prediction with noisy labels, and has better results for individual diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Individual-level Predictor for Various Disease It is observed our evolution-aware EATLM per- forms the best in the individual-level classifier to determine whether the patient suffering from SARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Besides, PALMs significantly outperform PPLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' The results is shown in Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' 23 Table 13: Sequence-level predictor for disease diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content=' Sequence-level SARS HIV AUC F1 MCC AUC F1 MCC No Pretrain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='894±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='801±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='637±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf'} +page_content='893±0.' metadata={'source': 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B¨oker +,1 T. L. Beck +,2 S. M. Birkmann +,1 G. Giardino +,3 C. Keyes +,2 N. Kumari +,4 J. Muzerolle,2 +T. Rawle +,1 P. Zeidler +,4 Y. Abul-Huda,2 C. Alves de Oliveira +,5 S. Arribas +,6 K. Bechtold +,2 +R. Bhatawdekar +,7 N. Bonaventura +,8 A. J. Bunker,9 A. J. Cameron +,9 S. Carniani +,10 S. Charlot +,11 +M. Curti +,12, 13 N. Espinoza +,2, 14 P. Ferruit +,5 M. Franx +,15 P. Jakobsen +,8 D. Karakla,2 +M. L´opez-Caniego +,16, 17 N. L¨utzgendorf +,1 R. Maiolino +,12, 13 E. Manjavacas +,4 A. P. Marston +,5 +S. H. Moseley,18 P. Ogle +,2 M. Perna +,6 M. Pe˜na-Guerrero +,2 N. Pirzkal,4 R. Plesha +,2 +C. R. Proffitt +,2 B. J. Rauscher +,19 H.-W. Rix,20 B. Rodr´ıguez del Pino +,6 Z. Rustamkulov,21 E. Sabbi +,2 +D. K. Sing +,14, 21 M. Sirianni,1 M. te Plate,1 L. ´Ubeda +,2 G. M. Wahlgren +,2 E. Wislowski,2 R. Wu,2 and +Chris J. Willott +22 +1European Space Agency, c/o STScI, 3700 San Martin Drive, Baltimore, MD 21218, USA +2Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +3ATG Europe for the European Space Agency, ESTEC, Noordwijk, The Netherlands +4AURA for the European Space Agency, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +5European Space Agency, ESAC, Villanueva de la Ca˜nada, E-28692 Madrid, Spain +6Centro de Astrobiolog´ıa (CAB), CSIC–INTA, Cra. de Ajalvir Km. 4, 28850- Torrej´on de Ardoz, Madrid, Spain +7European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands +8Cosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200, Denmark +9Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK +10Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy +11Sorbonne Universit´e, UPMC-CNRS, UMR7095, Institut d’Astrophysique de Paris, F-75014 Paris, France +12Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Ave., Cambridge CB3 0HE, UK +13Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +14Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA +15Leiden Observatory, Leiden University, PO Box 9513, 2300RA Leiden, The Netherlands +16Aurora Technology for the European Space Agency, Villanueva de la Ca˜nada, E-28692 Madrid, Spain. +17Universidad Europea de Madrid, 28670, Madrid, Spain. +18Quantum Circuits, Inc., New Haven, Connecticut, USA +19NASA Goddard Space Flight Center, Observational Cosmology Laboratory, Greenbelt, USA +20Max-Planck Institute for Astronomy, K¨onigstuhl 17, 69117 Heidelberg, Germany +21Department of Earth & Planetary Sciences, Johns Hopkins University, Baltimore, MD 21218, USA +22NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada +ABSTRACT +The Near-Infrared Spectrograph (NIRSpec) is one of the four focal plane instruments on the James +Webb Space Telescope. In this paper, we summarize the in-orbit performance of NIRSpec, as derived +from data collected during its commissioning campaign and the first few months of nominal science +operations. More specifically, we discuss the performance of some critical hardware components such as +the two NIRSpec Hawaii-2RG (H2RG) detectors, wheel mechanisms, and the micro-shutter array. We +also summarize the accuracy of the two target acquisition procedures used to accurately place science +targets into the slit apertures, discuss the current status of the spectrophotometric and wavelength +calibration of NIRSpec spectra, and provide the ’as measured’ sensitivity in all NIRSpec science modes. +Finally, we point out a few important considerations for the preparation of NIRSpec science programs. +Keywords: Instrumentation: spectrographs - Space vehicles: instruments +1. INTRODUCTION AND BACKGROUND +Many, if not most, of the science goals for the James +Webb Space Telescope (JWST) rely on the ability to ac- +arXiv:2301.13766v1 [astro-ph.IM] 31 Jan 2023 + +ID2 +quire high-quality near-infrared (NIR) spectra of astro- +nomical targets with a wide range of luminosities, from +the faintest galaxies at high redshift to the bright host +stars of nearby exoplanets. The NearInfraRed Spectro- +graph (NIRSpec) onboard JWST offers the necessary +versatility to enable observations of such a wide range +of targets, thanks to a variety of sophisticated observ- +ing modes, some of which are available for the first time +in space and at near-infrared wavelengths (e.g. multi- +object and integral field spectroscopy). +In this paper, we summarize the status of the NIRSpec +performance characterization, using mostly data ac- +quired during the JWST commissioning campaign. The +analysis of these data sets (which in many cases have +since been complemented by additional data acquired +during Cycle 1) is still ongoing, and many performance- +related measurements and products are preliminary. +Therefore, this paper can only provide a snapshot of +the performance assessment as of mid-Oct. 2022. +2. OVERVIEW OF THE NIRSPEC INSTRUMENT +The design of and scientific motivation for the NIR- +Spec instrument have been extensively discussed in +Jakobsen et al. (2022). For convenience, we show here +again the optical design of the NIRSpec instrument, to- +gether with the layout of the mechanical assembly in +Figure 1. +In brief, NIRSpec is designed to be capable of per- +forming both single- and multi-object spectroscopic ob- +servations over the 0.6 − 5.3 µm NIR wavelength range +at three spectral resolutions that cover a range of sci- +ence applications: a low-resolution (R ≃ 30–330) mode +intended for obtaining exploratory continuum spectra +and redshifts of remote galaxies; an intermediate resolu- +tion (R ≃ 1000) mode for accurately measuring atomic +and molecular emission lines, and a higher resolution +(R ≃ 2700, i.e. ≃ 110 km/s) mode primarily for kine- +matic studies using these emission lines. Table 1 sum- +marizes the various NIRSpec modes and their typical +science applications, together with their slit apertures +and wavelength coverage, and available spectral resolu- +tion. More detailed descriptions of the NIRSpec observ- +ing modes can be found in Ferruit et al. (2022) for the +Multi-Object Spectroscopy (MOS) mode, B¨oker et al. +(2022a) for the Integral-Field Spectroscopy (IFS) mode, +and Birkmann et al. (2022a) for the Bright Object Tran- +sit Spectroscopy (BOTS) mode. +3. THE NIRSPEC COMMISSIONING CAMPAIGN +As discussed in more detail in B¨oker et al. (2022b), +the NIRSpec commissioning campaign successfully con- +cluded 185 days after launch, when the last of its four +Table 1. NIRSpec instrument modes. +long-pass filter. + +Table 2-1 NIRSpec instrument modes +Mode +Target type +Wavelength +range +Aperture mask +Resolving +Power +MSA +spectroscopy +(MOS) +rich fields or +very extended +object +0.6 – 5.3 µm +any config. of +0.2² x 0.46² +micro-shutters +R=30-330 or +R»1000 or +R»2700 +Fixed Slit +Spectroscopy +(FS) +single +compact object 0.6 – 5.3 µm +1.6² x 1.6² or +0.2² x 3.2² or +0.4² x 3.65² FS +R=30-330 or +R»1000 or +R»2700 +Bright- +Object Time +Series +(BOTS) +bright point +sources +0.6 – 5.3 µm +1.6² x 1.6² +R=30-330 or +R»1000 or +R»2700 +Integral-field +Spectroscopy +(IFS) +moderately +extended +objects +0.6 – 5.3 µm +3.0² x 3.0² IFU +R=30-330 or +R»1000 or +R»2700 +Target +Acquisition +(TA) +reference stars +(n~10-20) +0.8 – 2.0 µm +all shutters open +(except around +bright targets) +undispersed +imaging +(MIRROR) +Internal +Calibrations +none +0.6 – 5.3 µm +any config. of +microshutters +and FS/IFU +undispersed +imaging or +R=30-330 or +R»1000 or +R»2700 + +Every instrument mode listed in Table 2-1 is implemented in the flight software and can be used +observing modes was formally declared ‘ready for sci- +ence’. A total of 33 NIRSpec commissioning activity re- +quests (CARs) were required to activate the instrument +electronics, to commission the various observing modes, +and to obtain a first level of calibration and performance +evaluation. While the activities were scattered through- +out the six months of JWST commissioning, they had +to be executed in a carefully planned sequence in order +to derive the various intermediate products required for +subsequent, and increasingly more complex, activities. +The major milestones were the following: +1. The successful execution of an onboard script that +controls and maintains the temperature of the mi- +croshutter quadrants 10-15K above the environ- +ment in JWST’s Integrated Science Instrument +Module (ISIM), until the latter dropped below +140K, in order to avoid condensation of water ice +or other volatiles onto the micro-shutters. +2. The NIRSpec Optical Bench Assembly (OBA) +reaching sufficiently cold temperatures for the safe +operation of the various mechanisms (the magnet +arm of the microshutter array (MSA), the wheel +mechanisms, and the re-focus mechanism) and the +active control of the Focal Plane Assembly (FPA) +temperature. +3. The verification of the mechanical and operational +performance of these mechanisms and the various +lamps in the Calibration Assembly (CAA). +4. The detailed performance characterization of the +NIRSpec detectors and the ≈ 250000 individual +shutters of the MSA. The results, which are de- +tailed in Birkmann et al. (2022b) and Rawle et al. + +3 +OTE Primary +filter wheel pupil image +Grating pupil plane +OTE focus +MSA +FPA +131.4 m +F/ 20 +F / 12.5 +F/5.6 +FORE optics +COLL optics +CAM optics + +Fig 2.1 : Schematic representation of OTE + NIRSpec. + +Fig 2.1 shows that the OTE NIRSpec optical train has three field planes and three pupil planes. The +pupil planes are the OTE primary mirror (which is the limiting pupil stop of the system), the pupil +plane at the filter position and finally the pupil plane at the position of the grating. Not shown in the +picture above is the fact that the OTE primary is being re-imaged at the position of the Fine Steering +Mirror (FSM) which serves as the exit pupil of the OTE. The image planes are located at the OTE +focal surface, the Micro-Shutter Assembly (MSA) and finally at the Focal Plane Assembly (FPA). +The real optical lay-out of NIRSpec is shown in fig 2.2 below. + + +Figure 2.2: Optical lay-out of the NIRSpec instrument +Detector FPA +Grating Wheel +COLLimator optics + MSA + +IFU +Filter Wheel +FORE Optics +Pick-Off mirrors +CAMera Optics +Refoc +1000 mm +Proc. of SPIE 59040L-2 +Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 12 Jan 2020 +Terms of Use: https://www.spiedigitallibrary.org/terms-of-use +Disperser +Detector +Array +Camera +Collimator +Slit Plane +Foreoptics +Filter +Pick-off +Mirrors +Focus +Mirrors +16.4o +OTE Primary +filter wheel pupil image +Grating pupil plane +OTE focus +MSA +FPA +131.4 m +F/ 20 +F / 12.5 +F/5.6 +FORE optics +COLL optics +CAM optics + +Fig 2.1 : Schematic representation of OTE + NIRSpec. + +Fig 2.1 shows that the OTE NIRSpec optical train has three field planes and three pupil planes. The +pupil planes are the OTE primary mirror (which is the limiting pupil stop of the system), the pupil +plane at the filter position and finally the pupil plane at the position of the grating. Not shown in the +picture above is the fact that the OTE primary is being re-imaged at the position of the Fine Steering +Mirror (FSM) which serves as the exit pupil of the OTE. The image planes are located at the OTE +focal surface, the Micro-Shutter Assembly (MSA) and finally at the Focal Plane Assembly (FPA). +The real optical lay-out of NIRSpec is shown in fig 2.2 below. + + +Figure 2.2: Optical lay-out of the NIRSpec instrument +Detector FPA +Grating Wheel +COLLimator optics + MSA + +IFU +Filter Wheel +FORE Optics +Pick-Off mirrors +CAMera Optics +Refoc +1000 mm +Proc. of SPIE 59040L-2 +Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 12 Jan 2020 +Terms of Use: https://www.spiedigitallibrary.org/terms-of-use +Disperser +Detector +Array +Camera +Collimator +Slit Plane +Foreoptics +Filter +Pick-off +Mirrors +Focus +Mirrors +Filter +Wheel +Grating +Wheel +Detector +Array +Calibration +Source +Microshutter +Array +Figure 1. Left: The optical path through the NIRSpec instrument. The dispersion direction is out of the page. Right: CAD +rendering of NIRSpec with its major mechanisms identified. The dimensions of the NIRSpec optical assembly are 1.9 m × 1.4 m +× 0.7 m, with a total mass of 196 kg. +(2022), confirmed the expected in-flight perfor- +mance, and marked the start of Phase 2 of NIR- +Spec commissioning, i.e. the ability to obtain sci- +ence exposures via the onboard scripts. +5. The completion of the optical telescope element +(OTE) alignment described in McElwain et al. +(2022) enabled the first on-sky NIRSpec observa- +tions, in particular the ‘focus sweep’ which was +used to verify the optimal positioning of the in- +ternal focus adjustment mechanism, and allowed +a first assessment of the image quality and total +wavefront error across the NIRSpec field of view +(FOV). For this, the NIRSpec focus mechanism +was ’swept’ over a range of ±2000 motor steps, +with an exposure in the F110W filter taken ev- +ery 400 steps. The sweep range corresponds to a +physical mechanism movement of ±1 mm and a +defocus of ±3.6 mm in the OTE focal plane. +6. Once the optimal NIRSpec focus was established, +undispersed images of a dense star field with +known and accurate stellar astrometry were ob- +tained. These observations of the ‘astrometric ref- +erence field’ (Anderson et al. 2021) enabled precise +measurements of the magnification and distortions +of the NIRSpec optical train, which is a crucial +ingredient for the parametric instrument model +(Dorner et al. 2016; L¨utzgendorf et al. 2022), +which underlies the wavelength calibration of all +NIRSpec spectra (see Sec. 6.3), as well as the co- +ordinate transformations between detector, MSA, +and sky that are required to accurately place sci- +ence targets in the various NIRSpec apertures. +7. Once the accuracy of the instrument model was +confirmed via a series of successful target acqui- +sitions (TAs), the last few activities of the NIR- +Spec commissioning campaign obtained on-sky ob- +servations of stars and planetary nebulae for flux +and wavelength calibration, which enabled the cre- +ation of various ‘reference files’ and other products +needed for the data reduction pipeline. +4. OBSERVATORY PERFORMANCE +Because of the narrow slit apertures involved in most +of its observing modes, the NIRSpec science perfor- +mance is critically dependent on a number of telescope +and observatory characteristics. Here, we briefly list the +most important ones, without discussing them in detail +because they are the subject of accompanying papers in +this issue. +4.1. Optical Quality of the OTE +The quality of the image delivered by the OTE is of +fundamental importance for all JWST instruments, as +it limits the spatial resolution and sensitivity of images. +As discussed in other papers in this edition (Menzel +et al. 2022; McElwain et al. 2022), the in-orbit opti- +cal performance of the OTE is significantly better than +expected, with lower wavefront errors and therefore a +sharper point spread function (PSF) across the entire +FOV (Feinberg et al. 2022). More specifically, the JWST +PSF is diffraction limited already at 1.1 µm, and reaches +a Strehl ratio of 0.9 at 4 µm across the FOV. +While this is clearly good news for all JWST instru- +ments, the NIRSpec sensitivity stands to gain the most. +This is because a narrower PSF minimizes the ‘path +losses’ caused by the physical truncation and subsequent +diffraction of the PSF by the narrow apertures of the + +4 +NIRSpec fixed slits and microshutters. As discussed in +Giardino et al. (2022), this is the main reason for the +fact that the NIRSpec slitlosses are significantly smaller +than expected, with peak gains of more than 10% at +the shorter wavelengths. The resulting benefit to the +NIRSpec throughput is discussed in Section 6.1. +4.2. Cleanliness and thermal backgrounds +In addition to the low wavefront error, the OTE optics +are also free of water ice or other molecular contamina- +tion, resulting in a very high telescope throughput across +the entire NIRSpec wavelength range. +Moreover, the +good thermal performance of the sunshield, and the re- +sulting nominal temperatures of the OTE result in ther- +mal backgrounds that are at or below the pre-launch +expectations (Rigby et al. 2022a), also enhancing the +sensitivity of NIRSpec observations. +Taken together, +the outstanding optical and thermal performance of the +JWST OTE provides a much better wavefront to the +NIRSpec optics than expected, which is reflected in the +sensitivity numbers discussed in Section 6.1. +4.3. Attitude Control System +Another important factor for the quality of NIRSpec +observations is the pointing accuracy and stability of +the JWST Attitude Control System (ACS), which is +discussed in detail in Menzel et al. (2022). +Accurate +‘blind pointing’ is required to ensure that the science +target is indeed imaged onto the rather small detector +area used to identify the target by the onboard TA al- +gorithms, and that the corrective small-angle maneuvers +(SAMs) are not too large. The ability to accurately ex- +ecute those SAMs, and the absence of any drifts or low- +frequency jitter in the telescope pointing is crucial for +stable and precise target placement throughout the ex- +posure, which is particularly important for time series +observations of transiting exoplanets, as discussed fur- +ther in Section 6.4. On all these accounts, the JWST ob- +servatory meets or exceeds the pre-launch requirements, +which is the foundation for the problem-free execution +of the various NIRSpec observing templates, and the +outstanding NIRSpec science performance presented in +this paper. +5. NIRSPEC HARDWARE PERFORMANCE +5.1. Wheel Mechanisms +NIRSpec is equipped with two wheel mechanisms: i) +the Filter Wheel Assembly (FWA) with five long-pass +transmission filters (F070LP, F100LP, F170LP, F290LP, +and CLEAR) that define the wavelength ranges for the +dispersers, two filters used for the TA exposures (F110W +and F140X), and one position (OPAQUE) that serves +both as an instrument shutter towards the telescope side +and a coupling mirror for illumination from the CAA on +the instrument side, and ii) the Grating Wheel Assem- +bly (GWA) with the seven NIRSpec dispersers, and a +mirror for undispersed imaging (see also Table 6). The +functionality of both wheels was verified during com- +missioning, and their torque profiles were characterized +and found to be nominal, i.e. similar to those measured +during ground testing, and well within their expected +tolerances. +As discussed in Jakobsen et al. (2022), the limited me- +chanical angular reproducibility of the GWA ball bear- +ings causes small, but significant, variations in the po- +sition of NIRSpec spectra on the detector, especially in +dispersion direction. Since this has an obvious effect on +the wavelength calibration as well as the precision of +the TA algorithm, it is important that the actual GWA +position of any given NIRSpec exposure can be accu- +rately inferred from telemetry. To this end, the GWA +design includes two magneto-resistive position sensors +(a.k.a.‘tilt sensors’) which were extensively tested before +launch (De Marchi et al. 2012) to ensure that NIRSpec +can indeed achieve the required accuracy of its wave- +length calibration. +The tilt sensors have to be calibrated after each +cooldown, and their post-launch calibration was one of +the important goals of the NIRSpec commissioning cam- +paign. The accuracy of the in-orbit sensor calibration +has been discussed in detail by Alves de Oliveira et al. +(2022) : the remaining uncertainties are consistently be- +low 1/10th of a pixel, which is fully in line with expecta- +tions and sufficient to ensure a reliable wavelength cal- +ibration for all NIRSpec spectra, as demonstrated fur- +ther in Sec. 6.3. +The tilt sensor performance and the +stability of the wavelength calibration will be monitored +with a roughly semi-annual cadence as part of the over- +all JWST calibration plan, using a combination of stel- +lar spectra and the internal LINE and REF calibration +lamps (see Jakobsen et al. 2022). +5.2. Microshutter Array +The NIRSpec microshutter array (MSA) has per- +formed excellently throughout flight thus far, with no +unexpected hardware issues, overall multiplexing re- +maining very high, and an average success rate of ∼96% +for commanding shutters open in science-like patterns +(Rawle et al. 2022). Operability at the level of individ- +ual shutters is of course crucial for the MOS mode itself, +but less obviously also for the IFS mode, where problems +with the shutter array can become a source of parasitic +light, either contamination through stuck-open shutters + +5 +or thermal glow1 emanating from electrical shorts in the +circuitry. Our knowledge of this performance is critical +to allow mitigation, such as short-masking, and enable +users to generate the optimal target set around the in- +operable shutter population. +As described in detail by Rawle et al. (2022), the +marginal increase in inoperable shutters seen during +commissioning was in-line with pre-launch estimates. +The primary consequence of these trends is that the +masking required to mitigate electrical shorts is now the +dominant factor affecting the usability of unvignetted +shutters (Table 2). +While comparatively fewer shorts +have emerged after launch, with approximately one +row/column needing to be masked for every 50 shut- +ter array re-configurations compared to one masked per +20–25 during ground testing, shorts have continued to +appear throughout the latter stages of commissioning +and another two required masking in the first three +months of science observations. An added complication +apparent for two of the shorts seen since launch (and +never during ground testing) was that they produced +glow when applying the all-closed shutter configuration +for IFS observations. Although successfully mitigated +in the usual manner, the contamination before masking +affected both MOS and IFS exposures, increasing the +overall impact of the shorts. +In the regime that shorts may still occur every few +months, there are two operational issues to tackle. First, +to ensure that data still meet scientific objectives, users +need to be informed in a timely manner when exe- +cuted exposures are directly contaminated by a short +and also when upcoming observations are affected by +newly masked shutters. The former may now apply to +both MOS and IFS visits. Second, as the population of +shorts continues to grow, each removing several hundred +shutters from operation, this will eventually have a no- +ticeable impact on multiplexing. Shorts are believed to +be caused by particulate contamination of the complex +MSA control electronics, and those particles may shift +during reconfiguration, which is the origin of new shorts +but also potentially clears the cause of previously de- +tected shorts. Therefore, as discussed by Rawle et al. +(2022), re-checking whether older shorts still remain +may be an avenue to recover previously unusable shut- +ters and maintain the multiplexing of NIRSpec MOS at +the current high level. +1 while a full in-orbit characterization of the spectral characteristics +of MSA shorts is still pending, ground test data have indicated +that it is clearly thermal in nature. However, it is not a pure +blackbody spectrum, likely because it contains signatures from +the MSA coating materials. +Table 2. MSA operability report from the end of commis- +sioning, demonstrating that 82.5% of un-vignetted shutters +are available for use as science apertures. +Q1 +Q2 +Q3 +Q4 +Total +Total +62415 +62415 +62415 +62415 +249660 +Vignetted +6119 +5929 +6102 +5874 +24024 +Failed open +6 +3 +12 +1 +22 +Short-masked +7835 +5150 +3466 +7177 +23628 +Failed closed +1569 +3328 +5932 +5064 +15893 +Total Usable +46886 +48005 +46903 +44299 +186093 +5.3. Detector System +The NIR light collected by the JWST OTE and fed +through the NIRSpec optical train is registered by two +Hawaii-2RG (H2RG) sensor chip assemblies (SCAs), +which are described in detail by Rauscher et al. (2007). +The SCAs are operated at a temperature of 42.8 K, cho- +sen as the best compromise between pixel operability +and total noise of the detector system: a higher temper- +ature leads to an increased number of hot pixels, while +a lower temperature leads to higher noise in the signal +chain. +As +a +reminder2, +the +NIRSpec +SCAs +are +non- +destructively read “up-the-ramp”, using one of two fun- +damentally different readout modes: the so-called tra- +ditional readout mode (TRAD) or the improved refer- +ence sampling and subtraction (IRS2; pronounced “IRS- +square”; Rauscher et al. 2017) readout mode. A num- +ber of detector subarrays with different sizes such as, +for example, the ALLSLITS subarray are supported in +TRAD mode only, offering faster readouts and using a +slightly higher conversion gain to make the full physical +well depth of the pixels accessible, which is of particular +importance for time series observations of bright targets. +The in-orbit performance of the NIRSpec SCAs and +their readout electronics is an important factor for the +NIRSpec science performance, because the sensitivity +of most NIRSpec observing modes is limited by the to- +tal detector noise in a given exposure (see Appendix A +in Jakobsen et al. 2022). The analysis of the detector +performance data collected during the NIRSpec in-orbit +commissioning campaign characterization has been pre- +sented in detail by Birkmann et al. (2022b). Here, we +briefly summarize the most relevant results. +2 for details, see the NIRSpec User Documentation at https://jwst- +docs.stsci.edu/jwst-near-infrared-spectrograph/nirspec- +instrumentation/nirspec-detectors + +6 +5.3.1. Cosmetics +The vast majority of pixels in the NIRSpec detectors +can be considered operable. Non-operable or bad pixels +are, for example, those which have a very low quan- +tum efficiency or no response at all, are not connected +to the readout electronics, or exhibit a large / highly +non-linear dark signal. The number of non-operable / +bad pixels in the NIRSpec detectors as measured during +commissioning is summarized in Table 3. +5.3.2. Dark Current +The median dark signal of the two SCAs for the dif- +ferent NIRSpec readout modes is presented in Table 4. +Two general observations can be made: i) the in-orbit +dark signal for NRS1 is generally higher than for NRS2, +which is fully in line with previous measurements ob- +tained during the various ground testing campaigns. ii) +both NIRSpec SCAs exhibit a slightly elevated dark cur- +rent signal compared to on-ground measurements, with +a more pronounced increase towards the array edges +(see Fig. 2 in Birkmann et al. 2022b). +As discussed +in 5.3.4, this increase is likely related to the cosmic ray +environment at L2. However, even with this small in- +crease compared to pre-launch measurements, the dark +signal is still very low for most pixels, and not a driver +for the total noise (and thus the sensitivity) of NIRSpec +observations. +Multiplexer readout glow is thought to +dominate the observed dark signal (Regan & Bergeron +2020). +5.3.3. Noise Performance +The total noise for the different readout modes of the +two NIRSpec SCAs as a function of effective integration +time is summarized in Table 5. These numbers include +the effects of cosmic rays, i.e. broken ramps due to cos- +mic ray hits and early saturation for some pixels. +Figure 2 shows the noise behavior as a function of inte- +gration time in more detail. For all readout modes, the +total noise decreases with exposure length, up to integra- +tion times of ≈ 500 s where it levels out and then slowly +increases for longer integrations. Nevertheless, for obser- +vations of faint sources that are detector noise limited, +it is still beneficial to use longer integration times for +optimal signal-to-noise ratios. On the other hand, it is +always advisable to have multiple integrations per ob- +servation, ideally in the form of dithered exposures to +guard against early saturation after a strong cosmic ray +hit, as well as to enable a robust rejection of outliers. +The IRS2 readout mode shows the best total noise +performance, in particular for NRS1, and should be used +when observing faint targets. The ALLSLITS subarray +readout mode has noise levels comparable to those in +Figure 2. Median total noise signal (in e−) as a function of +effective integration time for the two NIRSpec detectors and +three readout modes: IRS2, traditional full frame (TRAD), +and ALLSLITS subarray. +traditional full frame mode (TRAD), and is best used +for observations of bright targets with short integrations, +because the effective full well depth is higher due to the +higher conversion gain, like for the other subarrays. +5.3.4. Cosmic Ray Environment +From the NIRSpec dark exposures obtained during +commissioning, we have measured an average CR hit +rate of about 5.5 cm−2 s−1, using the jump detection +algorithm in the ramps-to-slopes pipeline. The observed +hit rate is well in line with pre-launch predictions of the +proton fluence at solar minimum (Giardino et al. 2019). +Most cosmic ray events affect several pixels (partially +due to inter-pixel-capacitance), but are compact, with a +typical hit area of about 10.5 pixels. However, there can +also be longer streaks and so-called ‘snowballs’, affecting +many pixels at once. As described in Birkmann et al. +(2022b), snowballs have the following characteristics: +• a core of fully saturated pixels +• an extended ‘halo’ around the saturated core that +is detected at the same time (i.e. within the same +correlated double sample), with the intensity of +the halo dropping off towards larger radii +• often accompanied by a shower of more compact +cosmic ray events in the vicinity +• core and halo are often spherical in appearance, +but can be very elongated as well +The flux of snowballs varies, but stronger ones can +produce tens of millions of electrons within their satu- +rated core, indicating large energies of the involved par- +ticle(s). Snowballs are observed by all NIR instruments +on JWST (see e.g. Rieke et al. 2022), but their origin is +not yet fully understood. + +16 +NRS1 IRS2 +NRS2 IRS2 +NRS1 TRAD +NRS2 TRAD +14 - +NRS1 ALLSLITS +NRS2 ALLSLITS +Total noise [e-] +12 +10 +8 +6 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +TINTeff [s]7 +Table 3. Summary of bad pixel statistics for the two NIRSpec detectors. Note that the total number of bad pixels can be less +than the sum of the individual categories, as some bad pixels belong to several categories. Note that the fraction of operable +pixels listed in the last row is for the active area of each detector (2040 x 2040 pixels). +Bad pixel type +NRS1 +NRS2 +Notes +Open +294 +252 +Very low response, signal ends up in adjacent pixels (next row) +Adjacent open +1664 +1216 +Impacted by open neighbor pixel (previous row) +Dead +7757 +3938 +Does not respond to light +Low QE +1766 +887 +Low response to light +RC-like +3908 +1902 +Non-linear dark signal (RC-like ramp) +Hot +6062 +2123 +Large dark signal (> 1 e− s−1) +Total +16948 +8275 +Total number of non-operable pixels +Operable pixels [%] +99.59 +99.80 +Fraction of operable pixels +Table 4. Median dark signal of the two NIRSpec SCAs (in +e− per 1000s integration time) for traditional full frame, IRS2 +full frame, and ALLSLITS subarray readout modes, as mea- +sured during commissioning. For comparison, the equivalent +numbers obtained during the last ground test campaign are +listed in brackets. +Readout mode +SCA +TRAD +IRS2 +ALLSLITS +NSR1 +9.0 (7.7) +8.2 (7.1) +22.3 (13.6) +NRS2 +6.9 (5.1) +4.8 (4.0) +15.3 (13.7) +Table 5. Total noise of the two NIRSpec detectors for dif- +ferent readout modes and effective integration times as mea- +sured during commissioning. +Effective Integration Time +Readout / SCA +∼950 s +∼1700 s +∼3560 s +TRAD / NRS1 +6.9 +7.4 +N/A +TRAD / NRS2 +7.3 +7.7 +N/A +IRS2 / NRS1 +5.9 +6.6 +8.5 +IRS2 / NRS2 +7.2 +7.6 +9.2 +SUB / NRS1 +7.0 +7.8 +N/A +SUB / NRS2 +7.0 +7.5 +N/A +5.3.5. Persistence +Image persistence, or latency signal, is an unavoid- +able effect in NIR HgCdTe detectors. It manifests itself +as an ’afterglow’ in pixels that have been subjected to +strong illumination, resulting in a faint residual image in +the following integrations. Persistence is caused by elec- +trically active defects, so-called ’charge traps’, in the +detector material which accumulate charge that is only +released in later exposures. Past results from the ground +testing of the JWST NIR detectors has shown that their +persistence behavior is generally better than that of sim- +ilar detectors on previous NASA missions including the +Hubble Space Telescope: any persistence decays to be- +low the background level after about 2000 s (Rauscher +et al. 2014). While the detailed characterization of the +persistence behavior of the two NIRSpec SCAs is still +ongoing, the in-orbit data collected so far suggest that +persistence is not a major concern for most NIRSpec +science programs. +6. NIRSPEC SCIENCE PERFORMANCE +Many key parameters of the end-to-end science per- +formance of NIRSpec and the JWST OTE could only +be measured with on-sky data obtained after launch. In +this section, we discuss a number of aspects that are +relevant for the scientific performance of all NIRSpec +modes. +6.1. Photon Conversion Efficiency and Sensitivity +For NIRSpec, and in fact any optical instrument, +a critical performance parameter is its efficiency, i.e. +the fraction of photons incident on the primary mirror +that are being registered by the detector after passing +through the entire optical path. This metric, together +with the noise performance of the NIRSpec detectors, +drives the ultimate sensitivity for astronomical observa- +tions. +Because NIRSpec is a complex instrument that sup- +ports many different observing modes, its optical train +is rather complicated, as evident from Fig. 1. Except +for the order-separation filters and the low resolution +double-pass prism, the NIRSpec optics are reflective +throughout. Photons captured by the JWST primary +mirror undergo a total of 19 reflections before reach- +ing the NIRSpec detector array in the MOS, FS, and +BOTS observing modes. For IFS mode, there are an ad- +ditional 8 reflections in the IFU optics (see B¨oker et al. + +8 +Table 6. NIRSpec spectral configurations. +Band +Disperser +λ/δλ +Filter +λ range [ µm] +0 +G140M +G140H +1000 +2700 +F070LP +0.7 – 1.2a +I +G140M +G140H +1000 +2700 +F100LP +1.0 – 1.8 +II +G235M +G235H +1000 +2700 +F170LP +1.7 – 3.1 +III +G395M +G395H +1000 +2700 +F290LP +2.9 – 5.2 +n/a +PRISM +30-330 +CLEAR +0.6 – 5.3 +a +the Band 0 configurations using the F070LP filter will obtain +spectra over a much wider wavelength range which, however, con- +tain 2nd-order contamination beyond 1.2 µm. Users who are pre- +pared to deal with this contamination can potentially use Band +0 out to ≈ 1.8 µm. +2022a). Cleanliness and high reflectivity of the NIRSpec +optics therefore was of ultimate importance throughout +the construction and test phases. +By necessity, the metric of choice to measure the op- +tical throughput is the Photon Conversion Efficiency +(PCE), i.e. the ratio of photons incident on the JWST +primary mirror to electrons registered by the NIRSpec +detector system. It can be derived using observations +of a ‘flux standard’ (typically a well-characterized star) +with an accurately known spectral energy distribution +(SED). As can be seen from Table 6, NIRSpec has a to- +tal of nine spectral configurations, and the throughput +must be measured separately for each of them. +As described in more detail by Giardino et al. (2022), +the NIRSpec optical efficiency meets or exceeds the pre- +flight expectations. +In particular, significantly higher +than predicted PCEs are achieved for the high-resolution +configurations, for the MOS/FS mode, and for all con- +figurations below ∼ 2.6 µm for the IFS mode. +The +10-20% lower than expected efficiencies above ∼ 4 µm, +apparent in particular for the IFS mode, are (mostly) +explained by the more significant path losses at longer +wavelengths, which are reflected in the measurements +but not the predictions (see Fig. 3 in Giardino et al. +2022). +These in-orbit PCE measurements , together with the +detector noise performance discussed in Section 5.3, can +be used to calculate the in-flight NIRSpec sensitivity for +its various science modes and configuration, following +the methodology described in Appendix A of Jakobsen +et al. (2022). Figure 3 shows the results of the calcu- +lations in terms of continuum sensitivity curves for a +point source observed in the MOS and IFS modes, using +a bench-mark observation in all spectral configurations. +The achieved level of sensitivity is extremely impressive +when compared to other NIR spectrographs with simi- +lar observing modes and spectral resolutions: by at least +two orders of magnitude, NIRSpec is the most sensitive +NIR spectrograph currently available for astronomical +studies (see also Rigby et al. 2022b). +6.2. Spectro-Photometric Calibration +The full photometric calibration of NIRSpec requires +three major steps in the reduction pipeline, with their +associated reference files: i) the detector flat (D-Flat) +to capture the pixel-to-pixel response variations, and +derived from component-level ground test data of the +two NIRSpec SCAs, ii) the spectrograph flat (S-Flat) to +correct the throughput variations of the spectrograph +optics, and measured from exposures using the internal +calibration lamps in the CAA, and iii) the FORE optics +flat (F-Flat) to characterize any field- and wavelength- +dependent effects caused by the OTE and the NIRSpec +FORE Optics. For the last step, observations of spectro- +photometric standard stars are necessary to verify the +integrity of the entire NIRSpec optical system, in partic- +ular the pick-off and FORE optics (including the FWA), +neither of which can be illuminated with the CAA lamps +(see te Plate et al. 2005, for a detailed description of the +light path from the CAA). For a detailed overview of +the NIRSpec flat field strategy we refer to Rawle et al. +(2016). +The S-Flat step corrects any throughput variations in +the spectrograph, i.e. +introduced in the optical path +after the FWA and before the detector. The S-Flat ref- +erence files for all NIRSpec observing modes are derived +from internal lamp exposures and after correction for the +detector response (D-Flat) and spectral energy distribu- +tion of the lamp(s). Because the throughput is strongly +dependent on the properties of the chosen disperser and +slit aperture, different reference files are created per de- +tector for each combination of prism or grating and NIR- +Spec observing mode (FS, IFS, or MOS). For the FS +and IFS modes, the S-flat consists of two parts: (i) a +2D image capturing the pixel-to-pixel variations for all +slits/slices, and (ii) a vector for each slit and the IFU, to +correct the fast throughput variations with wavelength. +This approach is possible for the FS and IFS mode, be- +cause for a given aperture and spectral configuration, +each detector pixel is always illuminated by the same +wavelength (modulo the non-repeatability of the GWA, +which is corrected separately using the position sensors +described in Sec. 5.1). +For MOS mode, in contrast, the same detector pixel +can receive light of different wavelengths, depending on +the position of the open MSA shutter. Hence, the MOS + +9 +Figure 3. NIRSpec point-source continuum sensitivity in MOS/FS (left) and IFS (right) mode, derived from in-orbit mea- +surements and assuming a well-centered source in a microshutter or an IFU slice, respectively. The sensitivity for the S200 +slits in FS mode is similar to the one in MOS mode. The plots show, for each disperser and as a function of wavelength, the +flux required to reach S/N = 10 per spectral pixel in 10,000s. More specifically, the computations assume 10 NRSIRS2RAPID +exposures of 1006.7 s each (70 groups of 1 frame), using the methodology described in Appendix A of Jakobsen et al. (2022). +Figure 4. The spectra of standard stars 1808347 (left) and P177D (right), extracted with the NIRSpec-internal reduction +pipeline, and compared to the CALSPEC model to verify the flux calibration. Both spectra were taken through the S1600A1 +slit using the F290LP/G395H configuration. The bottom frames show the residuals. The observations of P177D were obtained +without a target acquisition step, and the systematically lower flux compared to the model is most likely due to imperfect +centering in the S1600A1 aperture, which can easily cause aperture losses of 1-2%. +S-flat reference consists of a data cube, sampling the +wavelength variation for each pixel, so that the specific +S-flat for any given MSA configuration can be derived +on-the-fly by the pipeline. Given the large number of +shutters and the finite amount of time available for com- +missioning, direct measurement of the throughput for +every shutter was impossible. Therefore, only a subset +of MSA shutters was observed, from which the complete +(smoothed) S-flat cube could be generated by interpola- +tion. +Similar to the S-Flat, the F-Flat must be created for +each of the filter and grating wheel combinations listed +in Table 1 and for each of the three observing modes +(FS, IFS, and MOS3), to take into account any wave- +length dependence on the throughput of the OTE and +FORE optics. Standard star observations were obtained +for all but the F100LP/G140M configuration, for which +calibration files based on simulated data will remain in +place until such observations are taken during Cycle 1. +We +used +the +A3V +star +1808347 +(2MASS +J18083474+6927286) for all gratings, while for the +CLEAR/PRISM configuration, we used either the A8III +3 The in-flight update for the MOS mode is still pending, but the +necessary data have been taken. + +3.5 +2.5 +data +data +3.0 +1808347 +P177D +2.0 +[gw/Mlb-0T> +F290LP/G395H S1600A1 +2.0 +1.5 +1.5 +xnd +1.0 +0.5 +0.5 +0.05 +0.05 +sjenp! +0.00 +0.00 +resi +-0.05 +3.00 +3.25 +3.50 +3.75 +4.00 +4.25 +4.50 +4.75 +3.0 +3.5 +4.0 +4.5 +wavelength [μm] +wavelength [μm]10 +star 1743045 (2MASS J17430448+6655015), or the G0- +5 star SNAP-2 (2MASS J16194609+5534178), because +1808347 would saturate the low-resolution spectra. For +the fixed slits, we used a 3-nod pattern (2-nod pattern +for S1600A1) to obtain the data, and a 4-nod pattern +with a 4 spaxel (0.4 arcsec) extraction radius for the IFS +observations. +The MOS program was executed using +a 3-shutter nod pattern for each of the filter/grating +combinations. +To derive the conversion to absolute flux units (which +is part of the F-Flat step), the extracted spectra were +divided by the resampled standard star templates avail- +able on the CALSPEC website4 (Bohlin et al. 2014). To +create a smooth F-Flat, any outliers such as remaining +hot pixels, stellar absorption lines, or the 0th order con- +tamination of the lamp spectra used to create the S-Flat, +were masked and a smooth vector was fitted, creating +the final F-Flat. +To verify the validity of the NIRSpec calibration +approach, we re-extracted the spectrum of the stan- +dard stars using the NIRSpec-internal data reduction +pipeline, and compared the resulting spectra to the +CALSPEC template, as shown in Fig. 4 for the case +of FS mode with the S1600A1 aperture. Overall, the +agreement is very good, with an RMS of the residu- +als well below 2%. For the other NIRSpec modes and +apertures, the results are of similar quality. Note that +this comparison only represents an estimate of the ‘best +case’ calibration accuracy, because it is performed on +the same star that was used to derive the pipeline refer- +ence files, and thus does not account for any systematic +uncertainties. +While an evaluation of the systematic errors must wait +for additional calibration data obtained during Cycle 1 +or later, commissioning spectra of two other standard +stars, WD1057+719 (a DA1.2 white dwarf, M-gratings) +and P177D (a G0V star, H-gratings), were obtained in +FS mode with the S1600A1 slit (see Table 1). These ob- +servations were executed early in commissioning and be- +fore the wide aperture TA procedure (see Section 6.5.1) +was available. Therefore, a proper centering is not guar- +anteed and (small) uncorrected aperture losses are possi- +ble. Nevertheless, we measure residuals between the flux +template and the extracted spectra of −2.12 ± 2.55%, +−3.76 ± 1.00%, −3.67 ± 1.57%, −5.09 ± 1.16%, −1.00 ± +1.21%, and −0.91 ± 1.97% for the F070LP/G140H, +F170LP/G235M, F170LP/G235H, F290LP/G395M , +4 https://www.stsci.edu/hst/instrumentation/ +reference-data-for-calibration-and-tools/astronomical-catalogs/ +calspec +F290LP/G395H, and CLEAR/PRISM configurations, +respectively. +These residuals are well below the pre-launch require- +ment (10% absolute photometric accuracy of all NIR- +Spec spectra, i.e. even after correction for ‘delta’ slit +losses due to imperfect source centering). While source +centering is most critical for the 200 mas wide slits, it is +worth mentioning here that after a successful target ac- +quisition procedure (see Section 6.5), the source is typi- +cally placed within 10 mas of the slit center. This magni- +tude of source displacement would result in insignificant +‘delta’ slit losses. +As noted above, the results presented here were +derived using the NIRSpec-internal data reduction +pipeline. The user pipeline provided by STScI is cur- +rently being checked and improved to deliver similar re- +sults, and the status is discussed further in Section 7. +6.3. Wavelength Calibration +The accuracy of the NIRSpec wavelength calibration +is determined by the performance of the parametric in- +strument model, which was a crucial product to be de- +rived from NIRSpec commissioning data. As described +in L¨utzgendorf et al. (2022), the residuals in the fi- +nal instrument model were well within the requirements +for the wavelength and astrometric calibration of NIR- +Spec data. +This is illustrated in Fig. 5, which shows +wavelength-calibrated spectra of the spectrophotomet- +ric standard 1808347, an A3V star with prominent hy- +drogen absorption features, taken through the 200 mas +wide fixed slits. +As can be seen, the NIRSpec instrument model al- +lows for a highly accurate wavelength calibration, with +residuals for internal lamp spectra well below the re- +quirement (1/8 of a resolution element) for all gratings +analyzed so far. Note that the full verification for all +pipeline products in the various NIRSpec modes is still +ongoing, especially in the case of the IFS mode. +6.4. Photometric Stability for Time-Series +Observations +The temporal stability of the instrument response to a +constant flux stimulus is a critical parameter for the ac- +curacy of measured light curves of astronomical targets, +e.g. +in the case of Time Series Observations (TSOs) +of transiting exoplanets. In order to quantify the sta- +bility of the NIRSpec response over various timescales, +we obtained a TSO of the star HAT-P-14 (PID 1118; +PI: Proffitt) during the transit of its well-characterized +exoplanet HAT-P-14 b using the G395H/F290LP grat- +ing/filter combination (see Table 6). The observations, +which also served the purpose of verifying the correct ex- +ecution of the BOTS observing template (see Birkmann + +11 +Figure 5. Wavelength-calibrated spectra of the spectrophotometric standard star 1808347, obtained in FS mode through the +S200A1 slit (see Table 1), and using the medium-resolution gratings in three of the four spectral configurations (F100LP/G140M +is not shown). These data were obtained during the NIRSpec commissioning program ‘Spectrophotometric Sensitivity and +Absolute Flux Calibration’ (PID1128). +et al. 2022a), were analyzed in detail by Espinoza et al. +(2022). +To recap, the transmission spectrum of HAT-P-14 b +could be measured with a precision of 50-60 ppm at +R = 100 (rebinned down from R = 2700), which is +in excellent agreement with pre-flight expectations, and +close to the photon-noise limit for a J = 9.094, F-type +star like HAT-P-14. +There were two noteworthy fea- +tures observed in this analysis. The first was a weak +linear trend in the white-light response as a function +of time, which was observed to be stronger in NRS1 +(-150 ppm/hour) than in NRS2 (30 ppm/hour). This +was shown to also be slightly wavelength dependant in +NRS1, but not in NRS2. These trends can be easily cor- +rected for, and are most likely caused by low-amplitude +instabilities in the detector signal chain. +The second +is the fact that binning pixels in the spectral direction +seems to not decrease the noise level in the time-series +as 1/�Nbin where Nbin is the size of the bin. +It is +likely this is related to unaccounted covariance between +pixels due to effects such as, e.g., 1/f noise. This can +also be accounted for by either simply working with the + +F170LP/G235M/S200A1 +2.00 +1.75 +1.50 +1.25 +(Arbitrary +1.00 +0.75 +Flux +0.50 +0.25 +0.00 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.0 +Wavelength (micron)F290LP/G395M/S200A1 +0.4 +Units) +0.3 +(Arbitrary +0.2 +0.1 +0.0 +3.0 +3.5 +4.0 +4.5 +5.0 +Wavelength (micron)F070LP/G140M/S200A1 +6 +5 +4 +3 +2 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +Wavelength (micron)12 +added pixels on each spectral channel or considering a +more complex spectral extraction and binning scheme +that includes this spatial covariance of pixels in the de- +tector (see, e.g., Schlawin et al. 2020). +Given the excellent performance of the BOTS mode, +and the absence of any strong variations and system- +atic trends in the combined response from telescope and +instrument over timescales of a few hours to a day, NIR- +Spec promises to fulfill its key role for cutting-edge tran- +siting exoplanet atmospheric science with JWST during +Cycle 1 and beyond. +6.5. Target Acquisition +Depending on their scientific needs, NIRSpec users +have a number of options for fine-tuning of the tele- +scope pointing after the completion of the initial slew to +the target field, which typically places the source within +100 mas of its intended position. +Programs for which +this level of accuracy is sufficient do not need a dedi- +cated TA procedure, and should select TA=NONE or +TA=VERIFY ONLY. The difference between these two +options is that (for MOS and IFS mode) the latter ob- +tains an undispersed image of the sky after the spectro- +scopic exposures in order to allow a determination of the +pointing post-facto. +Programs that require a more accurate target place- +ment of an individual source in either the IFU, one of +the fixed slit apertures, or the MOS single-point field +position should use the NIRSpec Wide Aperture Target +Acquisition (WATA) procedure. For MOS observations, +on the other hand, the added complexity of requiring a +roll angle optimization calls for the use of a dedicated +method called Micro-Shutter Array Target Acquisition +(MSATA). The details and in-orbit performance of both +of these TA methods are discussed in the following sub- +sections. +6.5.1. Wide Aperture Target Acquisition +The WATA procedure takes an image of an isolated, +point-like target through the 1.”6 × 1.”6 wide fixed slit +aperture (S1600A1). Using this image, the onboard soft- +ware then computes the centroid of the source emission +to determine its position after the initial ‘blind’ tele- +scope slew, and autonomously calculates the corrective +‘delta’ slew required to accurately position either this +target or another nearby target at the optimal location +in the NIRSpec science aperture. +The total duration +of the WATA procedure can be as short as 5 min, and +as long as 11 min, depending on the size of the detec- +tor area being used (subarray or FULL frame), and the +depth of the acquisition exposure. +Figure 6a shows the WATA image of a star used to ver- +ify the WATA process during commissioning, obtained +from Proposal ID 1118 (PID1118). This program con- +tained six successful WATA attempts5 that were used to +evaluate the onboard algorithms, the correct computa- +tion and execution of the offset slew, and the subsequent +science exposures. Figure 6b shows the position of the +source after the initial slew, i.e. +at the beginning of +each WATA. Because of the excellent slew and pointing +performance of the observatory, the WATA starting po- +sition is already quite close to the aperture center: on +average, the target is within 50 mas radial distance, or +one half of the size of a NIRSpec detector pixel. +Figure 6c shows the target position at the end of the +WATA procedure, i.e. after the corrective slew to the +center of the S1600A1 aperture was computed by the +onboard algorithm, and executed by the telescope guid- +ing system. For the six successful WATA observations in +PID1118, the average target position was within 2.5 mas +from the aperture center, which is almost ten times bet- +ter than the pre-launch requirement of 20 mas. +For observations that require the science target to be +placed in apertures other than the S1600 slit itself, fur- +ther tests executed during commissioning and the early +science program have demonstrated that the telescope +slew accuracy is sufficient to place the target at, e.g., the +intended location in the IFU aperture with an accuracy +of better than 10 mas, again well within the requirement. +6.5.2. Micro-Shutter Array Target Acquisition +The NIRSpec MOS mode requires that the images +of astrophysical targets are accurately placed onto the +MSA over the entire (3′ × 3′) FOV, such that they fall +within their dedicated microshutters. The onboard algo- +rithms to achieve this rely on highly precise coordinate +transformations between the detector, MSA, and sky +planes, which are another crucial output of the NIR- +Spec parametric instrument model (L¨utzgendorf et al. +2022). +As explained in detail by Keyes et al. (2018), the NIR- +Spec MSA target acquisition (MSATA) process uses a +set of 5 to 8 ‘reference stars’ that are imaged onto the +detector through the microshutter grid. +For MSATA +exposures, the MSA can be either in the all-open con- +figuration or in ‘protected’ mode, which closes the shut- +ters around bright stars to prevent persistence in the +subsequent science exposures. Two MSATA exposures +are acquired, separated by an offset equivalent to half +of the microshutter pitch (in both x and y direction), in +order to mitigate the effect of vignetting by the MSA +bars. +The centroid position of each reference star is +calculated autonomously by the on-board software, and +5 The data discussed here are from Observations 1-4, 6, and 8. + +13 +Figure 6. (a) The Wide Aperture Target Acquisition (WATA) exposure image (from PID 1118 Obs 3). The 32 pixel by 32 +pixel aperture is placed around the 1.”6 × 1.”6 (approximately 16 by 16 pixels) square aperture, but not exactly centered. (b) +The V2, V3 coordinate offset of the initial blind pointing position of the star that is calculated for the first exposure for each +WATA Observation test. The average blind pointing for the star is better than 50 mas radial from the wide aperture center. +(c) The post-target acquisition pointing showing the improved centering of the TA target. The average corrected position for +the stars is better than 2.5 mas radial. +the set of centroids is analyzed, outliers are clipped, and +their mean offset from the intended position is used to +correct the initial spacecraft pointing (‘pitch’ and ‘yaw’) +and position angle (‘roll’). +The duration of the MSATA procedure depends on +the number of reference stars used and the depth of the +acquisition exposures, ranging from 23 min for the min- +imum number of reference stars (5) and the shortest +readout pattern, to 35 min for 8 reference stars (the rec- +ommended number), and the deepest available readout +pattern. In cases where the MSATA algorithm does not +converge to a sufficiently accurate solution, it may be +repeated once, adding up to 22 min to the duration if +the maximum of 8 reference stars is used. +The NIRSpec MSATA process is the only operational +situation where the roll angle of the Webb telescope is +adjusted after the start of an observation. To provide an +impression of the complexities involved, Figure 7 shows +the image used to verify the NIRSpec MSATA process +during commissioning (PID 1117, Obs. 31). The tar- +get field is the astrometric calibration field (Anderson +et al. 2021) in the Large Magellanic Cloud (LMC) which +had been pre-observed with HST and ground-based tele- +scopes to provide accurate stellar coordinates for the as- +trometric calibration of the JWST focal plane. In such +highly crowded fields, careful planning is necessary to +identify the optimal MSATA reference stars, which must +not saturate in the TA exposure, and must be suitably +isolated for optimal centroid calculation by the on-board +algorithm. +To allow accurate calibration of MOS spectra, the +MSATA process must place the science targets with an +accuracy of just 20 mas across the NIRSpec FOV. The +most important driver for the MSATA accuracy is the +absolute anchoring of the planning catalog in rotation, +as it is critical for the proper derivation of the TA roll +solution. +If the desired accuracy of the target place- +ment in the shutters can be relaxed, the MSATA can +also be planned using catalogs with poorer astrometric +precision or rotation anchoring, or using galaxies with +compact central cores instead of stars. +To evaluate the performance and post-TA target po- +sitioning accuracy, we have analysed the 20 NIRSpec +MSATA procedures carried out so far. Figure 8 shows +their distribution of corrective TA slews (in V2, V3 and +roll). +We find all slew offset solutions to be within +100 mas radial offset and at an average −51′′ roll off- +set compared to the optimal pointing. These relatively +small corrections are a testament to the excellent ’blind’ +pointing accuracy of JWST which allows very accurate +target placement even without any TA. +To illustrate the improvement achieved by the MSATA +procedure, Fig. 8 shows the target placement after +MSATA execution for the same 20 observations. These +measurements were derived by running an offset analysis +on the reference image that is acquired after the final TA +slew is complete. As can be seen, the average radial off- +set over the ensemble of TA reference targets is 25.0 mas, +and the average roll offset from the optimal pointing so- +lution is 14′′. This is close to, but not yet fully in line +with the requirements, but it should be kept in mind +that the more critical displacement in cross-dispersion +direction should be a factor of +√ +2 smaller. +In addi- +tion, many of the observations analyzed so far have used + +WATA Image Observation 3 +Blind Pointing- WATA Exposure 1 +Post-TA Slew - WATA Exposure 2 +2000 +a) +★★★★★ +Obs 1 +b) +c) +Obs 2 +bs 2 +1750 +25 +60 +Obs 3 +60 +Obs 3 +Obs 4 +Obs 4 +Y Pixel Position +1500 +★ +Obs 6 +Obs 6 +40 +Obs 8 +Obs 8 +Counts (ADU) +★ +(spu +1250 +20 +20 +1000 +set( +-20 +-20 +3 +10 +-40 +40 +500 +-60 +-60 +-80 +-80 +-80 +-60 +-40 +-20 +20 +60 +80 +-60 +0 +8 +-40 +-20 +20 +40 +60 +80 +V2 Offset (milli-arcseconds) +V2 Offset (milli-arcseconds) +X Pixel Position14 +MSA – ALL OPEN +TA Reference Stars +Fixed Slits +Flux – electrons / sec +0 +56 +Figure 7. +Undispersed NIRSpec exposure of the JWST astrometric field (from PID 1117, Obs 31). +This crowded, but +astrometrically well calibrated stellar field was selected to perform the first in-orbit test of the MSATA procedure. To ensure +that enough stellar centroids were successfully measured, this observation used a special requirement to allow the use of 12 TA +reference stars (highlighted by the cyan boxes), instead of the default maximum of 8 stars. The NIRSpec Fixed Slit apertures +are highlighted in green. +catalogs with non-optimal astrometric accuracy, and/or +non-stellar objects as reference targets. +7. NIRSPEC SCIENCE CALIBRATION PIPELINE +NIRSpec data for all modes are processed by the +JWST Science Calibration Pipeline6 in three stages. +6 see https://jwst-docs.stsci.edu/jwst-science-calibration-pipeline- +overview +Stage 1 takes raw up-the-ramp integrations, applies var- +ious detector-level corrections such as dark subtraction +and linearity correction, flags jumps due to cosmic ray +hits, and fits a slope to the ramp. +Stage 2 calcu- +lates wavelength and spatial coordinates per pixel us- +ing the parametric instrument model, corrects for in- +strument throughput losses using the three-component +flat field reference files, and converts to physical flux +units (MJy/pixel for point sources, MJy/steradian for + +15 +Average radial offset = 101.2 +/- 27.5 mas +Q = +-51.0” ++/- 34.8” +Average radial offset = 25.0 +/- 14.7 mas +Q = +14.0” ++/- 78.9” +Figure 8. Left: computed offset correction (in V2/V3 and roll angle) computed by the onboard algorithm for the ensemble of +20 MSATA visits analyzed so far. Each star symbol contains the average over the n reference stars used for the computation +(color-coded by the value of 5 ≤ n ≤ 8). Right: the results after execution of the corrective slew computed via the onboard +MSATA algorithm, as measured from the subsequent reference image (which still has the TA filter in place). +extended sources). +The final stage combines multiple +exposures (such as from a nod or dither pattern) at the +2D level. High-level pipeline products include 1D and +2D calibrated spectra for FS and MOS modes and 3D +data cubes for IFS mode. +The JWST Science Calibration Pipeline is a continual +work in progress. The code is updated via four regu- +lar build releases per year with bug fixes and algorithm +enhancements (although development versions are avail- +able on a continuous basis). Reference files are updated +whenever new calibration data are available and indi- +cate evolution in detector or throughput performance. +As of the time of writing, there are some outstanding +issues that users may want to be aware of: +i) the signal produced by snowball events is only par- +tially corrected by the cosmic ray jump detection step. +An optional algorithm that flags additional pixels af- +fected by a snowball has recently become available, and +further enhancements are being investigated. +ii) the ‘fast’ throughput correction vector used during +the S-flat step must be normalized by the wavelength +interval falling onto a given detector pixel. The current +reference files are normalized using representative val- +ues, which do not take into account the variation over +the FOV. Tests are ongoing, but preliminary investi- +gations suggest that this simplification adds an addi- +tional systematic uncertainty to the flux calibration that +can be as high as ten percent. A future version of the +pipeline will include an on-the-fly pixel-by-pixel normal- +ization that should remove this source of uncertainty. +iii) in earlier versions of the pipeline, the resam- +pling algorithms used in creating IFS cubes have some- +times produced wavelength-dependent artifacts in cer- +tain cases. A recent bug fix seems to fix the issue at least +for point sources, but users should carefully inspect the +extracted spectra to look for any unexpected behavior. +Also note that because of the optical field distortion in +the NIRSpec camera optics (see Jakobsen et al. 2022, for +a more detailed explanation), all NIRSpec spectra are +curved relative to the detector pixel grid, which in the +case of compact or point sources leads to aliasing effects, +esp. in the case of the gratings. Therefore, the spectrum +of a single cube spaxel will show a sinusoidal variation +in the extracted 1D spectrum. When summing up over +a sufficiently large circular aperture, however, this ef- +fect should average out. For the same reason, the alias- +ing should be much reduced for extended sources. Ulti- +mately, though, it is important to use dithering when- +ever possible, as the combination of dithered exposures +will minimize such resampling artifacts. +iv) the 1D extraction apertures are automatically cen- +tered at the expected position of the source, based on sky +coordinates specified in the JWST Astronomer’s Pro- +posal Tools (APT). For reasons that are still under in- +vestigation, this centering is often offset from the true +position of the spectral trace, requiring manual specifi- +cation of the center location by users. +v) there are no aperture corrections yet available for +1D extractions. The default extraction apertures are set +to be consistent with those used for the generation of the +F-flat reference files. For different aperture widths, users +must currently compute their own aperture corrections +using available observations of point sources. +vi) the outlier detection step in stage 3 of the pipeline +is intended to catch any outliers missed by stage 1 by +means of comparing the multiple input exposures. How- +ever, testing to date has shown performance problems +when using the default thresholds. Users will need to + +300 +★★★★ +Nstars = 5 +Nstars = 6 +Nstars=7 +200 +Nstars = 8 +V3 Slew (milli-arcseconds) +100 ++ +0 +-100 +-200 +-300 +-300 +-200 +-100 +0 +100 +200 +300 +V2Slew(milli-arcseconds)200 +150 +100 +Roll Slew (arcseconds) +50 +0 +-50 +-100- +-150 +-200 +-200 +0 +200200 +150 +100 +Post TA Roll (arcseconds) +50 +0 +-50 +-100 +-150 +-200 +-200 +200300 +★★★★ +Nstars = 5 +Nstars = 6 +Nstars=7 +200 +Nstars = 8 +Post TA V3 Residual (milli-arcseconds) +100 +0 +-100 +-200 +-300 +-300 +-200 +-100 +0 +100 +200 +300 +Post TA V2 Residual (milli-arcseconds)16 +adjust the parameters to get a reasonable identification +of true outliers, although in many cases it may be prefer- +able to turn the step off entirely until the algorithm is +improved. +8. SUMMARY AND CONCLUSIONS +We have provided an overview of the NIRSpec perfor- +mance as derived during the JWST commissioning cam- +paign and the first few months of in-orbit operations. +From a hardware perspective, the NIRSpec instrument +performs nominally in all electrical, mechanical, and +thermal aspects. When combined with the outstanding +optical performance of the JWST OTE, and the better +than expected accuracy and stability of Webb’s Attitude +Control System, it is no surprise that the scientific per- +formance of NIRSpec meets or exceeds the pre-launch +expectations for all observing modes. +More specifically, we have shown that the complex on- +board target acquisition procedures work well, the mea- +sured sensitivities are excellent across the board, and +unrivalled by any other NIR spectrograph, and the pho- +tometric stability is very good which is important for +time-series observations of exoplanets. +In terms of the calibration accuracy of NIRSpec sci- +ence data, we have shown that the data acquired and the +methods used so far allow the creation of reference files +that are sufficient to meet the requirements for the pho- +tometric and wavelength calibration of NIRSpec spectra. +On the other hand, we have mentioned a number +of open issues with the NIRSpec Science Calibration +Pipeline, in particular for the flux calibration of NIR- +Spec spectra, the cube-building step for IFS data, and +1D spectral extraction. These issues are actively being +worked, and improvements to the quality of the higher- +level NIRSpec data in the Mikulski Archive for Space +Telescopes (MAST) can be expected soon. +There are a few important ‘lessons learned’ from these +results that NIRSpec users should take note of: +• the higher than expected PCE measured over most of +the NIRSpec wavelength range may cause some ex- +posures to saturate earlier than predicted from pre- +launch estimates. This may make some ‘bright tar- +get’ science less feasible and/or require modifications +to the observing strategies, especially the selection of +MSATA reference stars. +• for MOS mode and the associated MSATA planning, it +is essential that the catalog of MSATA reference stars +is properly registered to the Gaia frame, not just in +RA and Dec, but also in rotation. +• the initial telescope pointing after guide star acqui- +sition is accurate enough that for most IFS observa- +tions, foregoing TA (by using TA=NONE or VER- +IFY ONLY) is feasible. Because of the better than ex- +pected PSF quality and the resulting small slit losses +in the S1600A1 aperture, this may even be true for +some time-series observations. +To summarize, +the NIRSpec instrument onboard +JWST presents a quantum leap for NIR spectroscopy +in terms of sensitivity and multiplexing capability, and +promises to have a lasting impact on many research +fields. +We are deeply grateful to the large number of engineers +and scientists in Europe, Canada, and the US, whose +dedication and hard work over many years have turned +NIRSpec and the entire JWST mission from a mere vi- +sion into reality. +REFERENCES +Alves de Oliveira, C., L¨utzgendorf, N., Zeidler, P., et al. +2022, in Proc. SPIE, Vol. 12180, 121803S, +doi: 10.1117/12.2629911 +Anderson, J., Fall, M., et al. 2021, The JWST Calibration +Field, Tech. Rep. JWST-STScI-007716, Space Telescope +Science Institute, Baltimore, MD +Birkmann, S. 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+page_content=' 2201 AZ Noordwijk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The Netherlands 8Cosmic Dawn Center (DAWN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Niels Bohr Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' University of Copenhagen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Jagtvej 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' DK-2200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Denmark 9Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Denys Wilkinson Building,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Keble Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' OX1 3RH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' UK 10Scuola Normale Superiore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Piazza dei Cavalieri 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' I-56126 Pisa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Italy 11Sorbonne Universit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' UPMC-CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' UMR7095,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Institut d’Astrophysique de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' F-75014 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' France 12Cavendish Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 19 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Thomson Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=', Cambridge CB3 0HE, UK 13Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 14Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA 15Leiden Observatory, Leiden University, PO Box 9513, 2300RA Leiden, The Netherlands 16Aurora Technology for the European Space Agency, Villanueva de la Ca˜nada, E-28692 Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 17Universidad Europea de Madrid, 28670, Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 18Quantum Circuits, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=', New Haven, Connecticut, USA 19NASA Goddard Space Flight Center, Observational Cosmology Laboratory, Greenbelt, USA 20Max-Planck Institute for Astronomy, K¨onigstuhl 17, 69117 Heidelberg, Germany 21Department of Earth & Planetary Sciences, Johns Hopkins University, Baltimore, MD 21218, USA 22NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada ABSTRACT The Near-Infrared Spectrograph (NIRSpec) is one of the four focal plane instruments on the James Webb Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In this paper, we summarize the in-orbit performance of NIRSpec, as derived from data collected during its commissioning campaign and the first few months of nominal science operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' More specifically, we discuss the performance of some critical hardware components such as the two NIRSpec Hawaii-2RG (H2RG) detectors, wheel mechanisms, and the micro-shutter array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' We also summarize the accuracy of the two target acquisition procedures used to accurately place science targets into the slit apertures, discuss the current status of the spectrophotometric and wavelength calibration of NIRSpec spectra, and provide the ’as measured’ sensitivity in all NIRSpec science modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Finally, we point out a few important considerations for the preparation of NIRSpec science programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Keywords: Instrumentation: spectrographs - Space vehicles: instruments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' INTRODUCTION AND BACKGROUND Many, if not most, of the science goals for the James Webb Space Telescope (JWST) rely on the ability to ac- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='13766v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='IM] 31 Jan 2023 ID2 quire high-quality near-infrared (NIR) spectra of astro- nomical targets with a wide range of luminosities, from the faintest galaxies at high redshift to the bright host stars of nearby exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The NearInfraRed Spectro- graph (NIRSpec) onboard JWST offers the necessary versatility to enable observations of such a wide range of targets, thanks to a variety of sophisticated observ- ing modes, some of which are available for the first time in space and at near-infrared wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' multi- object and integral field spectroscopy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In this paper, we summarize the status of the NIRSpec performance characterization, using mostly data ac- quired during the JWST commissioning campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The analysis of these data sets (which in many cases have since been complemented by additional data acquired during Cycle 1) is still ongoing, and many performance- related measurements and products are preliminary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Therefore, this paper can only provide a snapshot of the performance assessment as of mid-Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' OVERVIEW OF THE NIRSPEC INSTRUMENT The design of and scientific motivation for the NIR- Spec instrument have been extensively discussed in Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For convenience, we show here again the optical design of the NIRSpec instrument, to- gether with the layout of the mechanical assembly in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In brief, NIRSpec is designed to be capable of per- forming both single- and multi-object spectroscopic ob- servations over the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 µm NIR wavelength range at three spectral resolutions that cover a range of sci- ence applications: a low-resolution (R ≃ 30–330) mode intended for obtaining exploratory continuum spectra and redshifts of remote galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' an intermediate resolu- tion (R ≃ 1000) mode for accurately measuring atomic and molecular emission lines, and a higher resolution (R ≃ 2700, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' ≃ 110 km/s) mode primarily for kine- matic studies using these emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Table 1 sum- marizes the various NIRSpec modes and their typical science applications, together with their slit apertures and wavelength coverage, and available spectral resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' More detailed descriptions of the NIRSpec observ- ing modes can be found in Ferruit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022) for the Multi-Object Spectroscopy (MOS) mode, B¨oker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022a) for the Integral-Field Spectroscopy (IFS) mode, and Birkmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022a) for the Bright Object Tran- sit Spectroscopy (BOTS) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' THE NIRSPEC COMMISSIONING CAMPAIGN As discussed in more detail in B¨oker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022b), the NIRSpec commissioning campaign successfully con- cluded 185 days after launch, when the last of its four Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' NIRSpec instrument modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' long-pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Table 2-1 NIRSpec instrument modes Mode Target type Wavelength range Aperture mask Resolving Power MSA spectroscopy (MOS) rich fields or very extended object 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 µm any config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2² x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='46² micro-shutters R=30-330 or R»1000 or R»2700 Fixed Slit Spectroscopy (FS) single compact object 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 µm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6² x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6² or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2² x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2² or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4² x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='65² FS R=30-330 or R»1000 or R»2700 Bright- Object Time Series (BOTS) bright point sources 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 µm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6² x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6² R=30-330 or R»1000 or R»2700 Integral-field Spectroscopy (IFS) moderately extended objects 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 µm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0² x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0² IFU R=30-330 or R»1000 or R»2700 Target Acquisition (TA) reference stars (n~10-20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 µm all shutters open (except around bright targets) undispersed imaging (MIRROR) Internal Calibrations none 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 µm any config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' of microshutters and FS/IFU undispersed imaging or R=30-330 or R»1000 or R»2700 Every instrument mode listed in Table 2-1 is implemented in the flight software and can be used observing modes was formally declared ‘ready for sci- ence’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' A total of 33 NIRSpec commissioning activity re- quests (CARs) were required to activate the instrument electronics, to commission the various observing modes, and to obtain a first level of calibration and performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' While the activities were scattered through- out the six months of JWST commissioning, they had to be executed in a carefully planned sequence in order to derive the various intermediate products required for subsequent, and increasingly more complex, activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The major milestones were the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The successful execution of an onboard script that controls and maintains the temperature of the mi- croshutter quadrants 10-15K above the environ- ment in JWST’s Integrated Science Instrument Module (ISIM), until the latter dropped below 140K, in order to avoid condensation of water ice or other volatiles onto the micro-shutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The NIRSpec Optical Bench Assembly (OBA) reaching sufficiently cold temperatures for the safe operation of the various mechanisms (the magnet arm of the microshutter array (MSA), the wheel mechanisms, and the re-focus mechanism) and the active control of the Focal Plane Assembly (FPA) temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The verification of the mechanical and operational performance of these mechanisms and the various lamps in the Calibration Assembly (CAA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The detailed performance characterization of the NIRSpec detectors and the ≈ 250000 individual shutters of the MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The results, which are de- tailed in Birkmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022b) and Rawle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 3 OTE Primary filter wheel pupil image Grating pupil plane OTE focus MSA FPA 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4 m F/ 20 F / 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 F/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 FORE optics COLL optics CAM optics Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1 : Schematic representation of OTE + NIRSpec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1 shows that the OTE NIRSpec optical train has three field planes and three pupil planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The pupil planes are the OTE primary mirror (which is the limiting pupil stop of the system), the pupil plane at the filter position and finally the pupil plane at the position of the grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Not shown in the picture above is the fact that the OTE primary is being re-imaged at the position of the Fine Steering Mirror (FSM) which serves as the exit pupil of the OTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The image planes are located at the OTE focal surface, the Micro-Shutter Assembly (MSA) and finally at the Focal Plane Assembly (FPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The real optical lay-out of NIRSpec is shown in fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2: Optical lay-out of the NIRSpec instrument Detector FPA Grating Wheel COLLimator optics MSA + IFU Filter Wheel FORE Optics Pick-Off mirrors CAMera Optics Refoc 1000 mm Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' of SPIE 59040L-2 Downloaded From: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='spiedigitallibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='org/conference-proceedings-of-spie on 12 Jan 2020 Terms of Use: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='spiedigitallibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='org/terms-of-use Disperser Detector Array Camera Collimator Slit Plane Foreoptics Filter Pick-off Mirrors Focus Mirrors 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4o OTE Primary filter wheel pupil image Grating pupil plane OTE focus MSA FPA 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4 m F/ 20 F / 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 F/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 FORE optics COLL optics CAM optics Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1 : Schematic representation of OTE + NIRSpec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1 shows that the OTE NIRSpec optical train has three field planes and three pupil planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The pupil planes are the OTE primary mirror (which is the limiting pupil stop of the system), the pupil plane at the filter position and finally the pupil plane at the position of the grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Not shown in the picture above is the fact that the OTE primary is being re-imaged at the position of the Fine Steering Mirror (FSM) which serves as the exit pupil of the OTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The image planes are located at the OTE focal surface, the Micro-Shutter Assembly (MSA) and finally at the Focal Plane Assembly (FPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The real optical lay-out of NIRSpec is shown in fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2: Optical lay-out of the NIRSpec instrument Detector FPA Grating Wheel COLLimator optics MSA + IFU Filter Wheel FORE Optics Pick-Off mirrors CAMera Optics Refoc 1000 mm Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' of SPIE 59040L-2 Downloaded From: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='spiedigitallibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='org/conference-proceedings-of-spie on 12 Jan 2020 Terms of Use: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='spiedigitallibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='org/terms-of-use Disperser Detector Array Camera Collimator Slit Plane Foreoptics Filter Pick-off Mirrors Focus Mirrors Filter Wheel Grating Wheel Detector Array Calibration Source Microshutter Array Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Left: The optical path through the NIRSpec instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The dispersion direction is out of the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Right: CAD rendering of NIRSpec with its major mechanisms identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The dimensions of the NIRSpec optical assembly are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='9 m × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4 m × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='7 m, with a total mass of 196 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022), confirmed the expected in-flight perfor- mance, and marked the start of Phase 2 of NIR- Spec commissioning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' the ability to obtain sci- ence exposures via the onboard scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The completion of the optical telescope element (OTE) alignment described in McElwain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022) enabled the first on-sky NIRSpec observa- tions, in particular the ‘focus sweep’ which was used to verify the optimal positioning of the in- ternal focus adjustment mechanism, and allowed a first assessment of the image quality and total wavefront error across the NIRSpec field of view (FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For this, the NIRSpec focus mechanism was ’swept’ over a range of ±2000 motor steps, with an exposure in the F110W filter taken ev- ery 400 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The sweep range corresponds to a physical mechanism movement of ±1 mm and a defocus of ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 mm in the OTE focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Once the optimal NIRSpec focus was established, undispersed images of a dense star field with known and accurate stellar astrometry were ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These observations of the ‘astrometric ref- erence field’ (Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2021) enabled precise measurements of the magnification and distortions of the NIRSpec optical train, which is a crucial ingredient for the parametric instrument model (Dorner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' L¨utzgendorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022), which underlies the wavelength calibration of all NIRSpec spectra (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3), as well as the co- ordinate transformations between detector, MSA, and sky that are required to accurately place sci- ence targets in the various NIRSpec apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Once the accuracy of the instrument model was confirmed via a series of successful target acqui- sitions (TAs), the last few activities of the NIR- Spec commissioning campaign obtained on-sky ob- servations of stars and planetary nebulae for flux and wavelength calibration, which enabled the cre- ation of various ‘reference files’ and other products needed for the data reduction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' OBSERVATORY PERFORMANCE Because of the narrow slit apertures involved in most of its observing modes, the NIRSpec science perfor- mance is critically dependent on a number of telescope and observatory characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Here, we briefly list the most important ones, without discussing them in detail because they are the subject of accompanying papers in this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Optical Quality of the OTE The quality of the image delivered by the OTE is of fundamental importance for all JWST instruments, as it limits the spatial resolution and sensitivity of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As discussed in other papers in this edition (Menzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' McElwain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022), the in-orbit opti- cal performance of the OTE is significantly better than expected, with lower wavefront errors and therefore a sharper point spread function (PSF) across the entire FOV (Feinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' More specifically, the JWST PSF is diffraction limited already at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1 µm, and reaches a Strehl ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='9 at 4 µm across the FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' While this is clearly good news for all JWST instru- ments, the NIRSpec sensitivity stands to gain the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This is because a narrower PSF minimizes the ‘path losses’ caused by the physical truncation and subsequent diffraction of the PSF by the narrow apertures of the 4 NIRSpec fixed slits and microshutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As discussed in Giardino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022), this is the main reason for the fact that the NIRSpec slitlosses are significantly smaller than expected, with peak gains of more than 10% at the shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The resulting benefit to the NIRSpec throughput is discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Cleanliness and thermal backgrounds In addition to the low wavefront error, the OTE optics are also free of water ice or other molecular contamina- tion, resulting in a very high telescope throughput across the entire NIRSpec wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Moreover, the good thermal performance of the sunshield, and the re- sulting nominal temperatures of the OTE result in ther- mal backgrounds that are at or below the pre-launch expectations (Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022a), also enhancing the sensitivity of NIRSpec observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Taken together, the outstanding optical and thermal performance of the JWST OTE provides a much better wavefront to the NIRSpec optics than expected, which is reflected in the sensitivity numbers discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Attitude Control System Another important factor for the quality of NIRSpec observations is the pointing accuracy and stability of the JWST Attitude Control System (ACS), which is discussed in detail in Menzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Accurate ‘blind pointing’ is required to ensure that the science target is indeed imaged onto the rather small detector area used to identify the target by the onboard TA al- gorithms, and that the corrective small-angle maneuvers (SAMs) are not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The ability to accurately ex- ecute those SAMs, and the absence of any drifts or low- frequency jitter in the telescope pointing is crucial for stable and precise target placement throughout the ex- posure, which is particularly important for time series observations of transiting exoplanets, as discussed fur- ther in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' On all these accounts, the JWST ob- servatory meets or exceeds the pre-launch requirements, which is the foundation for the problem-free execution of the various NIRSpec observing templates, and the outstanding NIRSpec science performance presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' NIRSPEC HARDWARE PERFORMANCE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Wheel Mechanisms NIRSpec is equipped with two wheel mechanisms: i) the Filter Wheel Assembly (FWA) with five long-pass transmission filters (F070LP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' F100LP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' F170LP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' F290LP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' and CLEAR) that define the wavelength ranges for the dispersers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' two filters used for the TA exposures (F110W and F140X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' and one position (OPAQUE) that serves both as an instrument shutter towards the telescope side and a coupling mirror for illumination from the CAA on the instrument side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' and ii) the Grating Wheel Assem- bly (GWA) with the seven NIRSpec dispersers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' and a mirror for undispersed imaging (see also Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The functionality of both wheels was verified during com- missioning, and their torque profiles were characterized and found to be nominal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' similar to those measured during ground testing, and well within their expected tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As discussed in Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022), the limited me- chanical angular reproducibility of the GWA ball bear- ings causes small, but significant, variations in the po- sition of NIRSpec spectra on the detector, especially in dispersion direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Since this has an obvious effect on the wavelength calibration as well as the precision of the TA algorithm, it is important that the actual GWA position of any given NIRSpec exposure can be accu- rately inferred from telemetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To this end, the GWA design includes two magneto-resistive position sensors (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='‘tilt sensors’) which were extensively tested before launch (De Marchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2012) to ensure that NIRSpec can indeed achieve the required accuracy of its wave- length calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The tilt sensors have to be calibrated after each cooldown, and their post-launch calibration was one of the important goals of the NIRSpec commissioning cam- paign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The accuracy of the in-orbit sensor calibration has been discussed in detail by Alves de Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022) : the remaining uncertainties are consistently be- low 1/10th of a pixel, which is fully in line with expecta- tions and sufficient to ensure a reliable wavelength cal- ibration for all NIRSpec spectra, as demonstrated fur- ther in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The tilt sensor performance and the stability of the wavelength calibration will be monitored with a roughly semi-annual cadence as part of the over- all JWST calibration plan, using a combination of stel- lar spectra and the internal LINE and REF calibration lamps (see Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Microshutter Array The NIRSpec microshutter array (MSA) has per- formed excellently throughout flight thus far, with no unexpected hardware issues, overall multiplexing re- maining very high, and an average success rate of ∼96% for commanding shutters open in science-like patterns (Rawle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Operability at the level of individ- ual shutters is of course crucial for the MOS mode itself, but less obviously also for the IFS mode, where problems with the shutter array can become a source of parasitic light, either contamination through stuck-open shutters 5 or thermal glow1 emanating from electrical shorts in the circuitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Our knowledge of this performance is critical to allow mitigation, such as short-masking, and enable users to generate the optimal target set around the in- operable shutter population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As described in detail by Rawle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022), the marginal increase in inoperable shutters seen during commissioning was in-line with pre-launch estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The primary consequence of these trends is that the masking required to mitigate electrical shorts is now the dominant factor affecting the usability of unvignetted shutters (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' While comparatively fewer shorts have emerged after launch, with approximately one row/column needing to be masked for every 50 shut- ter array re-configurations compared to one masked per 20–25 during ground testing, shorts have continued to appear throughout the latter stages of commissioning and another two required masking in the first three months of science observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' An added complication apparent for two of the shorts seen since launch (and never during ground testing) was that they produced glow when applying the all-closed shutter configuration for IFS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Although successfully mitigated in the usual manner, the contamination before masking affected both MOS and IFS exposures, increasing the overall impact of the shorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In the regime that shorts may still occur every few months, there are two operational issues to tackle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' First, to ensure that data still meet scientific objectives, users need to be informed in a timely manner when exe- cuted exposures are directly contaminated by a short and also when upcoming observations are affected by newly masked shutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The former may now apply to both MOS and IFS visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Second, as the population of shorts continues to grow, each removing several hundred shutters from operation, this will eventually have a no- ticeable impact on multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Shorts are believed to be caused by particulate contamination of the complex MSA control electronics, and those particles may shift during reconfiguration, which is the origin of new shorts but also potentially clears the cause of previously de- tected shorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Therefore, as discussed by Rawle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022), re-checking whether older shorts still remain may be an avenue to recover previously unusable shut- ters and maintain the multiplexing of NIRSpec MOS at the current high level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 1 while a full in-orbit characterization of the spectral characteristics of MSA shorts is still pending, ground test data have indicated that it is clearly thermal in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' However, it is not a pure blackbody spectrum, likely because it contains signatures from the MSA coating materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' MSA operability report from the end of commis- sioning, demonstrating that 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5% of un-vignetted shutters are available for use as science apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Q1 Q2 Q3 Q4 Total Total 62415 62415 62415 62415 249660 Vignetted 6119 5929 6102 5874 24024 Failed open 6 3 12 1 22 Short-masked 7835 5150 3466 7177 23628 Failed closed 1569 3328 5932 5064 15893 Total Usable 46886 48005 46903 44299 186093 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Detector System The NIR light collected by the JWST OTE and fed through the NIRSpec optical train is registered by two Hawaii-2RG (H2RG) sensor chip assemblies (SCAs), which are described in detail by Rauscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The SCAs are operated at a temperature of 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 K, cho- sen as the best compromise between pixel operability and total noise of the detector system: a higher temper- ature leads to an increased number of hot pixels, while a lower temperature leads to higher noise in the signal chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As a reminder2, the NIRSpec SCAs are non- destructively read “up-the-ramp”, using one of two fun- damentally different readout modes: the so-called tra- ditional readout mode (TRAD) or the improved refer- ence sampling and subtraction (IRS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' pronounced “IRS- square”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Rauscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2017) readout mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' A num- ber of detector subarrays with different sizes such as, for example, the ALLSLITS subarray are supported in TRAD mode only, offering faster readouts and using a slightly higher conversion gain to make the full physical well depth of the pixels accessible, which is of particular importance for time series observations of bright targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The in-orbit performance of the NIRSpec SCAs and their readout electronics is an important factor for the NIRSpec science performance, because the sensitivity of most NIRSpec observing modes is limited by the to- tal detector noise in a given exposure (see Appendix A in Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The analysis of the detector performance data collected during the NIRSpec in-orbit commissioning campaign characterization has been pre- sented in detail by Birkmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Here, we briefly summarize the most relevant results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2 for details, see the NIRSpec User Documentation at https://jwst- docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='edu/jwst-near-infrared-spectrograph/nirspec- instrumentation/nirspec-detectors 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Cosmetics The vast majority of pixels in the NIRSpec detectors can be considered operable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Non-operable or bad pixels are, for example, those which have a very low quan- tum efficiency or no response at all, are not connected to the readout electronics, or exhibit a large / highly non-linear dark signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The number of non-operable / bad pixels in the NIRSpec detectors as measured during commissioning is summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Dark Current The median dark signal of the two SCAs for the dif- ferent NIRSpec readout modes is presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Two general observations can be made: i) the in-orbit dark signal for NRS1 is generally higher than for NRS2, which is fully in line with previous measurements ob- tained during the various ground testing campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' ii) both NIRSpec SCAs exhibit a slightly elevated dark cur- rent signal compared to on-ground measurements, with a more pronounced increase towards the array edges (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2 in Birkmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As discussed in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4, this increase is likely related to the cosmic ray environment at L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' However, even with this small in- crease compared to pre-launch measurements, the dark signal is still very low for most pixels, and not a driver for the total noise (and thus the sensitivity) of NIRSpec observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Multiplexer readout glow is thought to dominate the observed dark signal (Regan & Bergeron 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Noise Performance The total noise for the different readout modes of the two NIRSpec SCAs as a function of effective integration time is summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These numbers include the effects of cosmic rays, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' broken ramps due to cos- mic ray hits and early saturation for some pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Figure 2 shows the noise behavior as a function of inte- gration time in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For all readout modes, the total noise decreases with exposure length, up to integra- tion times of ≈ 500 s where it levels out and then slowly increases for longer integrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Nevertheless, for obser- vations of faint sources that are detector noise limited, it is still beneficial to use longer integration times for optimal signal-to-noise ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' On the other hand, it is always advisable to have multiple integrations per ob- servation, ideally in the form of dithered exposures to guard against early saturation after a strong cosmic ray hit, as well as to enable a robust rejection of outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The IRS2 readout mode shows the best total noise performance, in particular for NRS1, and should be used when observing faint targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The ALLSLITS subarray readout mode has noise levels comparable to those in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Median total noise signal (in e−) as a function of effective integration time for the two NIRSpec detectors and three readout modes: IRS2, traditional full frame (TRAD), and ALLSLITS subarray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' traditional full frame mode (TRAD), and is best used for observations of bright targets with short integrations, because the effective full well depth is higher due to the higher conversion gain, like for the other subarrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Cosmic Ray Environment From the NIRSpec dark exposures obtained during commissioning, we have measured an average CR hit rate of about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 cm−2 s−1, using the jump detection algorithm in the ramps-to-slopes pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The observed hit rate is well in line with pre-launch predictions of the proton fluence at solar minimum (Giardino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Most cosmic ray events affect several pixels (partially due to inter-pixel-capacitance), but are compact, with a typical hit area of about 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' However, there can also be longer streaks and so-called ‘snowballs’, affecting many pixels at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As described in Birkmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022b), snowballs have the following characteristics: a core of fully saturated pixels an extended ‘halo’ around the saturated core that is detected at the same time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' within the same correlated double sample), with the intensity of the halo dropping off towards larger radii often accompanied by a shower of more compact cosmic ray events in the vicinity core and halo are often spherical in appearance, but can be very elongated as well The flux of snowballs varies, but stronger ones can produce tens of millions of electrons within their satu- rated core, indicating large energies of the involved par- ticle(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Snowballs are observed by all NIR instruments on JWST (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Rieke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022), but their origin is not yet fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 16 NRS1 IRS2 NRS2 IRS2 NRS1 TRAD NRS2 TRAD 14 - NRS1 ALLSLITS NRS2 ALLSLITS Total noise [e-] 12 10 8 6 0 500 1000 1500 2000 2500 3000 3500 TINTeff [s]7 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Summary of bad pixel statistics for the two NIRSpec detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Note that the total number of bad pixels can be less than the sum of the individual categories, as some bad pixels belong to several categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Note that the fraction of operable pixels listed in the last row is for the active area of each detector (2040 x 2040 pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Bad pixel type NRS1 NRS2 Notes Open 294 252 Very low response, signal ends up in adjacent pixels (next row) Adjacent open 1664 1216 Impacted by open neighbor pixel (previous row) Dead 7757 3938 Does not respond to light Low QE 1766 887 Low response to light RC-like 3908 1902 Non-linear dark signal (RC-like ramp) Hot 6062 2123 Large dark signal (> 1 e− s−1) Total 16948 8275 Total number of non-operable pixels Operable pixels [%] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='59 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='80 Fraction of operable pixels Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Median dark signal of the two NIRSpec SCAs (in e− per 1000s integration time) for traditional full frame, IRS2 full frame, and ALLSLITS subarray readout modes, as mea- sured during commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For comparison, the equivalent numbers obtained during the last ground test campaign are listed in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Readout mode SCA TRAD IRS2 ALLSLITS NSR1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='7) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6) NRS2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='9 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='7) Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Total noise of the two NIRSpec detectors for dif- ferent readout modes and effective integration times as mea- sured during commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Effective Integration Time Readout / SCA ∼950 s ∼1700 s ∼3560 s TRAD / NRS1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4 N/A TRAD / NRS2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='7 N/A IRS2 / NRS1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 IRS2 / NRS2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 SUB / NRS1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 N/A SUB / NRS2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 N/A 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Persistence Image persistence, or latency signal, is an unavoid- able effect in NIR HgCdTe detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' It manifests itself as an ’afterglow’ in pixels that have been subjected to strong illumination, resulting in a faint residual image in the following integrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Persistence is caused by elec- trically active defects, so-called ’charge traps’, in the detector material which accumulate charge that is only released in later exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Past results from the ground testing of the JWST NIR detectors has shown that their persistence behavior is generally better than that of sim- ilar detectors on previous NASA missions including the Hubble Space Telescope: any persistence decays to be- low the background level after about 2000 s (Rauscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' While the detailed characterization of the persistence behavior of the two NIRSpec SCAs is still ongoing, the in-orbit data collected so far suggest that persistence is not a major concern for most NIRSpec science programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' NIRSPEC SCIENCE PERFORMANCE Many key parameters of the end-to-end science per- formance of NIRSpec and the JWST OTE could only be measured with on-sky data obtained after launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In this section, we discuss a number of aspects that are relevant for the scientific performance of all NIRSpec modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Photon Conversion Efficiency and Sensitivity For NIRSpec, and in fact any optical instrument, a critical performance parameter is its efficiency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' the fraction of photons incident on the primary mirror that are being registered by the detector after passing through the entire optical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This metric, together with the noise performance of the NIRSpec detectors, drives the ultimate sensitivity for astronomical observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Because NIRSpec is a complex instrument that sup- ports many different observing modes, its optical train is rather complicated, as evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Except for the order-separation filters and the low resolution double-pass prism, the NIRSpec optics are reflective throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Photons captured by the JWST primary mirror undergo a total of 19 reflections before reach- ing the NIRSpec detector array in the MOS, FS, and BOTS observing modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For IFS mode, there are an ad- ditional 8 reflections in the IFU optics (see B¨oker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 8 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' NIRSpec spectral configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Band Disperser λ/δλ Filter λ range [ µm] 0 G140M G140H 1000 2700 F070LP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='7 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2a I G140M G140H 1000 2700 F100LP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 II G235M G235H 1000 2700 F170LP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='7 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1 III G395M G395H 1000 2700 F290LP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='9 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 n/a PRISM 30-330 CLEAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 a the Band 0 configurations using the F070LP filter will obtain spectra over a much wider wavelength range which, however, con- tain 2nd-order contamination beyond 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Users who are pre- pared to deal with this contamination can potentially use Band 0 out to ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Cleanliness and high reflectivity of the NIRSpec optics therefore was of ultimate importance throughout the construction and test phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' By necessity, the metric of choice to measure the op- tical throughput is the Photon Conversion Efficiency (PCE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' the ratio of photons incident on the JWST primary mirror to electrons registered by the NIRSpec detector system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' It can be derived using observations of a ‘flux standard’ (typically a well-characterized star) with an accurately known spectral energy distribution (SED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As can be seen from Table 6, NIRSpec has a to- tal of nine spectral configurations, and the throughput must be measured separately for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As described in more detail by Giardino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022), the NIRSpec optical efficiency meets or exceeds the pre- flight expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In particular, significantly higher than predicted PCEs are achieved for the high-resolution configurations, for the MOS/FS mode, and for all con- figurations below ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 µm for the IFS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The 10-20% lower than expected efficiencies above ∼ 4 µm, apparent in particular for the IFS mode, are (mostly) explained by the more significant path losses at longer wavelengths, which are reflected in the measurements but not the predictions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 3 in Giardino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These in-orbit PCE measurements , together with the detector noise performance discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3, can be used to calculate the in-flight NIRSpec sensitivity for its various science modes and configuration, following the methodology described in Appendix A of Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Figure 3 shows the results of the calcu- lations in terms of continuum sensitivity curves for a point source observed in the MOS and IFS modes, using a bench-mark observation in all spectral configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The achieved level of sensitivity is extremely impressive when compared to other NIR spectrographs with simi- lar observing modes and spectral resolutions: by at least two orders of magnitude, NIRSpec is the most sensitive NIR spectrograph currently available for astronomical studies (see also Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Spectro-Photometric Calibration The full photometric calibration of NIRSpec requires three major steps in the reduction pipeline,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' with their associated reference files: i) the detector flat (D-Flat) to capture the pixel-to-pixel response variations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' and derived from component-level ground test data of the two NIRSpec SCAs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' ii) the spectrograph flat (S-Flat) to correct the throughput variations of the spectrograph optics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' and measured from exposures using the internal calibration lamps in the CAA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' and iii) the FORE optics flat (F-Flat) to characterize any field- and wavelength- dependent effects caused by the OTE and the NIRSpec FORE Optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For the last step, observations of spectro- photometric standard stars are necessary to verify the integrity of the entire NIRSpec optical system, in partic- ular the pick-off and FORE optics (including the FWA), neither of which can be illuminated with the CAA lamps (see te Plate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2005, for a detailed description of the light path from the CAA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For a detailed overview of the NIRSpec flat field strategy we refer to Rawle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The S-Flat step corrects any throughput variations in the spectrograph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' introduced in the optical path after the FWA and before the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The S-Flat ref- erence files for all NIRSpec observing modes are derived from internal lamp exposures and after correction for the detector response (D-Flat) and spectral energy distribu- tion of the lamp(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Because the throughput is strongly dependent on the properties of the chosen disperser and slit aperture, different reference files are created per de- tector for each combination of prism or grating and NIR- Spec observing mode (FS, IFS, or MOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For the FS and IFS modes, the S-flat consists of two parts: (i) a 2D image capturing the pixel-to-pixel variations for all slits/slices, and (ii) a vector for each slit and the IFU, to correct the fast throughput variations with wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This approach is possible for the FS and IFS mode, be- cause for a given aperture and spectral configuration, each detector pixel is always illuminated by the same wavelength (modulo the non-repeatability of the GWA, which is corrected separately using the position sensors described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For MOS mode, in contrast, the same detector pixel can receive light of different wavelengths, depending on the position of the open MSA shutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Hence, the MOS 9 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' NIRSpec point-source continuum sensitivity in MOS/FS (left) and IFS (right) mode, derived from in-orbit mea- surements and assuming a well-centered source in a microshutter or an IFU slice, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The sensitivity for the S200 slits in FS mode is similar to the one in MOS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The plots show, for each disperser and as a function of wavelength, the flux required to reach S/N = 10 per spectral pixel in 10,000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' More specifically, the computations assume 10 NRSIRS2RAPID exposures of 1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='7 s each (70 groups of 1 frame), using the methodology described in Appendix A of Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The spectra of standard stars 1808347 (left) and P177D (right), extracted with the NIRSpec-internal reduction pipeline, and compared to the CALSPEC model to verify the flux calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Both spectra were taken through the S1600A1 slit using the F290LP/G395H configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The bottom frames show the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The observations of P177D were obtained without a target acquisition step, and the systematically lower flux compared to the model is most likely due to imperfect centering in the S1600A1 aperture, which can easily cause aperture losses of 1-2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' S-flat reference consists of a data cube, sampling the wavelength variation for each pixel, so that the specific S-flat for any given MSA configuration can be derived on-the-fly by the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Given the large number of shutters and the finite amount of time available for com- missioning, direct measurement of the throughput for every shutter was impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Therefore, only a subset of MSA shutters was observed, from which the complete (smoothed) S-flat cube could be generated by interpola- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Similar to the S-Flat, the F-Flat must be created for each of the filter and grating wheel combinations listed in Table 1 and for each of the three observing modes (FS, IFS, and MOS3), to take into account any wave- length dependence on the throughput of the OTE and FORE optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Standard star observations were obtained for all but the F100LP/G140M configuration, for which calibration files based on simulated data will remain in place until such observations are taken during Cycle 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' We used the A3V star 1808347 (2MASS J18083474+6927286) for all gratings, while for the CLEAR/PRISM configuration, we used either the A8III 3 The in-flight update for the MOS mode is still pending, but the necessary data have been taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 data data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 1808347 P177D 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 [gw/Mlb-0T> F290LP/G395H S1600A1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 xnd 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='05 sjenp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='00 resi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 wavelength [μm] wavelength [μm]10 star 1743045 (2MASS J17430448+6655015), or the G0- 5 star SNAP-2 (2MASS J16194609+5534178), because 1808347 would saturate the low-resolution spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For the fixed slits, we used a 3-nod pattern (2-nod pattern for S1600A1) to obtain the data, and a 4-nod pattern with a 4 spaxel (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4 arcsec) extraction radius for the IFS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The MOS program was executed using a 3-shutter nod pattern for each of the filter/grating combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To derive the conversion to absolute flux units (which is part of the F-Flat step), the extracted spectra were divided by the resampled standard star templates avail- able on the CALSPEC website4 (Bohlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To create a smooth F-Flat, any outliers such as remaining hot pixels, stellar absorption lines, or the 0th order con- tamination of the lamp spectra used to create the S-Flat, were masked and a smooth vector was fitted, creating the final F-Flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To verify the validity of the NIRSpec calibration approach, we re-extracted the spectrum of the stan- dard stars using the NIRSpec-internal data reduction pipeline, and compared the resulting spectra to the CALSPEC template, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 4 for the case of FS mode with the S1600A1 aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Overall, the agreement is very good, with an RMS of the residu- als well below 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For the other NIRSpec modes and apertures, the results are of similar quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Note that this comparison only represents an estimate of the ‘best case’ calibration accuracy, because it is performed on the same star that was used to derive the pipeline refer- ence files, and thus does not account for any systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' While an evaluation of the systematic errors must wait for additional calibration data obtained during Cycle 1 or later, commissioning spectra of two other standard stars, WD1057+719 (a DA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 white dwarf, M-gratings) and P177D (a G0V star, H-gratings), were obtained in FS mode with the S1600A1 slit (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These ob- servations were executed early in commissioning and be- fore the wide aperture TA procedure (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1) was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Therefore, a proper centering is not guar- anteed and (small) uncorrected aperture losses are possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Nevertheless, we measure residuals between the flux template and the extracted spectra of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='12 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='55%, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='76 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='00%, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='67 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='57%, −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='09 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='16%, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='21%, and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='97% for the F070LP/G140H, F170LP/G235M, F170LP/G235H, F290LP/G395M , 4 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='edu/hst/instrumentation/ reference-data-for-calibration-and-tools/astronomical-catalogs/ calspec F290LP/G395H, and CLEAR/PRISM configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These residuals are well below the pre-launch require- ment (10% absolute photometric accuracy of all NIR- Spec spectra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' even after correction for ‘delta’ slit losses due to imperfect source centering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' While source centering is most critical for the 200 mas wide slits, it is worth mentioning here that after a successful target ac- quisition procedure (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5), the source is typi- cally placed within 10 mas of the slit center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This magni- tude of source displacement would result in insignificant ‘delta’ slit losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As noted above, the results presented here were derived using the NIRSpec-internal data reduction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The user pipeline provided by STScI is cur- rently being checked and improved to deliver similar re- sults, and the status is discussed further in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Wavelength Calibration The accuracy of the NIRSpec wavelength calibration is determined by the performance of the parametric in- strument model, which was a crucial product to be de- rived from NIRSpec commissioning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As described in L¨utzgendorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022), the residuals in the fi- nal instrument model were well within the requirements for the wavelength and astrometric calibration of NIR- Spec data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 5, which shows wavelength-calibrated spectra of the spectrophotomet- ric standard 1808347, an A3V star with prominent hy- drogen absorption features, taken through the 200 mas wide fixed slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As can be seen, the NIRSpec instrument model al- lows for a highly accurate wavelength calibration, with residuals for internal lamp spectra well below the re- quirement (1/8 of a resolution element) for all gratings analyzed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Note that the full verification for all pipeline products in the various NIRSpec modes is still ongoing, especially in the case of the IFS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Photometric Stability for Time-Series Observations The temporal stability of the instrument response to a constant flux stimulus is a critical parameter for the ac- curacy of measured light curves of astronomical targets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' in the case of Time Series Observations (TSOs) of transiting exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In order to quantify the sta- bility of the NIRSpec response over various timescales, we obtained a TSO of the star HAT-P-14 (PID 1118;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' PI: Proffitt) during the transit of its well-characterized exoplanet HAT-P-14 b using the G395H/F290LP grat- ing/filter combination (see Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The observations, which also served the purpose of verifying the correct ex- ecution of the BOTS observing template (see Birkmann 11 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Wavelength-calibrated spectra of the spectrophotometric standard star 1808347, obtained in FS mode through the S200A1 slit (see Table 1), and using the medium-resolution gratings in three of the four spectral configurations (F100LP/G140M is not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These data were obtained during the NIRSpec commissioning program ‘Spectrophotometric Sensitivity and Absolute Flux Calibration’ (PID1128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022a), were analyzed in detail by Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To recap, the transmission spectrum of HAT-P-14 b could be measured with a precision of 50-60 ppm at R = 100 (rebinned down from R = 2700), which is in excellent agreement with pre-flight expectations, and close to the photon-noise limit for a J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='094, F-type star like HAT-P-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' There were two noteworthy fea- tures observed in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The first was a weak linear trend in the white-light response as a function of time, which was observed to be stronger in NRS1 (-150 ppm/hour) than in NRS2 (30 ppm/hour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This was shown to also be slightly wavelength dependant in NRS1, but not in NRS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These trends can be easily cor- rected for, and are most likely caused by low-amplitude instabilities in the detector signal chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The second is the fact that binning pixels in the spectral direction seems to not decrease the noise level in the time-series as 1/�Nbin where Nbin is the size of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' It is likely this is related to unaccounted covariance between pixels due to effects such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=', 1/f noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This can also be accounted for by either simply working with the F170LP/G235M/S200A1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='25 (Arbitrary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='75 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 Wavelength (micron)F290LP/G395M/S200A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='4 Units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='3 (Arbitrary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 Wavelength (micron)F070LP/G140M/S200A1 6 5 4 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 Wavelength (micron)12 added pixels on each spectral channel or considering a more complex spectral extraction and binning scheme that includes this spatial covariance of pixels in the de- tector (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=', Schlawin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Given the excellent performance of the BOTS mode, and the absence of any strong variations and system- atic trends in the combined response from telescope and instrument over timescales of a few hours to a day, NIR- Spec promises to fulfill its key role for cutting-edge tran- siting exoplanet atmospheric science with JWST during Cycle 1 and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Target Acquisition Depending on their scientific needs, NIRSpec users have a number of options for fine-tuning of the tele- scope pointing after the completion of the initial slew to the target field, which typically places the source within 100 mas of its intended position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Programs for which this level of accuracy is sufficient do not need a dedi- cated TA procedure, and should select TA=NONE or TA=VERIFY ONLY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The difference between these two options is that (for MOS and IFS mode) the latter ob- tains an undispersed image of the sky after the spectro- scopic exposures in order to allow a determination of the pointing post-facto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Programs that require a more accurate target place- ment of an individual source in either the IFU, one of the fixed slit apertures, or the MOS single-point field position should use the NIRSpec Wide Aperture Target Acquisition (WATA) procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For MOS observations, on the other hand, the added complexity of requiring a roll angle optimization calls for the use of a dedicated method called Micro-Shutter Array Target Acquisition (MSATA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The details and in-orbit performance of both of these TA methods are discussed in the following sub- sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Wide Aperture Target Acquisition The WATA procedure takes an image of an isolated, point-like target through the 1.”6 × 1.”6 wide fixed slit aperture (S1600A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Using this image, the onboard soft- ware then computes the centroid of the source emission to determine its position after the initial ‘blind’ tele- scope slew, and autonomously calculates the corrective ‘delta’ slew required to accurately position either this target or another nearby target at the optimal location in the NIRSpec science aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The total duration of the WATA procedure can be as short as 5 min, and as long as 11 min, depending on the size of the detec- tor area being used (subarray or FULL frame), and the depth of the acquisition exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Figure 6a shows the WATA image of a star used to ver- ify the WATA process during commissioning, obtained from Proposal ID 1118 (PID1118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This program con- tained six successful WATA attempts5 that were used to evaluate the onboard algorithms, the correct computa- tion and execution of the offset slew, and the subsequent science exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Figure 6b shows the position of the source after the initial slew, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' at the beginning of each WATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Because of the excellent slew and pointing performance of the observatory, the WATA starting po- sition is already quite close to the aperture center: on average, the target is within 50 mas radial distance, or one half of the size of a NIRSpec detector pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Figure 6c shows the target position at the end of the WATA procedure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' after the corrective slew to the center of the S1600A1 aperture was computed by the onboard algorithm, and executed by the telescope guid- ing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For the six successful WATA observations in PID1118, the average target position was within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 mas from the aperture center, which is almost ten times bet- ter than the pre-launch requirement of 20 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For observations that require the science target to be placed in apertures other than the S1600 slit itself, fur- ther tests executed during commissioning and the early science program have demonstrated that the telescope slew accuracy is sufficient to place the target at, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=', the intended location in the IFU aperture with an accuracy of better than 10 mas, again well within the requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Micro-Shutter Array Target Acquisition The NIRSpec MOS mode requires that the images of astrophysical targets are accurately placed onto the MSA over the entire (3′ × 3′) FOV, such that they fall within their dedicated microshutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The onboard algo- rithms to achieve this rely on highly precise coordinate transformations between the detector, MSA, and sky planes, which are another crucial output of the NIR- Spec parametric instrument model (L¨utzgendorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As explained in detail by Keyes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (2018), the NIR- Spec MSA target acquisition (MSATA) process uses a set of 5 to 8 ‘reference stars’ that are imaged onto the detector through the microshutter grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For MSATA exposures, the MSA can be either in the all-open con- figuration or in ‘protected’ mode, which closes the shut- ters around bright stars to prevent persistence in the subsequent science exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Two MSATA exposures are acquired, separated by an offset equivalent to half of the microshutter pitch (in both x and y direction), in order to mitigate the effect of vignetting by the MSA bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The centroid position of each reference star is calculated autonomously by the on-board software, and 5 The data discussed here are from Observations 1-4, 6, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 13 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (a) The Wide Aperture Target Acquisition (WATA) exposure image (from PID 1118 Obs 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The 32 pixel by 32 pixel aperture is placed around the 1.”6 × 1.”6 (approximately 16 by 16 pixels) square aperture, but not exactly centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (b) The V2, V3 coordinate offset of the initial blind pointing position of the star that is calculated for the first exposure for each WATA Observation test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The average blind pointing for the star is better than 50 mas radial from the wide aperture center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' (c) The post-target acquisition pointing showing the improved centering of the TA target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The average corrected position for the stars is better than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 mas radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' the set of centroids is analyzed, outliers are clipped, and their mean offset from the intended position is used to correct the initial spacecraft pointing (‘pitch’ and ‘yaw’) and position angle (‘roll’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The duration of the MSATA procedure depends on the number of reference stars used and the depth of the acquisition exposures, ranging from 23 min for the min- imum number of reference stars (5) and the shortest readout pattern, to 35 min for 8 reference stars (the rec- ommended number), and the deepest available readout pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In cases where the MSATA algorithm does not converge to a sufficiently accurate solution, it may be repeated once, adding up to 22 min to the duration if the maximum of 8 reference stars is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The NIRSpec MSATA process is the only operational situation where the roll angle of the Webb telescope is adjusted after the start of an observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To provide an impression of the complexities involved, Figure 7 shows the image used to verify the NIRSpec MSATA process during commissioning (PID 1117, Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The tar- get field is the astrometric calibration field (Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2021) in the Large Magellanic Cloud (LMC) which had been pre-observed with HST and ground-based tele- scopes to provide accurate stellar coordinates for the as- trometric calibration of the JWST focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In such highly crowded fields, careful planning is necessary to identify the optimal MSATA reference stars, which must not saturate in the TA exposure, and must be suitably isolated for optimal centroid calculation by the on-board algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To allow accurate calibration of MOS spectra, the MSATA process must place the science targets with an accuracy of just 20 mas across the NIRSpec FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The most important driver for the MSATA accuracy is the absolute anchoring of the planning catalog in rotation, as it is critical for the proper derivation of the TA roll solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' If the desired accuracy of the target place- ment in the shutters can be relaxed, the MSATA can also be planned using catalogs with poorer astrometric precision or rotation anchoring, or using galaxies with compact central cores instead of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To evaluate the performance and post-TA target po- sitioning accuracy, we have analysed the 20 NIRSpec MSATA procedures carried out so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Figure 8 shows their distribution of corrective TA slews (in V2, V3 and roll).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' We find all slew offset solutions to be within 100 mas radial offset and at an average −51′′ roll off- set compared to the optimal pointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These relatively small corrections are a testament to the excellent ’blind’ pointing accuracy of JWST which allows very accurate target placement even without any TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To illustrate the improvement achieved by the MSATA procedure, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 8 shows the target placement after MSATA execution for the same 20 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These measurements were derived by running an offset analysis on the reference image that is acquired after the final TA slew is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As can be seen, the average radial off- set over the ensemble of TA reference targets is 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 mas, and the average roll offset from the optimal pointing so- lution is 14′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This is close to, but not yet fully in line with the requirements, but it should be kept in mind that the more critical displacement in cross-dispersion direction should be a factor of √ 2 smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In addi- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' many of the observations analyzed so far have used ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='WATA Image Observation 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Blind Pointing- WATA Exposure 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Post-TA Slew - WATA Exposure 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='★★★★★ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Obs 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Obs 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Obs 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Obs 8 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='V2 Offset (milli-arcseconds) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='V2 Offset (milli-arcseconds) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='X Pixel Position14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='MSA – ALL OPEN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='TA Reference Stars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Fixed Slits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Flux – electrons / sec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Undispersed NIRSpec exposure of the JWST astrometric field (from PID 1117, Obs 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This crowded, but astrometrically well calibrated stellar field was selected to perform the first in-orbit test of the MSATA procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To ensure that enough stellar centroids were successfully measured, this observation used a special requirement to allow the use of 12 TA reference stars (highlighted by the cyan boxes), instead of the default maximum of 8 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The NIRSpec Fixed Slit apertures are highlighted in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' catalogs with non-optimal astrometric accuracy, and/or non-stellar objects as reference targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' NIRSPEC SCIENCE CALIBRATION PIPELINE NIRSpec data for all modes are processed by the JWST Science Calibration Pipeline6 in three stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 6 see https://jwst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='edu/jwst-science-calibration-pipeline- overview Stage 1 takes raw up-the-ramp integrations, applies var- ious detector-level corrections such as dark subtraction and linearity correction, flags jumps due to cosmic ray hits, and fits a slope to the ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Stage 2 calcu- lates wavelength and spatial coordinates per pixel us- ing the parametric instrument model, corrects for in- strument throughput losses using the three-component flat field reference files, and converts to physical flux units (MJy/pixel for point sources, MJy/steradian for 15 Average radial offset = 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='2 +/- 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='5 mas Q = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0” +/- 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='8” Average radial offset = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 +/- 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='7 mas Q = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0” +/- 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='9” Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Left: computed offset correction (in V2/V3 and roll angle) computed by the onboard algorithm for the ensemble of 20 MSATA visits analyzed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Each star symbol contains the average over the n reference stars used for the computation (color-coded by the value of 5 ≤ n ≤ 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Right: the results after execution of the corrective slew computed via the onboard MSATA algorithm, as measured from the subsequent reference image (which still has the TA filter in place).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' extended sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The final stage combines multiple exposures (such as from a nod or dither pattern) at the 2D level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' High-level pipeline products include 1D and 2D calibrated spectra for FS and MOS modes and 3D data cubes for IFS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The JWST Science Calibration Pipeline is a continual work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The code is updated via four regu- lar build releases per year with bug fixes and algorithm enhancements (although development versions are avail- able on a continuous basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Reference files are updated whenever new calibration data are available and indi- cate evolution in detector or throughput performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' As of the time of writing, there are some outstanding issues that users may want to be aware of: i) the signal produced by snowball events is only par- tially corrected by the cosmic ray jump detection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' An optional algorithm that flags additional pixels af- fected by a snowball has recently become available, and further enhancements are being investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' ii) the ‘fast’ throughput correction vector used during the S-flat step must be normalized by the wavelength interval falling onto a given detector pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The current reference files are normalized using representative val- ues, which do not take into account the variation over the FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Tests are ongoing, but preliminary investi- gations suggest that this simplification adds an addi- tional systematic uncertainty to the flux calibration that can be as high as ten percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' A future version of the pipeline will include an on-the-fly pixel-by-pixel normal- ization that should remove this source of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' iii) in earlier versions of the pipeline, the resam- pling algorithms used in creating IFS cubes have some- times produced wavelength-dependent artifacts in cer- tain cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' A recent bug fix seems to fix the issue at least for point sources, but users should carefully inspect the extracted spectra to look for any unexpected behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Also note that because of the optical field distortion in the NIRSpec camera optics (see Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 2022, for a more detailed explanation), all NIRSpec spectra are curved relative to the detector pixel grid, which in the case of compact or point sources leads to aliasing effects, esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' in the case of the gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Therefore, the spectrum of a single cube spaxel will show a sinusoidal variation in the extracted 1D spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' When summing up over a sufficiently large circular aperture, however, this ef- fect should average out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For the same reason, the alias- ing should be much reduced for extended sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Ulti- mately, though, it is important to use dithering when- ever possible, as the combination of dithered exposures will minimize such resampling artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' iv) the 1D extraction apertures are automatically cen- tered at the expected position of the source, based on sky coordinates specified in the JWST Astronomer’s Pro- posal Tools (APT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For reasons that are still under in- vestigation, this centering is often offset from the true position of the spectral trace, requiring manual specifi- cation of the center location by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' v) there are no aperture corrections yet available for 1D extractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' The default extraction apertures are set to be consistent with those used for the generation of the F-flat reference files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' For different aperture widths, users must currently compute their own aperture corrections using available observations of point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' vi) the outlier detection step in stage 3 of the pipeline is intended to catch any outliers missed by stage 1 by means of comparing the multiple input exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' How- ever, testing to date has shown performance problems when using the default thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Users will need to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='★★★★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Nstars = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Nstars = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Nstars=7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Nstars = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='V3 Slew (milli-arcseconds) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='V2Slew(milli-arcseconds)200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Roll Slew (arcseconds) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Post TA Roll (arcseconds) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='★★★★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Nstars = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Nstars = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Nstars=7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Nstars = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Post TA V3 Residual (milli-arcseconds) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='Post TA V2 Residual (milli-arcseconds)16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='adjust the parameters to get a reasonable identification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='of true outliers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' although in many cases it may be prefer- able to turn the step off entirely until the algorithm is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS We have provided an overview of the NIRSpec perfor- mance as derived during the JWST commissioning cam- paign and the first few months of in-orbit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' From a hardware perspective, the NIRSpec instrument performs nominally in all electrical, mechanical, and thermal aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' When combined with the outstanding optical performance of the JWST OTE, and the better than expected accuracy and stability of Webb’s Attitude Control System, it is no surprise that the scientific per- formance of NIRSpec meets or exceeds the pre-launch expectations for all observing modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' More specifically, we have shown that the complex on- board target acquisition procedures work well, the mea- sured sensitivities are excellent across the board, and unrivalled by any other NIR spectrograph, and the pho- tometric stability is very good which is important for time-series observations of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' In terms of the calibration accuracy of NIRSpec sci- ence data, we have shown that the data acquired and the methods used so far allow the creation of reference files that are sufficient to meet the requirements for the pho- tometric and wavelength calibration of NIRSpec spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' On the other hand, we have mentioned a number of open issues with the NIRSpec Science Calibration Pipeline, in particular for the flux calibration of NIR- Spec spectra, the cube-building step for IFS data, and 1D spectral extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' These issues are actively being worked, and improvements to the quality of the higher- level NIRSpec data in the Mikulski Archive for Space Telescopes (MAST) can be expected soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' There are a few important ‘lessons learned’ from these results that NIRSpec users should take note of: the higher than expected PCE measured over most of the NIRSpec wavelength range may cause some ex- posures to saturate earlier than predicted from pre- launch estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' This may make some ‘bright tar- get’ science less feasible and/or require modifications to the observing strategies, especially the selection of MSATA reference stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' for MOS mode and the associated MSATA planning, it is essential that the catalog of MSATA reference stars is properly registered to the Gaia frame, not just in RA and Dec, but also in rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' the initial telescope pointing after guide star acqui- sition is accurate enough that for most IFS observa- tions, foregoing TA (by using TA=NONE or VER- IFY ONLY) is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' Because of the better than ex- pected PSF quality and the resulting small slit losses in the S1600A1 aperture, this may even be true for some time-series observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' To summarize, the NIRSpec instrument onboard JWST presents a quantum leap for NIR spectroscopy in terms of sensitivity and multiplexing capability, and promises to have a lasting impact on many research fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' We are deeply grateful to the large number of engineers and scientists in Europe, Canada, and the US, whose dedication and hard work over many years have turned NIRSpec and the entire JWST mission from a mere vi- sion into reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=' REFERENCES Alves de Oliveira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=', L¨utzgendorf, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=', Zeidler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content=', et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} +page_content='620772' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFST4oBgHgl3EQfSjhy/content/2301.13766v1.pdf'} diff --git a/y9E3T4oBgHgl3EQfPglf/content/tmp_files/2301.04403v1.pdf.txt b/y9E3T4oBgHgl3EQfPglf/content/tmp_files/2301.04403v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6b547a717dfeef2387a20b6ffd048be2ba7ecd4 --- /dev/null +++ b/y9E3T4oBgHgl3EQfPglf/content/tmp_files/2301.04403v1.pdf.txt @@ -0,0 +1,2634 @@ +arXiv:2301.04403v1 [math.NA] 11 Jan 2023 +A semi-discrete first-order low regularity exponential +integrator for the “good” Boussinesq equation +without loss of regularity +Hang Li · Chunmei Su* +Abstract In this paper, we propose a semi-discrete first-order low regularity +exponential-type integrator (LREI) for the “good” Boussinesq equation. It is +shown that the method is convergent linearly in the space Hr for solutions +belonging to Hr+p(r) where 0 ≤ p(r) ≤ 1 is non-increasing with respect to r, +which means less additional derivatives might be needed when the numerical +solution is measured in a more regular space. Particularly, the LREI presents +the first-order accuracy in Hr with no assumptions of additional derivatives +when r > 5/2. This is the first time to propose a low regularity method which +achieves the optimal first-order accuracy without loss of regularity for the GB +equation. The convergence is confirmed by extensive numerical experiments. +Keywords “good” Boussinesq equation · low regularity · error estimate · +first-order integrator · without loss of regularity +Mathematics Subject Classification (2020) 35Q35 · 65M12 · 65M15 · +65M70 +1 Introduction +We consider the following periodic boundary value problem of the “good” +Boussinesq (GB) equation: +� +ztt + zxxxx − zxx − (z2)xx = 0, +x ∈ T, +t > 0, +z(0, x) = φ0(x), +zt(0, x) = ψ0(x), +(1.1) +C. Su (*Corresponding author) +Yau Mathematical Sciences Center, Tsinghua University, Beijing, China +Tel.: +123-45-678910 +Fax: +123-45-678910 +E-mail: sucm@tsinghua.edu.cn +H. Li +Yau Mathematical Sciences Center, Tsinghua University, Beijing, China + +2 +Hang Li, Chunmei Su* +in a torus T = [−π, π], where φ0(x) and ψ0(x) are given initial data. The GB +equation was originally founded by Joseph Boussinesq [6] to describe the prop- +agation of dispersive shallow water waves. Furthermore, it was also extended +by replacing the quadratic nonlinearity with a general function of z to model +small oscillations of nonlinear beams [40] or the two-way propagation of water +waves in a channel. There have been many applications for the GB equation +in physics [14,16] and oceanographic engineering [39]. +Analytically, similar to the well-know Korteweg-de Vries (KdV) equation, +the nonlinear Schr¨odinger (NLS) equation, and other dispersive equations, +the GB equation admits abundant soliton solutions, see [9,12,19,24,25]. How- +ever, the GB equation has some special characteristics that make it different +from the KdV or NLS equations, e.g., two solitons can merge into one soli- +ton or develop into the so-called antisolitons [25]. For less smooth solutions, +Kishimoto [17] gave a sharp locally well-posed result by using the fix-point +theory together with low regularity bilinear estimates in Bourgain spaces, +also known as the dispersive Sobolev spaces [38]. The main result in [17] is +that for any (φ0, ψ0) ∈ Hs × Hs−2, s ≥ −1/2, there exist a positive time +T (∥ψ0∥Hs, ∥ψ0∥Hs−2) > 0 and a unique solution of the GB equation in a +certain Banach space of functions X ⊂ C([0, T ]; Hs × Hs−2), however, this +equation is ill-posed when s < −1/2. We refer to [10,11,17,27,42] for more +detailed theoretical results of the GB equation. +Along the numerical part, a large variety of classical numerical schemes +for approximating the time dynamics of the GB equation have been proposed +and analyzed, including the pseudospectral methods [8,12], finite difference +methods [7,28], the exponential integrators [37] and splitting methods [46]. +However, as a result of the fourth-order spatial derivative in (1.1), these tradi- +tional schemes can not reach their ideal convergence rates when the solution is +not smooth enough. For example, an explicit finite difference scheme was con- +structed in [28], which strictly requires the boundedness of ∂6 +xz and ∂4 +t z and +a time step restriction of ∆t = O(∆x2), where ∆t and ∆x represent the time +and space step, respectively. Unfortunately, the solutions involved in practical +applications become rough due to the interference of noise. Thus it is nec- +essary to find appropriate methods which can achieve the ideal convergence +even for rough solutions. To this aim, some low regularity exponential inte- +grators (LREIs) requiring low additional regularity have been established by +introducing the concepts of twisted variable w(t) := eit∂2 +xu(t) and Duhamel’s +formula, see [22,31]. Compared to the classical methods, e.g., classical expo- +nential integrators, these strategies give rise to some numerical schemes that +still converge even when the solution is rough. Specifically, Ostermann & Su +[31] gave a first-order and a second-order LREIs and obtain the linear and +quadratic convergence in Hr (r > 1/2) by requiring one and three additional +derivatives, respectively. This demand is weaker than that of the operator split- +ting method [46] and the spectral method [8], the latter of which requires the +boundedness of at least four additional temporal and spatial derivatives to at- +tain the second-order convergence in time. Recently, the authors [22] proposed +a new first-order and second-order LREIs, which converge with less additional + +FIRST-ORDER LREI FOR GB +3 +derivatives required than those in [31]. In particular, the second-order LREI +in [22] converges quadratically with two additional derivatives required, which +is weaker than that of [31]. +In this article, we will introduce a newly developed LREI which has first- +order accuracy in Hr by requiring the boundedness of additional spatial deriva- +tives at the order of p(r), where p(r) is non-increasing with respect to r, i.e., +∥un − u(tn)∥r ≲ τ, +for +u ∈ L∞(0, T ; Hr+p(r)). +Particularly, p(r) = 0 for all r > 5/2, which means the method is convergent +at the first order in Hr with no additional regularity needed. The first-order +LREI is established by the following strategy: +(i) In the first step, we rewrite the GB equation as a first-order system +� z +zt +� +t += +� +0 +1 +−∂4 +x + ∂2 +x +0 +� � z +zt +� ++ +� +0 +(z2)xx +� +. +Then we diagonalize the above matrix in Fourier space and introduce a +new complex variable u(t) involving z and zt so that the GB equation +equivalents to a Schr¨odinger-type equation. +(ii) In the second step, we extract the dominant term in the linear part of +the equation involving u(t) and introduce the so-called twisted variable +w(t) = eit∂2 +xu(t). +An appropriate approximation is used to integrate the Duhamel’s for- +mula on the new variable w. +(iii) Finally, we twist the variable back and obtain an approximation to u. +The integral for the nonlinear term is approximated so that the iteration +can be efficiently calculated in physical space or Fourier space. +Remark 1 The method of twisted variable is firstly introduced by Ostermann +and Schratz [30] to design low-regularity numerical schemes for the nonlinear +Schr¨odinger equation. Since then this technique has been extensively applied +for the nonlinear Schr¨odinger equation [18,20,29,43,33], KdV equation [13,32, +44,45], Klein-Gordon equation [3,41] and other equations [34,35]. Compared to +classical numerical methods, this type of low regularity integrators can achieve +the same convergence when the solutions are less regular. +Below we present our idea to design the new LERI briefly. The approach is +based on the phase space analysis of the nonlinear dynamics. Specifically, we +are devoted to finding a suitable approximation for the following time integral +� τ +0 +e−is(k2+k2 +1+k2 +2)ds, +with +k1 + k2 = k. +The leading term −2k2 is kept and integrated exactly in [31], i.e., +� τ +0 +e−is(k2+k2 +1+k2 +2)ds = +� τ +0 +e−2isk2+2isk1k2ds ≈ +� τ +0 +e−2isk2ds. + +4 +Hang Li, Chunmei Su* +This finally leads to a first-order scheme with one additional order of regularity +required [31]. To weaken the constraint on regularity, the authors applied the +identity 1 = k1+k2 +k +together with the property +k2 + k2 +1 + k2 +2 = 2k2 +2 + 2kk1 = 2k2 +1 + 2kk2 = 2k2 − 2k1k2, +(1.2) +and decompose the integral as +� τ +0 +e−is(k2+k2 +1+k2 +2)ds = k1 +k +� τ +0 +e−is(k2+k2 +1+k2 +2)ds + k2 +k +� τ +0 +e−is(k2+k2 +1+k2 +2)ds += k1 +k +� τ +0 +e−2is(k2 +2+kk1)ds + k2 +k +� τ +0 +e−2is(k2 +1+kk2)ds +≈ k1 +k +� τ +0 +� +e−2isk2 +2 + e−2iskk1 − 1 +� +ds + k2 +k +� τ +0 +� +e−2isk2 +1 + e−2iskk2 − 1 +� +ds, +where the the integrals in the last line can be integrated exactly in phase space. +In this work, we apply the identity +k2 +1 + k2 +2 + 2k1k2 +k2 += 1 +(1.3) +instead and the decomposition follows as +e−is(k2+k2 +1+k2 +2) = k2 +1 +k2 e−2is(k2 +2+kk1) + k2 +2 +k2 e−2is(k2 +1+kk2) + 2k1k2 +k2 +e−2is(k2−k1k2) +≈ k2 +1 +k2 (e−2isk2 +2 + e−2iskk1 − 1) + k2 +2 +k2 (e−2isk2 +1 + e−2iskk2 − 1) ++ 2k1k2 +k2 +(e−2isk2 + e2isk1k2 − 1), +(1.4) +where all three terms in the approximation can be exactly integrated. In this +way we are able to establish the numerical flow as follows +zn = 1 +2(un + un) + atn + b, +zn +t = i +2⟨∂2 +x⟩(un − un) + a, +(1.5) +where +a = F0(zt(0, ·)) = 1 +2π +� +T +ψ0(x)dx, +b = F0(z(0, ·)) = 1 +2π +� +T +φ0(x)dx, +(1.6) +and +un+1 = Ψ τ +1 (un), +n ≥ 0, +u0 = u(0, x) = φ(x)−b−i⟨∂2 +x⟩−1(ψ(x)−a), (1.7) +with ⟨∂2 +x⟩−1 defined in Section 2 and +Ψ τ +1 (f) = eiτ⟨∂2 +x⟩f − i +4Bτ +� +− i∂−2 +x +�� +e2iτ∂2 +x∂−2 +x f +� � +∂2 +xf +�� ++ i∂−2 +x +�� +∂−2 +x f +�2� +− ieiτ∂2 +x∂−3 +x +�� +eiτ∂2 +x∂xf +� � +e−iτ∂2 +x∂xf +�� ++ i∂−3 +x +�� +∂xf +� +f +� +− 2τ∂−2 +x +�� +∂2 +xf +� +f +� + +FIRST-ORDER LREI FOR GB +5 +− i∂−4 +x +� +e2iτ∂2 +x − 1 +� � +∂xf +�2 + i∂−2 +x e−iτ∂2 +x +� +eiτ∂2 +xf +�2 +− i∂−2 +x +� +f +�2 +− 2τ∂−2 +x +� +∂xf +�2 + i +2 +� +(∂−1 +x f)2 − eiτ∂2 +x(e−iτ∂2 +x∂−1 +x f)2� +− ieiτ∂2 +x∂−1 +x +� +(e−iτ∂2 +xf)(eiτ∂2 +x∂−1 +x f) +� ++ i∂−1 +x +� +f(∂−1 +x f) +�� +− iτ(atn + b)Bτ� +f + ψ1(2iτ∂2 +x)f +� +, +(1.8) +with ψ1 and Bτ given by (2.5) and (3.10) with (2.7), respectively. It can be +easily seen that the scheme is explicit and easy to implement if one applies +Fourier spectral method for spatial discretization. +Now we state the main theorem concerning the convergence of the above +scheme (1.8). Before that, we define a function p(r): +p(r) = + + + + + + + + + + + + + + + + + + + + + + + + + +1, +r = 1; +(3 − 2r)+, +1 < r ≤ 7/6; +2/3, +7/6 < r ≤ 17/12; +(7/2 − 2r)+, +17/12 < r ≤ 3/2; +5/4 − r/2, +3/2 < r < 5/2; +0+, +r = 5/2; +0, +r ≥ 5/2, +where c+ means c + ε for any sufficiently small ε > 0. +Theorem 1 For r ≥ 1, suppose that the exact solution of (1.1) satisfies z +∈ C(0, T ; Hr+p(r)) and zt ∈ C(0, T ; Hr+p(r)−2). Then there exists a constant +τ0 > 0 such that for all step size 0 < τ ≤ τ0 and tn ≤ T , we have +∥z(tn) − zn∥r + ∥zt(tn) − zn +t ∥r−2 ≤ Cτ , +where C > 0 depends on T , ∥z∥L∞(0,T ;Hr+p(r)) and ∥zt∥L∞(0,T ;Hr+p(r)−2). +It is clear p(r) represents the order of additional regularity required to +promise the first-order convergence of the numerical solution in Hr. Fig. 1 +displays the plot of p(r), from which we observe that p(r) is non-increasing. +Particularly, p(r) ≡ 0 when r > 5/2, which means the scheme is convergent +linearly in Hr without loss of regularity when r > 5/2. +Compared to the convergence results of the scheme in [31], which converges +in Hr (r > 1/2) at the first order when the solution belongs to Hr+1, and the +method in [22], which achieves the first-order convergence in Hr for r > 7/6 as +the solution lies in Hr+2/3, it is obvious that our newly proposed scheme (1.8) +requires less regularity to attain the ideal first-order convergence. Furthermore, +for the convergence without smoothness assumptions, i.e., p(r) = 0, compared +to the first-order LREIs proposed in [22] and [31] which converge at the order of +1/2 and +r−1/2 +3r+1/2−, respectively, our newly developed first-order LREI presents +a linear convergence without additional regularity assumptions when r > 5/2. + +6 +Hang Li, Chunmei Su* +Additional Derivative +To Achieve Linear Convergence +1 +7 +6 +17 +12 +3 +2 +5 +2 +1 +2 +3 +1 +2 +1 +4 +0 +(1, 1) +( 7 +6, 2 +3) ( 17 +12, 2 +3) +( 3 +2, 1 +2) +( 5 +2, 0) +r +Fig. 1 Additional order of regularity required to achieve the first-order convergence. Par- +ticularly, for the domain (1, 7/6] and (17/12, 3/2], we plot it by a dash-dotted line or hollow +points to mean that a plus sign exists in p(r), e.g., the scheme is convergent linearly in H3/2 +for solutions in H(3/2+1/2)+. +This is the first time to establish the optimal linear convergence without loss +of regularity for the GB equation. On the other hand, we have to admit that +the deficiency is that we have to impose r ≥ 1 due to the stability analysis (cf. +Section 4). Thus the analysis in Hr for r ≤ 1 is absent at the moment. +The rest of the paper is organized as follows. In Section 2, we present some +notions and powerful technical tools. The first-order LREI is constructed in +Section 3. Section 4 is devoted to establishing the error estimate of the scheme. +Some numerical results are presented to confirm the theoretical analysis in +Section 5 and conclusions are drawn in Section 6. +2 Preliminary +In this section, we introduce some notations and present some useful technical +lemmas which are of vital importance to design the method or to establish the +error estimates. +2.1 Notations +In this paper, we use the notation X ≲ Y to denote that there exists a constant +C > 0 which may be different from line to line but is independent of the time +step τ such that |X| ≤ CY . The Fourier transform of a function f on a +torus T is defined by the coordinate representation { �fk}+∞ +k=−∞ under the basis + +FIRST-ORDER LREI FOR GB +7 +{eikx}+∞ +k=−∞ in L2(T), where +Fk(f) = �fk = 1 +2π +� +T +f(x)e−ikxdx. +Thus f(x) = � +k∈Z +�fkeikx is the inverse Fourier transform. The norm and inner +product in L2 are defined respectively by +∥f∥ := ∥f∥L2 = +� � +k∈Z +| �fk|2�1/2, +(f, g) = +� +k∈Z +�fk�gk = 1 +2π +� +T +f(x)g(x)dx. +(2.1) +Moreover, we define several operators given in Fourier space as +∂−1 +x f = +� +k̸=0 +1 +ik +�fkeikx, +�f = +� +k∈Z +�� �fk +��eikx, +|∂x|αf = +� +k̸=0 +|k|α �fkeikx, +Jαf = +� +k∈Z +(1 + k2)α/2 �fkeikx, +α ∈ R. +(2.2) +Similarly, we define ⟨∂2 +x⟩ := +� +−∂2x + ∂4x and its inverse by +⟨∂2 +x⟩f = +� +k∈Z +� +k2 + k4 �fkeikx, +⟨∂2 +x⟩−1f = +� +k̸=0 +1 +√ +k2 + k4 �fkeikx. +Furthermore, we introduce the Sobolev space Hα with α ∈ R, which con- +sists of the functions f = � +k∈Z +�fkeikx such that ∥f∥α = ∥Jαf∥ < ∞, where +∥f∥2 +α = ∥Jαf∥2 = +� +k∈Z +(1 + k2)α| �fk|2. +It is clear that for f with zero mean value, i.e., �fk = 0, it holds ∥f∥α ≲ +∥|∂x|αf∥ ≲ ∥f∥α. For α = 0, it is clear that the space reduces to L2 and the +corresponding norm is simply denoted as ∥ · ∥ which agrees with (2.1). +We say that R = R(u, t, τ, ξ) ∈ Rθ(τ ν) if and only if +∥R(u, t, τ, ξ)∥r ≤ Cτ ν, +where R(u, t, τ, ξ) depends on the value u(t + ξ), 0 ≤ ξ ≤ τ, and C relies +on +sup +0≤s≤τ +∥u(t + s)∥r+θ. We write f = g + Rθ(τ ν) whenever f = g + R with +R ∈ Rθ(τ ν). + +8 +Hang Li, Chunmei Su* +2.2 Preliminary tools +To begin with, we introduce the Kato-Ponce inequalities, which was previously +proved in [15,5,23] in the whole space R and extended to the periodic case by +Li and Wu [21] recently. +Lemma 1 (The Kato-Ponce inequalities) (i) If r > 1/2 and f, g ∈ Hr, then +we have +∥fg∥r ≲ ∥f∥r∥g∥r. +(2.3) +(ii) If s > 0, 1 < p ≤ ∞, 1 < p1, p3 < ∞, 1 < p2, p4 ≤ ∞ satisfying 1 +p = +1 +p1 + 1 +p2 +and 1 +p = +1 +p3 + 1 +p4 , then we have the following inequality +∥Js(fg)∥Lp ≲ ∥Jsf∥Lp1 ∥g∥Lp2 + ∥Jsg∥Lp3 ∥f∥Lp4 . +(2.4) +Next we present Hardy-Littlewood-Sobolev type inequality and Sobolev +embedding theorem on the torus T, which provides a new approach for the +subsequent estimate of local truncation errors. We refer to [1,2,4,26,36] and +references therein. +Lemma 2 (i) (Hardy-Littlewood-Sobolev type inequality) Let s ∈ [0, 1/2). +Then there exists a constant C = C(s) > 0 such that +∥f∥−s ≤ C∥f∥ +L +2 +1+2s (T), +for any f ∈ L +2 +1+2s (T). +(ii) (Sobolev embedding theorem) Let s ∈ (0, 1/2). The inclusion +Hs(T) ⊆ Lq(T) +is continuous for any q ∈ +� +1, +2 +1−2s +� +. +Lemma 3 (i) For all x, y ∈ R and 0 ≤ θ ≤ 1, we have +|eix − 1| ≤ 21−θ|x|θ, +|eix − 1 − ix| ≤ 21−2θ|x|1+θ. +(ii) For t ∈ R, r ≥ 0 and f ∈ Hr, we have +∥ψ1(it∂2 +x)f∥r ≤ ∥f∥r, +where +ψ1(y) = +� 1 +0 +eysds, +for +y ∈ C. +(2.5) +For the details of the proof, we refer to [31]. Moreover, we illustrate a lemma +which was introduced by [22,20]. + +FIRST-ORDER LREI FOR GB +9 +Lemma 4 (i) For f ∈ Hr with r ≥ 0, t ∈ R, it holds +∥⟨∂2 +x⟩−1f∥r ≤ ∥f∥r, ∥Af∥r ≤ ∥f∥r, ∥Bf∥r ≤ ∥f∥r, ∥(eitA − 1)f∥r ≤ |t|∥f∥r, +(2.6) +where A and B are given by +A := ⟨∂2 +x⟩ + ∂2 +x, +B := ⟨∂2 +x⟩−1∂2 +x. +(2.7) +(ii) For r ≥ 0 and 0 ≤ γ ≤ 1, f ∈ Hr+2γ, one has +∥(eit∂2 +x − 1)f∥r ≲ |t|γ∥f∥r+2γ, +∥(eit⟨∂2 +x⟩ − 1)f∥r ≲ |t|γ∥f∥r+2γ. +(2.8) +(iii) If f, g ∈ H1, then it holds +��J−1 (g(Jf)) +�� ≲ min{∥f∥∥g∥1, ∥f∥1∥g∥}. +(2.9) +(iv) If f, g ∈ Hr, r > 1/2 then we have +��J−1 (g(Jf)) +�� +r ≲ ∥f∥r∥g∥r. +(2.10) +Lemma 5 For f, g ∈ Hr with r ≥ 1, it holds +�� |∂x|−2 � +(|∂x| g)(|∂x| f) +��� +r ≲ ∥f∥r∥g∥r. +(2.11) +Proof To show the above inequality for r > 1, we only need to confirm +�� |∂x|r−2 � +(|∂x| g)(|∂x| f) +��� ≲ ∥f∥r∥g∥r. +According to the duality principle in L2, it suffices to prove +� +|∂x|r−2 � +(|∂x| g)(|∂x| f) +� +, φ +� +≲ ∥f∥r∥g∥r∥φ∥, +∀φ ∈ L2, +which is equivalent to +� +k̸=0 +� +k1+k2=k +|k|r−2|k1||k2| �fk1�gk2 �φk ≲ ∥f∥r∥g∥r∥φ∥. +To this aim, we divide the above formula into two parts by discussing the +relationship between Fourier coefficients k and k1, i.e., +� +k̸=0 +� +k1+k2=k +|k|r−2|k1||k2| �fk1�gk2 �φk = +� +k̸=0 +� +k1+k2=k +|k1|≤2|k| +|k|r−2|k1||k2| �fk1�gk2 �φk ++ +� +k̸=0 +� +k1+k2=k +|k1|>2|k| +|k|r−2|k1||k2| �fk1�gk2 �φk. +(2.12) +For the first term in (2.12), we have +|k2| = |k − k1| ≤ |k| + |k1| ≤ 3|k|. + +10 +Hang Li, Chunmei Su* +By using Plancherel’s identity and the bilinear estimate, the first term can be +bounded as +� +k̸=0 +� +k1+k2=k +|k1|≤2|k| +|k|r−2|k1||k2| �fk1�gk2 �φk ≲ +� +k̸=0 +� +k1+k2=k +|k1|≤2|k| +|k|r�� �fk1 +�� |�gk2| +���φk +�� +≲ (|∂x|r( �f�g), �φ) ≲ ∥ �f�g∥r∥�φ∥ ≲ ∥ �f∥r∥�g∥r∥�φ∥ ≲ ∥f∥r∥g∥r∥φ∥, +where �f, �g and �φ are defined in (2.2). +For the second term in (2.12), thanks to |k1| > 2|k|, we are led to +|k2| = |k1 − k| ≥ |k1| − |k| > |k|. +For r > 1, it holds +|k|r−2|k1||k2| = |k|−r|k|2r−2|k1||k2| ≲ |k|−r|k1|r|k2|r, +which implies +� +k̸=0 +� +k1+k2=k +|k1|>2|k| +|k|r−2|k1||k2| �fk1�gk2 �φk ≲ +� +k̸=0 +� +k1+k2=k +|k1|>2|k| +|k|−r|k1|r|k2|r�� �fk1 +�� |�gk2| +���φk +�� +≲ +� +k̸=0 +Fk(|∂x|r �f|∂x|r�g)|k|−r|�φk| +≲ max +k +���� +� +T +|∂x|r �f(x)|∂x|r�g(x)e−ikxdx +���� +� +k̸=0 +|k|−r|�φk| +≲ ∥|∂x|r �f|∂x|r�g∥L1∥(|k|−r)0̸=k∈Z∥l2∥(|�φk|)0̸=k∈Z∥l2 +≲ ∥|∂x|r �f∥∥|∂x|r�g∥∥�φ∥ +≲ ∥f∥r∥g∥r∥φ∥. +The proof is completed for the case of r > 1. For the case of r = 1, by using +the result in Lemma 4 (iii) +∥|∂x|−1(g|∂x|f)∥ ≲ ∥J−1(g|∂x|f)∥ ≲ ∥J−1(�g(J �f))∥ ≲ ∥f∥1∥g∥, +we are led to +∥ |∂x|−2 � +(|∂x| g)(|∂x| f) +� +∥1 ≲ ∥ |∂x|−1 � +(|∂x| g)(|∂x| f) +� +∥ +≲ ∥f∥1∥|∂x|g∥ ≲ ∥f∥1∥g∥1. +This completes the proof. +3 First-order exponential-type integrator Ψ τ +1 +In the following part, we construct the first-order LREI based on the idea in +(1.3)–(1.4). + +FIRST-ORDER LREI FOR GB +11 +3.1 Homogenization and reformulation of the GB equation +As can be seen blow, we will frequently encounter the operator ∂−1 +x +or ∂−2 +x +during the process of integration, which makes the mean value of the obtained +function zero. Hence usually the zero-mode needs to be treated separately. +Fortunately, thanks to the periodic boundary conditions, the zero-mode of +z can be integrated exactly so that it remains to investigate other nonzero +Fourier modes. +By the periodicity of the solution, one easily gets +F0(ztt) = ∂ttF0(z) = 0, +which immediately gives F0(z) = at + b, where a and b are defined as (1.6). +Setting z = F0(z) + ˇz and plugging it into (1.1), we derive that +� +ˇztt + ˇzxxxx − (2at + 2b + 1)ˇzxx − (ˇz2)xx = 0, +x ∈ T, +t > 0, +ˇz(0, x) = φ(x) − b, +ˇzt(0, x) = ψ(x) − a. +(3.1) +Diagonalize the equivalent first-order system +� ˇz +ˇzt +� +t += +� +0 +1 +−∂4 +x + ∂2 +x +0 +� � ˇz +zt +� ++ +� +0 +(ˇz2)xx + (2at + 2b)ˇzxx +� +, +and set +⟨∂2 +x⟩ = +� +∂2x + ∂4x, +u = ˇz − i⟨∂2 +x⟩−1ˇzt, +v = ˇz − i⟨∂2 +x⟩−1ˇzt, +we are led to the following coupled system + + + + + + + +i∂tu = −⟨∂2 +x⟩u + B +�1 +4(u + ¯v)2 + (at + b)(u + ¯v) +� +, +i∂tv = −⟨∂2 +x⟩v + B +�1 +4(¯u + v)2 + (at + b)(¯u + v) +� +, +(3.2) +where the operator B is defined in (2.7). Recalling that z is a real function, +this implies u = v and (3.2) reduces to a single first-order equation involving +a complex variable + + + + + +i∂tu = −⟨∂2 +x⟩u + B +�1 +4(u + u)2 + (at + b)(u + u) +� +, +u(0, x) = ˇz(0, x) − i⟨∂2 +x⟩−1ˇzt(0, x). +(3.3) +While ˇz and ˇzt can be recovered through +ˇz = 1 +2(u + u), +ˇzt = i +2⟨∂2 +x⟩(u − u). +(3.4) +Noticing that the leading term of ⟨∂2 +x⟩ is −∂2 +x, we introduce the so-called +twisted variable +w(t) = eit∂2 +xu(t). + +12 +Hang Li, Chunmei Su* +Plugging it into (3.3) yields +∂tw = iAw − i +4eit∂2 +xB(e−it∂2 +xw + eit∂2 +xw)2 − i(at + b)eit∂2 +xB(e−it∂2 +xw + eit∂2 +xw). +(3.5) +Applying Duhamel’s formula of (3.5), we obtain +w(tn + σ) = eiσAw(tn) − iB +4 +� σ +0 +ei(σ−s)Aei(tn+s)∂2 +x(g1(w(tn), s))2ds +− iB +� σ +0 +ei(σ−s)Aei(tn+s)∂2 +x[a(tn + s) + b]g1(w(tn), s)ds, +(3.6) +where A and B are defined in (2.7), and +g1(w(tn), s) = e−i(tn+s)∂2 +xw(tn + s) + ei(tn+s)∂2 +xw(tn + s). +Based on this, a first-order approximation can be easily derived [31] +∥w(tn + σ) − w(tn)∥r ≤ Cσ, +r > 1/2, +(3.7) +where C only depends on sup +0≤s≤σ +∥u(tn + s)∥r. Setting σ = τ and approximating +w(tn + s) by w(tn) in the integral of (3.6), applying (3.7) and Lemma 4 (i), +we get +w(tn + τ) = eiτAw(tn) − i +4BeiτA +� τ +0 +ei(tn+s)∂2 +x(g2(w(tn), s))2ds +− iBeiτA +� τ +0 +ei(tn+s)∂2 +x(atn + b)g2(w(tn), s)ds + R0(τ 2), +(3.8) +where g2(w(tn), s) = e−i(tn+s)∂2 +xw(tn) + ei(tn+s)∂2 +xw(tn). +Twisting the variable back, we obtain an approximation of u(tn + τ) with +a local error of order two +u(tn + τ) = eiτ⟨∂2 +x⟩u(tn) − i +4Beiτ⟨∂2 +x⟩ +� τ +0 +eis∂2 +x(e−is∂2 +xu(tn) + eis∂2 +xu(tn))2ds +− i(atn + b)Beiτ⟨∂2 +x⟩ +� τ +0 +eis∂2 +x(e−is∂2 +xu(tn) + eis∂2 +xu(tn))ds + R0(τ 2) += eiτ⟨∂2 +x⟩u(tn) − i +4Bτ� +Iτ +0 (u(tn)) + Iτ +1 (u(tn)) + 2Iτ +2 (u(tn)) +� +− iτ(atn + b)Bτ� +u(tn) + ψ1(2iτ∂2 +x)u(tn) +� ++ R0(τ 2), +(3.9) +where ψ1 is given by (2.5) and +Bτ(f) = Beiτ⟨∂2 +x⟩f = ⟨∂2 +x⟩−1∂2 +xeiτ⟨∂2 +x⟩f, +Iτ +0 (f) = +� τ +0 +eis∂2 +x� +eis∂2 +xf +�2ds, +(3.10) +Iτ +1 (f) = +� τ +0 +eis∂2 +x� +e−is∂2 +xf +�2ds, +Iτ +2 (f) = +� τ +0 +eis∂2 +x��e−is∂2 +xf +��2ds. +(3.11) + +FIRST-ORDER LREI FOR GB +13 +Now we calculate the terms Iτ +j in (3.9) respectively. Firstly for f satisfying +F0(f) = 0, as was shown in [31], Iτ +1 (f) and Iτ +2 (f) can be calculated exactly as +Iτ +1 (f) = +� +k +� +k1+k2=k +� τ +0 +e−is(k2−k2 +1−k2 +2)ds �fk1 �fk2eikx += +� � +k +� +k1+k2=k +k1̸=0,k2̸=0 +e−2iτk1k2 − 1 +−2ik1k2 ++ +� +k +� +k1+k2=k +k1=0 or k2=0 +� τ +0 +ds +� +�fk1 �fk2eikx += +� +k +� +k1+k2=k +k1̸=0,k2̸=0 +e−2iτk1k2 − 1 +−2ik1k2 +�fk1 �fk2eikx + 2τ �f0 +� +k∈Z +�fkeikx − τ �f 2 +0 += i +2 +� +(∂−1 +x f)2 − eiτ∂2 +x(e−iτ∂2 +x∂−1 +x f)2 +� +, +(3.12) +Iτ +2 (f) = +� +k1,k2∈Z +� τ +0 +eis(k2 +1−k2 +2−(k1−k2)2)ds �fk1 �fk2ei(k1−k2)x += − i +2eiτ∂2 +x∂−1 +x +� +(e−iτ∂2 +xf)(eiτ∂2 +x∂−1 +x f) +� ++ i +2∂−1 +x +� +f(∂−1 +x f) +� ++ τ∥f∥2. (3.13) +It remains to calculate Iτ +0 (f) which reads as +Iτ +0 (f) = +� +k∈Z +� +k1+k2=k +� τ +0 +e−isΦds�fk1�f k2eikx +with Φ = k2 + k2 +1 + k2 +2. (3.14) +Different from the above two terms Iτ +1 (f) and Iτ +2 (f), in which the obtained +integration is a function with separable variables k, k1 and k2 that enables us +to compute the obtained convolution efficiently in physical space or Fourier +space, it is impossible to compute the exact integral of Iτ +0 efficiently in any +space. To overcome this difficulty, in [22], we proposed an approximation of Iτ +0 +by applying the identity 1 = k1+k2 +k +and an appropriate approximation which +can be computed efficiently. In this paper, we utilize similar idea based on the +identity +1 = k2 +1 + k2 +2 + 2k1k2 +k2 +, +Φ = 2k2 +2 + 2kk1 = 2k2 +1 + 2kk2 = 2k2 − 2k1k2, +and approximate the corresponding integrals in a proper way. +3.2 The first-order exponential-type integrator Ψ τ +1 +Case I. When k = 0, the mean value of Iτ +0 (f) can be computed exactly and +efficiently by +F0 (Iτ +0 (f)) = i +2F0 +�� +eiτ∂2 +x∂−1 +x f +�2� +− i +2F0 +�� +∂−1 +x f +�2� ++ τ +��f0 +�2 = T τ +0 (f). (3.15) +Case II. When k ̸= 0, in order to balance the power of k1 and k2 as much +as possible in the following estimations, we use different forms of the phase + +14 +Hang Li, Chunmei Su* +function Φ as 2k2 +2 + 2kk1, 2k2 +1 + 2kk2, and 2k2 − 2k1k2 for coefficients k2 +1 +k2 , k2 +2 +k2 , +and 2k1k2 +k2 , respectively. Specifically, +� +k̸=0 +Fk (Iτ +0 (f)) eikx = +� +k̸=0 +� +k1+k2=k +� τ +0 +e−is(k2+k2 +1+k2 +2)ds�fk1 +�f k2eikx += +� +k̸=0 +� +k1+k2=k +k2 +1 +k2 +� τ +0 +e−2is(k2 +2+kk1)ds�f k1�f k2eikx ++ +� +k̸=0 +� +k1+k2=k +k2 +2 +k2 +� τ +0 +e−2is(k2 +1+kk2)ds�fk1 +�f k2eikx ++ +� +k̸=0 +� +k1+k2=k +2k1k2 +k2 +� τ +0 +e−2is(k2−k1k2)ds�f k1 +�f k2eikx += T τ +1 (f) + T τ +2 (f) + T τ +3 (f). +(3.16) +By symmetry, obviously we have T τ +1 (f) = T τ +2 (f) and it suffices to approximate +T τ +1 (f) and T τ +3 (f). To begin with, we decompose T τ +1 (f) as +T τ +1 (f) = +� +k̸=0 +� +k1+k2=k +k2 +1 +k2 +� τ +0 +e−2is(k2 +2+kk1)ds�f k1�f k2eikx += +� +k̸=0 +� +k1+k2=k +k2 +1 +k2 +� τ +0 +e−2isk2 +2ds�fk1 +�f k2eikx ++ +� +k̸=0 +� +k1+k2=k +k2 +1 +k2 +� τ +0 +� +e−2iskk1 − 1 +� +ds�fk1 +�f k2eikx ++ +� +k̸=0 +� +k1+k2=k +k2 +1 +k2 +� τ +0 +� +e−2isk2 +2 − 1 +� � +e−2iskk1 − 1 +� +ds�fk1 +�f k2eikx += Lτ +1(f) + Lτ +2(f) + P τ +1 (f). +(3.17) +For f with F0(f) = 0, similarly Lτ +1(f) and Lτ +2(f) can be integrated exactly as +Lτ +1(f) = +� +k̸=0 +� +k1+k2=k +k2=0 +k2 +1 +k2 +� τ +0 +e−2isk2 +2ds�fk1 +�f k2eikx ++ +� +k̸=0 +� +k1+k2=k +k2̸=0 +k2 +1 +k2 +� τ +0 +e−2isk2 +2ds�f k1 +�fk2eikx += − i +2∂−2 +x +�� +e2iτ∂2 +x∂−2 +x f +� � +∂2 +xf +�� ++ i +2∂−2 +x +�� +∂2 +xf +� � +∂−2 +x f +�� +, +(3.18) +Lτ +2(f) = − i +2eiτ∂2 +x∂−3 +x +�� +eiτ∂2 +x∂xf +� � +e−iτ∂2 +xf +�� ++ i +2∂−3 +x +�� +∂xf +� +f +� +− τ∂−2 +x +�� +∂2 +xf +� +f +� +. +(3.19) + +FIRST-ORDER LREI FOR GB +15 +The remainder term P τ +1 (f) will be thrown away in the scheme and the estimate +is postponed to the next section. Similarly T τ +3 (f) can be decomposed as +T τ +3 (f) = +� +k̸=0 +� +k1+k2=k +2k1k2 +k2 +� τ +0 +e−2is(k2−k1k2)ds�f k1 +�fk2eikx += +� +k̸=0 +� +k1+k2=k +2k1k2 +k2 +� τ +0 +e−2isk2ds�f k1 +�f k2eikx ++ +� +k̸=0 +� +k1+k2=k +2k1k2 +k2 +� τ +0 +� +e2isk1k2 − 1 +� +ds�f k1 +�f k2eikx ++ +� +k̸=0 +� +k1+k2=k +2k1k2 +k2 +� τ +0 +� +e−2isk2 − 1 +� � +e2isk1k2 − 1 +� +ds�f k1 +�f k2eikx += Lτ +3(f) + Lτ +4(f) + P τ +2 (f), +(3.20) +where Lτ +3(f) and Lτ +4(f) can be integrated exactly as +Lτ +3(f) = −i∂−4 +x +� +e2iτ∂2 +x − 1 +� � +∂xf +�2 , +(3.21) +Lτ +4(f) = i∂−2 +x e−iτ∂2 +x +� +eiτ∂2 +xf +�2 +− i∂−2 +x +� +f +�2 − 2τ∂−2 +x +� +∂xf +�2 . +(3.22) +Combining (3.9), (3.12), (3.13), (3.17) and (3.20), we obtain +u(tn + τ) = Ψ τ +1 (u(tn)) − i +2BτP τ +1 (u(tn)) − i +4BτP τ +2 (u(tn)) + R0(τ 2), +(3.23) +where +Ψ τ +1 (f) = eiτ⟨∂2 +x⟩f − i +4Bτ� +T τ +0 (f) + 2Lτ +1(f) + 2Lτ +2(f) + Lτ +3(f) + Lτ +4(f) + Iτ +1 (f) ++ 2Iτ +2 (f) +� +− iτ(atn + b)Bτ� +f + ψ1(2iτ∂2 +x)f +� +, +(3.24) +with operators Bτ, Iτ +1 , Iτ +2 , T τ +0 , Lτ +1, Lτ +2, Lτ +3, Lτ +4 defined in (3.10), (3.12), +(3.13), (3.15), (3.18), (3.19), (3.21), (3.22) respectively. Furthermore, notic- +ing BτT τ +0 (f) = 0, we can rewrite (3.24) simply as +Ψ τ +1 (f) = eiτ⟨∂2 +x⟩f − i +4Bτ� +2Lτ +1(f) + 2Lτ +2(f) + Lτ +3(f) + Lτ +4(f) ++ Iτ +1 (f) + 2Iτ +2 (f) +� +− iτ(atn + b)Bτ� +f + ψ1(2iτ∂2 +x)f +� +, +(3.25) +which is exactly (1.8) when Iτ +j and Lτ +j are plugged in. Recalling (3.4) and +z = F0(z) + ˇz, now we are able to propose the scheme +zn = 1 +2(un + un) + atn + b, +zn +t = i +2⟨∂2 +x⟩(un − un) + a, +(3.26) +where +un+1 = Ψ τ +1 (un), +n ≥ 0, +u0 = u(0, x). +(3.27) +The proposed scheme is fully explicit in time and it is easy to implement +efficiently if pseudospectral method is used for spatial discretization thanks to +FFT. + +16 +Hang Li, Chunmei Su* +4 Error estimates +In this section, we will establish the global error estimate concerning the first- +order scheme (3.25). +4.1 Local error estimate +In this part, we give the local error estimate of the scheme (3.25). Inspired by +(3.23) and (2.6), it remains to estimate P τ +1 (f) and P τ +2 (f). +Lemma 6 For r ≥ 1, f ∈ Hr+p(r), it holds +∥P τ +1 (f)∥r ≲ τ2∥f∥2 +r+p(r). +Proof Firstly the k-th Fourier coefficient of P τ +1 (f) can be bounded as +|Fk (P τ +1 (f))| +≲ +� +k1+k2=k +|k|−2|k1|2 +���� +� τ +0 +� +e−2isk2 +2 − 1 +� � +e−2iskk1 − 1 +� +ds +���� +����f k1 +��� +����fk2 +��� +≲ τ +� +k1+k2=k +|k|−2|k1|2 sup +0≤s≤τ +���� +e−2isk2 +2 − 1 +��� ��� +e−2iskk1 − 1 +��� � ����fk1 +��� +����f k2 +��� +≲ τ 1+α+β +� +k1+k2=k +|k|−2|k1|2 |k2|2α |k|β |k1|β ����f k1 +��� +����f k2 +��� +≲ τ 1+α+β|k|−2+β +� +k1+k2=k +|k1|2+β |k2|2α ����fk1 +��� +����f k2 +��� , +(4.1) +where α, β ∈ [0, 1]. This gives +∥P τ +1 (f)∥r ≲ +���τ 1+α+β � +k̸=0 +|k|−2+β +� +k1+k2=k +|k1|2+β |k2|2α ����fk1 +��� +����f k2 +��� eikx��� +r +≲ τ1+α+β ���|∂x|−2+β �� +|∂x|2+β �f +� � +|∂x|2α �f +����� +r . +(4.2) +Now we give several estimates for P τ +1 (f) which might be valid in different +regimes. +(1) For r ≥ 1, setting α = 1 and β = 0 in (4.2), applying the inequalities +in Lemma 5 yields +∥P τ +1 (f)∥r ≲ τ 2 ���|∂x|−2�� +|∂x|2 �f +�� +|∂x|2 �f +����� +r ≲ τ2�� |∂x| �f +��2 +r = τ 2∥f∥2 +r+1, +(4.3) +which implies a second-order local error by requiring one additional derivative. +(2) By applying Lemma 4 (iv), for r + β − 1 > 1/2, we get +∥P τ +1 (f)∥r ≲ τ1+α+β ���|∂x|−1 �� +|∂x|2+β �f +� � +|∂x|2α �f +����� +r+β−1 + +FIRST-ORDER LREI FOR GB +17 +≲ τ1+α+β ��� +� +|∂x|1+β �f +���� +r+β−1 +��� +� +|∂x|2α �f +���� +r+β−1 +≲ τ1+α+β∥f∥r+2β∥f∥r+2α+β−1. +To get a local error bound of order two, setting α + β = 1, one obtains +∥P τ +1 (f)∥r ≲ τ 2∥f∥2 +r+2β, +with +β ∈ [1/3, 1/2], +r > 3/2 − β. +(4.4) +We clearly see that compared to the estimate in (1), this decreases the addi- +tional regularity required when r > 1 and the least order of additional regular- +ity can be decreased to 2/3 which is valid when r > 7/6. On the other hand, +w observe that it is possible to require less additional regularity by choosing +smaller β, however, we have to pay extra price that the error itself is estimated +in a much more regular space by noticing the constraint r > 3/2 − β. +(3) When r +β −2 ∈ [0, 1/2), by applying Lemma 2 and H¨older inequality, +one gets +∥P τ +1 (f)∥r ≲ τ 1+α+β ��� +� +|∂x|2+β �f +� � +|∂x|2α �f +���� +r+β−2 +≲ τ 1+α+β ��� +� +|∂x|2+β �f +� � +|∂x|2α �f +���� +L +2 +5−2r−2β +≲ τ 1+α+β ��� +� +|∂x|2+β �f +���� +L +4 +5−2r−2β +��� +� +|∂x|2α �f +���� +L +4 +5−2r−2β +≲ τ 1+α+β ∥f∥ 2r+2β−3 +4 ++2+β ∥f∥ 2r+2β−3 +4 ++2α +≲ τ 1+α+β ∥f∥2 +r +2 + 3 +2 β+ 5 +4 . +Similarly setting α + β = 1, one derives +∥P τ +1 (f)∥r ≲ τ 2 ∥f∥2 +r +2 + 3 +2 β+ 5 +4 , +r ∈ (3/2 − β, 2 − β], +β ∈ [0, 1]. +(4.5) +(4) On the other hand, we can estimate P τ +1 (f) by employing the inequality +(2.4) in Lemma 1, by setting β = 0, α = 1 in (4.1), +∥P τ +1 (f)∥r ≲ τ 2 ���Jr−2 � +|∂x|2 �f +� � +|∂x|2 �f +���� +≲ τ 2 ��� +� +Jr�f +���� +Lp1 +��� +� +|∂x|2 �f +���� +Lp2 , +(4.6) +where 2 ≤ p1 < ∞, 2 < p2 ≤ ∞ and +1 +p1 + +1 +p2 = 1 +2, when r > 2. Applying the +Sobolev embedding theorem in Lemma 2, we get +∥P τ +1 (f)∥r ≲ τ 2∥f∥r− 1 +p1 + 1 +2 ∥f∥ 5 +2 − 1 +p2 , +2 < p1, p2 < ∞. +To obtain a lower spatial regularity requirement, it is natural to choose r− 1 +p1 + +1 +2 = 5 +2 − 1 +p2 for +1 +p1 + 1 +p2 = 1 +2 with p1 ∈ (2, ∞) and p2 ∈ (2, ∞), i.e., p1 = +1 +r +2 − 3 +4 +and p2 = +1 +5 +4 − r +2 with r ∈ (2, 5/2), which yields the local error estimate as +∥P τ +1 (f)∥r ≲ τ 2∥f∥2 +r +2 + 5 +4 , +r ∈ (2, 5/2). +(4.7) + +18 +Hang Li, Chunmei Su* +(5) Finally, using the bilinear inequality (2.3) in Lemma 1, for α + β = 1, +one has +∥P τ +1 (f)∥r ≲ τ 1+α+β ��� +� +|∂x|2+β �f +� � +|∂x|2α �f +���� +r+β−2 +≲ τ 2 ��� +� +|∂x|2+β �f +���� +r+β−2 +��� +� +|∂x|2α �f +���� +r+β−2 +≲ τ 2∥f∥r+2β∥f∥r+2α+β−2 +≲ τ 2∥f∥2 +r+2β, +(4.8) +for r > 5/2−β with 0 ≤ β ≤ 1. This implies an error without loss of regularity +when r > 5/2 by choosing β = 0. +Taking β = 0 in (4.5), one gets +∥P τ +1 (f)∥r ≲ τ 2 ∥f∥2 +r +2 + 5 +4 , +r ∈ (3/2, 2], +which combines with (4.7) and (4.8) gives +∥P τ +1 (f)∥r ≲ τ 2 ∥f∥2 +r +2 + 5 +4 , +r ∈ (3/2, 5/2); +∥P τ +1 (f)∥r ≲ τ 2 ∥f∥2 +r+ , +r = 5/2; +∥P τ +1 (f)∥r ≲ τ 2 ∥f∥2 +r , +r > 5/2. +(4.9) +For r ≤ 3 +2, by (4.5), we have to choose β = ( 3 +2 − r)+, i.e., β = 3 +2 − r + ε with +any sufficiently small ε > 0 which reads as +∥P τ +1 (f)∥r ≲ τ2 ∥f∥2 +r +2 + 5 +4 + 3 +2 ( 3 +2 −r)+ = ∥f∥2 +( 7 +2 −r)+ , +r ∈ [1, 3/2]. +(4.10) +Similarly (4.4) equivalents to the estimate +∥P τ +1 (f)∥r ≲ τ2 ∥f∥2 +(3−r)+ , +r ∈ (1, 7 +6]; +∥P τ +1 (f)∥r ≲ τ2 ∥f∥2 +r+ 2 +3 , +r > 7 +6. +(4.11) +Lemma 6 is concluded by taking the minimum of the required order of regu- +larity for (4.9), (4.10), (4.11) and (4.4). +Concerning the term P τ +2 (f), we have the following estimate. +Lemma 7 For r ≥ 1, we have +∥P τ +2 (f)∥r ≲ τ2∥f∥2 +r+q(r), +where +q(r) = +� +5/4 − r/2, +1 ≤ r ≤ 5/2; +0, +r > 5/2. + +FIRST-ORDER LREI FOR GB +19 +Proof It follows from (3.20) that +|Fk (P τ +2 (f))| +≲ +� +k1+k2=k +|k|−2|k1||k2| +���� +� τ +0 +� +e−2isk2 − 1 +� � +e2isk1k2 − 1 +� +ds +���� +����f k1 +��� +����fk2 +��� +≲ τ +� +k1+k2=k +|k|−2|k1||k2| sup +0≤s≤τ +���� +e−2isk2 − 1 +��� ��� +e2isk1k2 − 1 +��� � ����f k1 +��� +����fk2 +��� +≲ τ 1+α+β +� +k1+k2=k +|k|−2|k1||k2| |k|2α |k1|β |k2|β ����f k1 +��� +����f k2 +��� +≲ τ 2|k|−2+2α +� +k1+k2=k +|k1|1+β |k2|1+β ����fk1 +��� +����f k2 +��� , +(4.12) +for α, β ∈ [0, 1] satisfying α + β = 1. Using similar approach applied in the +proof of Lemma 6, we establish several estimates by applying various tools. +(1) When r + 2α − 2 ∈ (−1/2, 0], by applying Lemma 2 and H¨older in- +equality, one gets +∥P τ +2 (f)∥r ≲ τ2 ��� +� +|∂x|1+β �f +� � +|∂x|1+β �f +���� +r+2α−2 +≲ τ2 ��� +� +|∂x|1+β �f +� � +|∂x|1+β �f +���� +L +2 +5−2r−4α +≲ τ2 ��� +� +|∂x|1+β �f +���� +2 +L +4 +5−2r−4α +≲ τ2 ∥f∥2 +r +2 + 5 +4 . +Noticing the constraint r ∈ (3/2 − 2α, 2 − 2α] and α ∈ [0, 1], we immediately +get +∥P τ +2 (f)∥r ≲ τ 2 ∥f∥2 +r +2 + 5 +4 , +r ∈ [1, 2]. +(4.13) +(2) Setting α = 0 and β = 1, one gets +∥P τ +2 (f)∥r ≲ τ 2 ���|∂x|−2 � +|∂x|2 �f +� � +|∂x|2 �f +���� +r ≲ τ2 ���Jr−2 � +|∂x|2 �f +� � +|∂x|2 �f +���� , +which is exactly the same as in (4.6). Hence (4.7) also holds for P τ +2 (f), i.e., +∥P τ +2 (f)∥r ≲ τ 2∥f∥2 +r +2 + 5 +4 , +r ∈ (2, 5/2]. +(4.14) +(3) It remains to give a bound of ∥P τ +2 (f)∥r for r > 5/2. Employing the +bilinear estimate (2.3), one easily gets +∥P τ +2 (f)∥r ≲ τ 2 ��� +� +|∂x|1+β �f +� � +|∂x|1+β �f +���� +r+2α−2 +≲ τ 2 ��� +� +|∂x|1+β �f +���� +2 +r+2α−2 +≲ τ 2∥f∥2 +r+α, +for +r > 5/2 − 2α, + +20 +Hang Li, Chunmei Su* +which implies +∥P τ +2 (f)∥r ≲ τ2∥f∥2 +r, +for +r > 5/2; +∥P τ +2 (f)∥r ≲ τ2∥f∥2 +( r +2 + 5 +4 )+, +r ∈ [1, 5/2]. +This together with (4.13) and (4.14) concludes Lemma 7. +It is easy to see q(r) ≤ p(r) by direct computations. In spirit of (3.23), +Lemma 6 and Lemma 7, we obtain the local error of the scheme (3.25). +Lemma 8 Suppose r ≥ 1, u ∈ L∞(0, T ; Hr+p(r)). Then we have +∥u(tn + τ) − Ψ τ +1 (u(tn))∥r ≤ Lτ 2, +where L depends on ∥u∥L∞(0,T ;Hr+p(r)). +4.2 Stability +Lemma 9 Suppose r ≥ 1 and f, g ∈ Hr. Then for τ > 0, we have +∥Ψ τ +1 (f) − Ψ τ +1 (g)∥r ≤ (1 + Mτ)∥f − g∥r, +(4.15) +where M depends on r and ∥f∥r + ∥g∥r. +Proof To begin with, by using (2.3), (3.10) and (3.11), one can easily check +that +∥Iτ +1 (f) − Iτ +1 (g)∥r ≤ Cτ sup +τ≥s≥0 +���eis∂2 +x +� +(e−is∂2 +xf)2 − (e−is∂2 +xg)2���� +r +≤ Cτ sup +τ≥s≥0 +∥e−is∂2 +x(f + g)∥r∥e−is∂2 +x(f − g)∥r +≤ Cτ(∥f∥r + ∥g∥r)∥f − g∥r. +(4.16) +Similar discussions for Iτ +1 (f) and Iτ +2 (f) yield that +∥Iτ +2 (f) − Iτ +2 (g)∥r ≤ Cτ(∥f∥r + ∥g∥r)∥f − g∥r, +∥Iτ +0 (f) − Iτ +0 (g)∥r ≤ Cτ(∥f∥r + ∥g∥r)∥f − g∥r. +(4.17) +Noticing the decomposition of T τ +i (f) (i = 1, 2, 3) in (3.15), (3.17) and (3.20), +we have +Iτ +0 (f) = T τ +0 (f) + 2Lτ +1(f) + 2Lτ +2(f) + 2P τ +1 (f) + Lτ +3(f) + Lτ +4(f) + P τ +2 (f), +(4.18) +which yields +∥W τ(f) − W τ(g)∥r = ∥Iτ +0 (f) − Iτ +0 (g) − 2P τ +1 (f) + 2P τ +1 (g) − P τ +2 (f) + P τ +2 (g)∥r +≤ ∥Iτ +0 (f) − Iτ +0 (g)∥r + 2∥P τ +1 (f) − P τ +1 (g)∥r + ∥P τ +2 (f) − P τ +2 (g)∥r, (4.19) +where W τ(f) := T τ +0 (f) + 2Lτ +1(f) + 2Lτ +2(f) + Lτ +3(f) + Lτ +4(f). + +FIRST-ORDER LREI FOR GB +21 +It remains to deal with the terms P τ +1 (f) and P τ +2 (f). According to Lemmas +1, 4, 5 and the definition of P τ +1 in (3.17), for r ≥ 1, we have +∥P τ +1 (f) − P τ +1 (g)∥r +≲ +��� +� +k̸=0 +� +k1+k2=k +k2 +1 +k2 +� τ +0 +� +e−2isk2 +2 − 1 +� � +e−2iskk1 − 1 +� +ds +��f k1 +�fk2 − �gk1�gk2 +� +eikx��� +r +≲ τ +��� +� +k̸=0 +� +k1+k2=k +k−2k2 +1 +����f k1�fk2 − �f k1�gk2 + �fk1�gk2 − �gk1�gk2 +��� eikx��� +r +≲ τ +��� +� +k̸=0 +� +k1+k2=k +k−2k2 +1 +����f k1 +��� +����fk2 − �gk2 +��� + +����f k1 − �gk1 +��� +����gk2 +��� eikx��� +r +≲ τ +�����∂−2 +x +� +(∂2 +x�f)( � +f − g) +����� +r ++ +����∂−2 +x +� +∂2 +x( � +f − g)�g +����� +r +� += τ +� ����∂−1 +x +� +(∂x�f)( � +f − g) +� +− ∂−2 +x +� +(∂x�f)∂x( � +f − g) +����� +r ++ +����∂−1 +x +� +∂x( � +f − g)�g +� +− ∂−2 +x +� +∂x( � +f − g)∂x�g +����� +r +� +≲ τ(∥f∥r + ∥g∥r)∥f − g∥r, +(4.20) +where we have used the modified version of Newton-Leibniz formula +∂−2 +x [(∂2 +xf)g] = ∂−1 +x [(∂xf)g] − ∂−2 +x [(∂xf)(∂xg)], +which can be obviously derived by the decomposition +k−2k2 +1 = k−2k1(k − k2) = k−1k1 − k−2k1k2. +Recalling the definition of P τ +2 (f) in (3.20), we can establish +∥P τ +2 (f) − P τ +2 (g)∥r ≲ τ(∥f∥r + ∥g∥r)∥f − g∥r, +(4.21) +by employing similar arguments as above. Combining (4.16)–(4.21), applying +Lemma 1 and Lemma 4, for r ≥ 1, we have +∥Ψ τ +1 (f) − Ψ τ +1 (g)∥r = +����eiτ⟨∂2 +x⟩(f − g) − i +4Bτ� +W τ(f) − W τ(g) + Iτ +1 (f) − Iτ +1 (g) ++ 2 (Iτ +2 (f) − Iτ +2 (g)) +� +− iτ(atn + b)Bτ� +f − g + ψ1(2iτ∂2 +x)(f − g) +����� +r +≤ ∥f − g∥r + +���W τ(f) − W τ(g) +�� +r + ∥Iτ +1 (f) − Iτ +1 (g)∥r ++ 2 ∥Iτ +2 (f) − Iτ +2 (g)∥r +� ++ Crτ(∥f − g∥r + ∥f − g∥r) +≤ (1 + Mτ)∥f − g∥r. +(4.22) +where M depends on r and ∥f∥r + ∥g∥r and the proof is complete. + +22 +Hang Li, Chunmei Su* +4.3 Proof of Theorem 1 +Proof In spirit of (3.27), it suffices to show +∥u(tn) − un∥r ≤ Cτ. +Combining the local error estimate in Lemma 8 and stability inequality (4.15), +we are led to +∥u(tn+1) − un+1∥r ≤ ∥u(tn+1) − Ψ τ +1 (u(tn))∥r + ∥Ψ τ +1 (u(tn)) − Ψ τ +1 (un)∥r +≤ Lτ 2 + ∥Ψ τ +1 (u(tn)) − Ψ τ +1 (un)∥r +≤ Lτ 2 + eτM∥u(tn) − un∥r, +where L depends on ∥u∥L∞(0,T ;Hr+p(r)) (or equivalently ∥z∥L∞(0,T ;Hr+p(r)) + +∥zt∥L∞(0,T ;Hr+p(r)−2)), and M depends on ∥u(tn)∥r and ∥un∥r. Then the as- +sertion follows by a standard induction argument [18,30,31]. +5 Numerical experiments +In this section, we present some numerical experiments of the newly proposed +first-order LREI Ψ τ +1 to justify our theoretical convergence results. The numer- +ical investigations of convergence of the first-order LREIs in [22,31] will be +provided as comparisons. Furthermore, the Fourier pseudospectral method is +used for spatial discretization so that each iteration can be calculated by FFT +via O(MlogM) operations, where M represents the number of grid points +in space. We choose the spatial mesh size ∆x = 1/26 for soliton solutions +and ∆x = π/215 for rough solutions. Moreover, we define the error in Hr as +∥z(tn) − zn∥r + ∥∂tz(tn) − zn +t ∥r−2. +5.1 Soliton solutions +In the first experiment, we numerically verify the temporal convergence of the +first-order LREI Ψ τ +1 (3.27) for the soliton solution [24] of (1.1) +z(x, t) = −Asech2[(ω/2)(x − vt + ζ0)], +(5.1) +where ζ0 ∈ R, 0 < ω ≤ 1, and the relations among the amplitude A, velocity +v and frequency ω are given as follows +A = 3ω2/2, +v = ±(1 − ω2)1/2. +It can be clearly observed that (5.1) decays exponentially at far field, which +enables us to impose the periodic boundary conditions on a bounded domain +[−x0, x0] when x0 is chosen large enough. Fig. 2 shows the error in H2, for the +numerical solution obtained by (3.27) at tn = 1, where x0 = 80. From Fig. 2 +we can see that the scheme converges at the first order in time. + +FIRST-ORDER LREI FOR GB +23 +10-3 +10-2 +10-1 +10-6 +10-5 +10-4 +Fig. 2 Linear convergence of the LREI scheme Ψτ +1 for the soliton solution with ζ0 = 0, +ω = 1/2, v = +√ +3/2. Here we select the spatial mesh size as ∆x = 1/26. +5.2 Rough solutions +In the second experiment, we apply the first-order LREI Ψ τ +1 (3.27) to (1.1) +under nonsmooth initial data. We numerically compare the results with those +of the first-order LREIs in [22] and [31]. +Following the construction method of the initial data with desired regular- +ity in [31], we choose the spatial mesh size ∆x = 2π/M with M = 216 and the +grid points xj = −π + j∆x, 0 ≤ j < M. Moreover, a uniformly distributed +random vector rand(1, M) can be taken in the computer, which is denoted +by Z = (z0, . . . , zM−1) = rand(1, M). Then we define the inverse derivative +operator |∂x,M|−θ as a truncation of the operator |∂x|−θ (2.2), which maps a +function f ∈ L2(T) to Hθ(T) +|∂x,M|−θf = +M/2−1 +� +k=−M/2,k̸=0 +|k|−θ �fkeikx, +θ ∈ R. +Then we define +Z0 = +Z1 + c ∗ ∥Z1∥∞ +∥Z1 + c ∗ ∥Z1∥∞∥ , +where c := rand(1) is a random number and Z1 := |∂x,M|−θZ, x ∈ T. Finally, +we get Z0 ∈ Hθ(T). For instance, Fig. 3 displays the initial data obtained as +above for θ = 2 and θ = 2.5, respectively. +Figs. 4-6 show the errors of the scheme Ψ τ +1 and those in [22,31] in Hr with +various r at the final time tn = T = 1 for different rough initial data, where +the reference solution is obtained by the first-order LREI Ψ τ +1 (3.27) with a tiny +time step τ = 3 × 10−5. Specifically, Figs. 4–6 display the temporal errors of +the three schemes when the given initial data has additional order of regularity +0, 1/4, 1/2, 2/3 and 1, respectively. From the numerical results shown in Figs. +4 to 6, it can be clearly observed that: + +24 +Hang Li, Chunmei Su* +-3 +-2 +-1 +0 +1 +2 +3 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +-3 +-2 +-1 +0 +1 +2 +3 +-0.5 +0 +0.5 +1 +Fig. 3 Left: initial value z0 ∈ H2 with θ = 2. Right: initial value z(0, x) ∈ H2.5 with +θ = 2.5. +(1) The newly developed first-order LREI scheme in (3.27) has first-order +convergence in all cases, which demonstrates the theoretical results pre- +sented in Theorem 1. +(2) Compared to the other two schemes in [22,31], the newly proposed scheme +behaves most regularly and the oscillations are the weakest while the +method in [31] behaves most irregularly and might suffer an order re- +duction (cf. Fig. 4). Furthermore, the method presented in this paper is +the most accurate when the time step is small enough. This shows the +superiority of the newly proposed method (1.8). +10-3 +10-2 +10-1 +10-3 +10-2 +10-1 +100 +10-3 +10-2 +10-1 +10-3 +10-2 +10-1 +Fig. 4 Numerical errors in H2.5 (left) and H2 (right) of three first-order schemes at the +final time T = 1 with rough solution in H2.5 and H2.25, respectively. + +FIRST-ORDER LREI FOR GB +25 +10-3 +10-2 +10-1 +10-4 +10-3 +10-2 +10-3 +10-2 +10-1 +10-4 +10-3 +10-2 +Fig. 5 Numerical errors in H1.5 and H17/12 of three first-order schemes at the final time +T = 1 with rough solution in H2 and H25/12, respectively. +10-3 +10-2 +10-1 +10-4 +10-3 +10-2 +10-3 +10-2 +10-1 +10-4 +10-3 +10-2 +Fig. 6 Numerical errors in H7/6 and H1 of the three first-order schemes at the final time +T = 1 with rough solution in H11/6 and H2, respectively. +6 Conclusions +In this work, we developed a new first-order low regularity exponential-type +integrator for the “good” Boussinesq equation with rough initial data. The +method is based on a twisted variable and the phase space analysis of the +nonlinear dynamics. By applying the Kato-Ponce inequalities, the Hardy- +Littlewood-Sobolev type inequality and Sobolev embedding theorem, we es- +tablished the linear convergence in Hr with solutions in Hr+p(r) for r ≥ 1, +where p(r) is non-increasing with respect to r. Particularly, the first-order ac- +curacy can be achieved in Hr for solutions in Hr when r ≥ 5/2. This is the +lowest regularity requirement of the existing methods for the GB equation so +far. The analytical result is supported by extensive numerical experiments. +Acknowledgements This work was supported by the NSFC 12201342. + +26 +Hang Li, Chunmei Su* +Data Availability Data sharing not applicable to this article as no datasets +were generated or analyzed during the current study. +Declarations +Conflict of interest The author declares no conflict of interest. +References +1. Adams, R. A., Fournier, J. J.: Sobolev Spaces. 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Wu, Y., Zhao, X.: Embedded exponential-type low-regularity integrators for KdV equa- +tion under rough data. BIT Numer. Math. 62 (3), 1049–1090 (2022) +46. Zhang, C., Wang, H., Huang, J., Wang, C., Yue, X.: A second order operator splitting +numerical scheme for the “good” Boussinesq equation. Appl. Numer. Math. 119, 179– +193 (2017) + diff --git a/y9E3T4oBgHgl3EQfPglf/content/tmp_files/load_file.txt b/y9E3T4oBgHgl3EQfPglf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e48cb1b5a8ce47076684407ce4919c300b813aa5 --- /dev/null +++ b/y9E3T4oBgHgl3EQfPglf/content/tmp_files/load_file.txt @@ -0,0 +1,1000 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf,len=999 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='04403v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='NA] 11 Jan 2023 A semi-discrete first-order low regularity exponential integrator for the “good” Boussinesq equation without loss of regularity Hang Li · Chunmei Su* Abstract In this paper, we propose a semi-discrete first-order low regularity exponential-type integrator (LREI) for the “good” Boussinesq equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' It is shown that the method is convergent linearly in the space Hr for solutions belonging to Hr+p(r) where 0 ≤ p(r) ≤ 1 is non-increasing with respect to r, which means less additional derivatives might be needed when the numerical solution is measured in a more regular space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Particularly, the LREI presents the first-order accuracy in Hr with no assumptions of additional derivatives when r > 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' This is the first time to propose a low regularity method which achieves the optimal first-order accuracy without loss of regularity for the GB equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The convergence is confirmed by extensive numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Keywords “good” Boussinesq equation · low regularity · error estimate · first-order integrator · without loss of regularity Mathematics Subject Classification (2020) 35Q35 · 65M12 · 65M15 · 65M70 1 Introduction We consider the following periodic boundary value problem of the “good” Boussinesq (GB) equation: � ztt + zxxxx − zxx − (z2)xx = 0, x ∈ T, t > 0, z(0, x) = φ0(x), zt(0, x) = ψ0(x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Su (*Corresponding author) Yau Mathematical Sciences Center, Tsinghua University, Beijing, China Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' : +123-45-678910 Fax: +123-45-678910 E-mail: sucm@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='cn H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Li Yau Mathematical Sciences Center, Tsinghua University, Beijing, China 2 Hang Li, Chunmei Su* in a torus T = [−π, π], where φ0(x) and ψ0(x) are given initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The GB equation was originally founded by Joseph Boussinesq [6] to describe the prop- agation of dispersive shallow water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Furthermore, it was also extended by replacing the quadratic nonlinearity with a general function of z to model small oscillations of nonlinear beams [40] or the two-way propagation of water waves in a channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' There have been many applications for the GB equation in physics [14,16] and oceanographic engineering [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Analytically, similar to the well-know Korteweg-de Vries (KdV) equation, the nonlinear Schr¨odinger (NLS) equation, and other dispersive equations, the GB equation admits abundant soliton solutions, see [9,12,19,24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' How- ever, the GB equation has some special characteristics that make it different from the KdV or NLS equations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', two solitons can merge into one soli- ton or develop into the so-called antisolitons [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' For less smooth solutions, Kishimoto [17] gave a sharp locally well-posed result by using the fix-point theory together with low regularity bilinear estimates in Bourgain spaces, also known as the dispersive Sobolev spaces [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The main result in [17] is that for any (φ0, ψ0) ∈ Hs × Hs−2, s ≥ −1/2, there exist a positive time T (∥ψ0∥Hs, ∥ψ0∥Hs−2) > 0 and a unique solution of the GB equation in a certain Banach space of functions X ⊂ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Hs × Hs−2), however, this equation is ill-posed when s < −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' We refer to [10,11,17,27,42] for more detailed theoretical results of the GB equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Along the numerical part, a large variety of classical numerical schemes for approximating the time dynamics of the GB equation have been proposed and analyzed, including the pseudospectral methods [8,12], finite difference methods [7,28], the exponential integrators [37] and splitting methods [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' However, as a result of the fourth-order spatial derivative in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1), these tradi- tional schemes can not reach their ideal convergence rates when the solution is not smooth enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' For example, an explicit finite difference scheme was con- structed in [28], which strictly requires the boundedness of ∂6 xz and ∂4 t z and a time step restriction of ∆t = O(∆x2), where ∆t and ∆x represent the time and space step, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Unfortunately, the solutions involved in practical applications become rough due to the interference of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Thus it is nec- essary to find appropriate methods which can achieve the ideal convergence even for rough solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' To this aim, some low regularity exponential inte- grators (LREIs) requiring low additional regularity have been established by introducing the concepts of twisted variable w(t) := eit∂2 xu(t) and Duhamel’s formula, see [22,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Compared to the classical methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', classical expo- nential integrators, these strategies give rise to some numerical schemes that still converge even when the solution is rough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Specifically, Ostermann & Su [31] gave a first-order and a second-order LREIs and obtain the linear and quadratic convergence in Hr (r > 1/2) by requiring one and three additional derivatives, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' This demand is weaker than that of the operator split- ting method [46] and the spectral method [8], the latter of which requires the boundedness of at least four additional temporal and spatial derivatives to at- tain the second-order convergence in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Recently, the authors [22] proposed a new first-order and second-order LREIs, which converge with less additional FIRST-ORDER LREI FOR GB 3 derivatives required than those in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' In particular, the second-order LREI in [22] converges quadratically with two additional derivatives required, which is weaker than that of [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' In this article, we will introduce a newly developed LREI which has first- order accuracy in Hr by requiring the boundedness of additional spatial deriva- tives at the order of p(r), where p(r) is non-increasing with respect to r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', ∥un − u(tn)∥r ≲ τ, for u ∈ L∞(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Hr+p(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Particularly, p(r) = 0 for all r > 5/2, which means the method is convergent at the first order in Hr with no additional regularity needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The first-order LREI is established by the following strategy: (i) In the first step, we rewrite the GB equation as a first-order system � z zt � t = � 0 1 −∂4 x + ∂2 x 0 � � z zt � + � 0 (z2)xx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Then we diagonalize the above matrix in Fourier space and introduce a new complex variable u(t) involving z and zt so that the GB equation equivalents to a Schr¨odinger-type equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (ii) In the second step, we extract the dominant term in the linear part of the equation involving u(t) and introduce the so-called twisted variable w(t) = eit∂2 xu(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' An appropriate approximation is used to integrate the Duhamel’s for- mula on the new variable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (iii) Finally, we twist the variable back and obtain an approximation to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The integral for the nonlinear term is approximated so that the iteration can be efficiently calculated in physical space or Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Remark 1 The method of twisted variable is firstly introduced by Ostermann and Schratz [30] to design low-regularity numerical schemes for the nonlinear Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Since then this technique has been extensively applied for the nonlinear Schr¨odinger equation [18,20,29,43,33], KdV equation [13,32, 44,45], Klein-Gordon equation [3,41] and other equations [34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Compared to classical numerical methods, this type of low regularity integrators can achieve the same convergence when the solutions are less regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Below we present our idea to design the new LERI briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The approach is based on the phase space analysis of the nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Specifically, we are devoted to finding a suitable approximation for the following time integral � τ 0 e−is(k2+k2 1+k2 2)ds, with k1 + k2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The leading term −2k2 is kept and integrated exactly in [31], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', � τ 0 e−is(k2+k2 1+k2 2)ds = � τ 0 e−2isk2+2isk1k2ds ≈ � τ 0 e−2isk2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 4 Hang Li, Chunmei Su* This finally leads to a first-order scheme with one additional order of regularity required [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' To weaken the constraint on regularity, the authors applied the identity 1 = k1+k2 k together with the property k2 + k2 1 + k2 2 = 2k2 2 + 2kk1 = 2k2 1 + 2kk2 = 2k2 − 2k1k2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2) and decompose the integral as � τ 0 e−is(k2+k2 1+k2 2)ds = k1 k � τ 0 e−is(k2+k2 1+k2 2)ds + k2 k � τ 0 e−is(k2+k2 1+k2 2)ds = k1 k � τ 0 e−2is(k2 2+kk1)ds + k2 k � τ 0 e−2is(k2 1+kk2)ds ≈ k1 k � τ 0 � e−2isk2 2 + e−2iskk1 − 1 � ds + k2 k � τ 0 � e−2isk2 1 + e−2iskk2 − 1 � ds, where the the integrals in the last line can be integrated exactly in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' In this work, we apply the identity k2 1 + k2 2 + 2k1k2 k2 = 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3) instead and the decomposition follows as e−is(k2+k2 1+k2 2) = k2 1 k2 e−2is(k2 2+kk1) + k2 2 k2 e−2is(k2 1+kk2) + 2k1k2 k2 e−2is(k2−k1k2) ≈ k2 1 k2 (e−2isk2 2 + e−2iskk1 − 1) + k2 2 k2 (e−2isk2 1 + e−2iskk2 − 1) + 2k1k2 k2 (e−2isk2 + e2isk1k2 − 1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4) where all three terms in the approximation can be exactly integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' In this way we are able to establish the numerical flow as follows zn = 1 2(un + un) + atn + b, zn t = i 2⟨∂2 x⟩(un − un) + a, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5) where a = F0(zt(0, ·)) = 1 2π � T ψ0(x)dx, b = F0(z(0, ·)) = 1 2π � T φ0(x)dx, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6) and un+1 = Ψ τ 1 (un), n ≥ 0, u0 = u(0, x) = φ(x)−b−i⟨∂2 x⟩−1(ψ(x)−a), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='with ⟨∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x⟩−1 defined in Section 2 and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='Ψ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 (f) = eiτ⟨∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x⟩f − i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4Bτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='− i∂−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e2iτ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x∂−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='xf ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='− ieiτ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x∂−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='eiτ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x∂xf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e−iτ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x∂xf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='+ i∂−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='∂xf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='− 2τ∂−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='xf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='FIRST-ORDER LREI FOR GB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='− i∂−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e2iτ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='∂xf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�2 + i∂−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x e−iτ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='eiτ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='xf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='− i∂−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='− 2τ∂−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='xf)(eiτ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x∂−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='+ i∂−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f(∂−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='− iτ(atn + b)Bτ� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f + ψ1(2iτ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x)f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8) with ψ1 and Bτ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='10) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' It can be easily seen that the scheme is explicit and easy to implement if one applies Fourier spectral method for spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Now we state the main theorem concerning the convergence of the above scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Before that, we define a function p(r): p(r) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1, r = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3 − 2r)+, 1 < r ≤ 7/6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 2/3, 7/6 < r ≤ 17/12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (7/2 − 2r)+, 17/12 < r ≤ 3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 5/4 − r/2, 3/2 < r < 5/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 0+, r = 5/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 0, r ≥ 5/2, where c+ means c + ε for any sufficiently small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Theorem 1 For r ≥ 1, suppose that the exact solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1) satisfies z ∈ C(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Hr+p(r)) and zt ∈ C(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Hr+p(r)−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Then there exists a constant τ0 > 0 such that for all step size 0 < τ ≤ τ0 and tn ≤ T , we have ∥z(tn) − zn∥r + ∥zt(tn) − zn t ∥r−2 ≤ Cτ , where C > 0 depends on T , ∥z∥L∞(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='Hr+p(r)) and ∥zt∥L∞(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='Hr+p(r)−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' It is clear p(r) represents the order of additional regularity required to promise the first-order convergence of the numerical solution in Hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 1 displays the plot of p(r), from which we observe that p(r) is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Particularly, p(r) ≡ 0 when r > 5/2, which means the scheme is convergent linearly in Hr without loss of regularity when r > 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Compared to the convergence results of the scheme in [31], which converges in Hr (r > 1/2) at the first order when the solution belongs to Hr+1, and the method in [22], which achieves the first-order convergence in Hr for r > 7/6 as the solution lies in Hr+2/3, it is obvious that our newly proposed scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8) requires less regularity to attain the ideal first-order convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Furthermore, for the convergence without smoothness assumptions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', p(r) = 0, compared to the first-order LREIs proposed in [22] and [31] which converge at the order of 1/2 and r−1/2 3r+1/2−, respectively, our newly developed first-order LREI presents a linear convergence without additional regularity assumptions when r > 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 6 Hang Li, Chunmei Su* Additional Derivative To Achieve Linear Convergence 1 7 6 17 12 3 2 5 2 1 2 3 1 2 1 4 0 (1, 1) ( 7 6, 2 3) ( 17 12, 2 3) ( 3 2, 1 2) ( 5 2, 0) r Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 1 Additional order of regularity required to achieve the first-order convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Par- ticularly, for the domain (1, 7/6] and (17/12, 3/2], we plot it by a dash-dotted line or hollow points to mean that a plus sign exists in p(r), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', the scheme is convergent linearly in H3/2 for solutions in H(3/2+1/2)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' This is the first time to establish the optimal linear convergence without loss of regularity for the GB equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' On the other hand, we have to admit that the deficiency is that we have to impose r ≥ 1 due to the stability analysis (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Thus the analysis in Hr for r ≤ 1 is absent at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' In Section 2, we present some notions and powerful technical tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The first-order LREI is constructed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Section 4 is devoted to establishing the error estimate of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Some numerical results are presented to confirm the theoretical analysis in Section 5 and conclusions are drawn in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 2 Preliminary In this section, we introduce some notations and present some useful technical lemmas which are of vital importance to design the method or to establish the error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 Notations In this paper, we use the notation X ≲ Y to denote that there exists a constant C > 0 which may be different from line to line but is independent of the time step τ such that |X| ≤ CY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The Fourier transform of a function f on a torus T is defined by the coordinate representation { �fk}+∞ k=−∞ under the basis FIRST-ORDER LREI FOR GB 7 {eikx}+∞ k=−∞ in L2(T), where Fk(f) = �fk = 1 2π � T f(x)e−ikxdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Thus f(x) = � k∈Z �fkeikx is the inverse Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The norm and inner product in L2 are defined respectively by ∥f∥ := ∥f∥L2 = � � k∈Z | �fk|2�1/2, (f, g) = � k∈Z �fk�gk = 1 2π � T f(x)g(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1) Moreover, we define several operators given in Fourier space as ∂−1 x f = � k̸=0 1 ik �fkeikx, �f = � k∈Z �� �fk ��eikx, |∂x|αf = � k̸=0 |k|α �fkeikx, Jαf = � k∈Z (1 + k2)α/2 �fkeikx, α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2) Similarly, we define ⟨∂2 x⟩ := � −∂2x + ∂4x and its inverse by ⟨∂2 x⟩f = � k∈Z � k2 + k4 �fkeikx, ⟨∂2 x⟩−1f = � k̸=0 1 √ k2 + k4 �fkeikx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Furthermore, we introduce the Sobolev space Hα with α ∈ R, which con- sists of the functions f = � k∈Z �fkeikx such that ∥f∥α = ∥Jαf∥ < ∞, where ∥f∥2 α = ∥Jαf∥2 = � k∈Z (1 + k2)α| �fk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' It is clear that for f with zero mean value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', �fk = 0, it holds ∥f∥α ≲ ∥|∂x|αf∥ ≲ ∥f∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' For α = 0, it is clear that the space reduces to L2 and the corresponding norm is simply denoted as ∥ · ∥ which agrees with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' We say that R = R(u, t, τ, ξ) ∈ Rθ(τ ν) if and only if ∥R(u, t, τ, ξ)∥r ≤ Cτ ν, where R(u, t, τ, ξ) depends on the value u(t + ξ), 0 ≤ ξ ≤ τ, and C relies on sup 0≤s≤τ ∥u(t + s)∥r+θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' We write f = g + Rθ(τ ν) whenever f = g + R with R ∈ Rθ(τ ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 8 Hang Li, Chunmei Su* 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2 Preliminary tools To begin with, we introduce the Kato-Ponce inequalities, which was previously proved in [15,5,23] in the whole space R and extended to the periodic case by Li and Wu [21] recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Lemma 1 (The Kato-Ponce inequalities) (i) If r > 1/2 and f, g ∈ Hr, then we have ∥fg∥r ≲ ∥f∥r∥g∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3) (ii) If s > 0, 1 < p ≤ ∞, 1 < p1, p3 < ∞, 1 < p2, p4 ≤ ∞ satisfying 1 p = 1 p1 + 1 p2 and 1 p = 1 p3 + 1 p4 , then we have the following inequality ∥Js(fg)∥Lp ≲ ∥Jsf∥Lp1 ∥g∥Lp2 + ∥Jsg∥Lp3 ∥f∥Lp4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4) Next we present Hardy-Littlewood-Sobolev type inequality and Sobolev embedding theorem on the torus T, which provides a new approach for the subsequent estimate of local truncation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' We refer to [1,2,4,26,36] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Lemma 2 (i) (Hardy-Littlewood-Sobolev type inequality) Let s ∈ [0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Then there exists a constant C = C(s) > 0 such that ∥f∥−s ≤ C∥f∥ L 2 1+2s (T), for any f ∈ L 2 1+2s (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (ii) (Sobolev embedding theorem) Let s ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The inclusion Hs(T) ⊆ Lq(T) is continuous for any q ∈ � 1, 2 1−2s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Lemma 3 (i) For all x, y ∈ R and 0 ≤ θ ≤ 1, we have |eix − 1| ≤ 21−θ|x|θ, |eix − 1 − ix| ≤ 21−2θ|x|1+θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (ii) For t ∈ R, r ≥ 0 and f ∈ Hr, we have ∥ψ1(it∂2 x)f∥r ≤ ∥f∥r, where ψ1(y) = � 1 0 eysds, for y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5) For the details of the proof, we refer to [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Moreover, we illustrate a lemma which was introduced by [22,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' FIRST-ORDER LREI FOR GB 9 Lemma 4 (i) For f ∈ Hr with r ≥ 0, t ∈ R, it holds ∥⟨∂2 x⟩−1f∥r ≤ ∥f∥r, ∥Af∥r ≤ ∥f∥r, ∥Bf∥r ≤ ∥f∥r, ∥(eitA − 1)f∥r ≤ |t|∥f∥r, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6) where A and B are given by A := ⟨∂2 x⟩ + ∂2 x, B := ⟨∂2 x⟩−1∂2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7) (ii) For r ≥ 0 and 0 ≤ γ ≤ 1, f ∈ Hr+2γ, one has ∥(eit∂2 x − 1)f∥r ≲ |t|γ∥f∥r+2γ, ∥(eit⟨∂2 x⟩ − 1)f∥r ≲ |t|γ∥f∥r+2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8) (iii) If f, g ∈ H1, then it holds ��J−1 (g(Jf)) �� ≲ min{∥f∥∥g∥1, ∥f∥1∥g∥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='9) (iv) If f, g ∈ Hr, r > 1/2 then we have ��J−1 (g(Jf)) �� r ≲ ∥f∥r∥g∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='10) Lemma 5 For f, g ∈ Hr with r ≥ 1, it holds �� |∂x|−2 � (|∂x| g)(|∂x| f) ��� r ≲ ∥f∥r∥g∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='11) Proof To show the above inequality for r > 1, we only need to confirm �� |∂x|r−2 � (|∂x| g)(|∂x| f) ��� ≲ ∥f∥r∥g∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' According to the duality principle in L2, it suffices to prove � |∂x|r−2 � (|∂x| g)(|∂x| f) � , φ � ≲ ∥f∥r∥g∥r∥φ∥, ∀φ ∈ L2, which is equivalent to � k̸=0 � k1+k2=k |k|r−2|k1||k2| �fk1�gk2 �φk ≲ ∥f∥r∥g∥r∥φ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' To this aim, we divide the above formula into two parts by discussing the relationship between Fourier coefficients k and k1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', � k̸=0 � k1+k2=k |k|r−2|k1||k2| �fk1�gk2 �φk = � k̸=0 � k1+k2=k |k1|≤2|k| |k|r−2|k1||k2| �fk1�gk2 �φk + � k̸=0 � k1+k2=k |k1|>2|k| |k|r−2|k1||k2| �fk1�gk2 �φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='12) For the first term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='12), we have |k2| = |k − k1| ≤ |k| + |k1| ≤ 3|k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 10 Hang Li, Chunmei Su* By using Plancherel’s identity and the bilinear estimate, the first term can be bounded as � k̸=0 � k1+k2=k |k1|≤2|k| |k|r−2|k1||k2| �fk1�gk2 �φk ≲ � k̸=0 � k1+k2=k |k1|≤2|k| |k|r�� �fk1 �� |�gk2| ���φk �� ≲ (|∂x|r( �f�g), �φ) ≲ ∥ �f�g∥r∥�φ∥ ≲ ∥ �f∥r∥�g∥r∥�φ∥ ≲ ∥f∥r∥g∥r∥φ∥, where �f, �g and �φ are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' For the second term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='12), thanks to |k1| > 2|k|, we are led to |k2| = |k1 − k| ≥ |k1| − |k| > |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' For r > 1, it holds |k|r−2|k1||k2| = |k|−r|k|2r−2|k1||k2| ≲ |k|−r|k1|r|k2|r, which implies � k̸=0 � k1+k2=k |k1|>2|k| |k|r−2|k1||k2| �fk1�gk2 �φk ≲ � k̸=0 � k1+k2=k |k1|>2|k| |k|−r|k1|r|k2|r�� �fk1 �� |�gk2| ���φk �� ≲ � k̸=0 Fk(|∂x|r �f|∂x|r�g)|k|−r|�φk| ≲ max k ���� � T |∂x|r �f(x)|∂x|r�g(x)e−ikxdx ���� � k̸=0 |k|−r|�φk| ≲ ∥|∂x|r �f|∂x|r�g∥L1∥(|k|−r)0̸=k∈Z∥l2∥(|�φk|)0̸=k∈Z∥l2 ≲ ∥|∂x|r �f∥∥|∂x|r�g∥∥�φ∥ ≲ ∥f∥r∥g∥r∥φ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The proof is completed for the case of r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' For the case of r = 1, by using the result in Lemma 4 (iii) ∥|∂x|−1(g|∂x|f)∥ ≲ ∥J−1(g|∂x|f)∥ ≲ ∥J−1(�g(J �f))∥ ≲ ∥f∥1∥g∥, we are led to ∥ |∂x|−2 � (|∂x| g)(|∂x| f) � ∥1 ≲ ∥ |∂x|−1 � (|∂x| g)(|∂x| f) � ∥ ≲ ∥f∥1∥|∂x|g∥ ≲ ∥f∥1∥g∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 3 First-order exponential-type integrator Ψ τ 1 In the following part, we construct the first-order LREI based on the idea in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' FIRST-ORDER LREI FOR GB 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 Homogenization and reformulation of the GB equation As can be seen blow, we will frequently encounter the operator ∂−1 x or ∂−2 x during the process of integration, which makes the mean value of the obtained function zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Hence usually the zero-mode needs to be treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Fortunately, thanks to the periodic boundary conditions, the zero-mode of z can be integrated exactly so that it remains to investigate other nonzero Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' By the periodicity of the solution, one easily gets F0(ztt) = ∂ttF0(z) = 0, which immediately gives F0(z) = at + b, where a and b are defined as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Setting z = F0(z) + ˇz and plugging it into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1), we derive that � ˇztt + ˇzxxxx − (2at + 2b + 1)ˇzxx − (ˇz2)xx = 0, x ∈ T, t > 0, ˇz(0, x) = φ(x) − b, ˇzt(0, x) = ψ(x) − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1) Diagonalize the equivalent first-order system � ˇz ˇzt � t = � 0 1 −∂4 x + ∂2 x 0 � � ˇz zt � + � 0 (ˇz2)xx + (2at + 2b)ˇzxx � , and set ⟨∂2 x⟩ = � ∂2x + ∂4x, u = ˇz − i⟨∂2 x⟩−1ˇzt, v = ˇz − i⟨∂2 x⟩−1ˇzt, we are led to the following coupled system \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 i∂tu = −⟨∂2 x⟩u + B �1 4(u + ¯v)2 + (at + b)(u + ¯v) � , i∂tv = −⟨∂2 x⟩v + B �1 4(¯u + v)2 + (at + b)(¯u + v) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2) where the operator B is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Recalling that z is a real function, this implies u = v and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2) reduces to a single first-order equation involving a complex variable \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 i∂tu = −⟨∂2 x⟩u + B �1 4(u + u)2 + (at + b)(u + u) � , u(0, x) = ˇz(0, x) − i⟨∂2 x⟩−1ˇzt(0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3) While ˇz and ˇzt can be recovered through ˇz = 1 2(u + u), ˇzt = i 2⟨∂2 x⟩(u − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4) Noticing that the leading term of ⟨∂2 x⟩ is −∂2 x, we introduce the so-called twisted variable w(t) = eit∂2 xu(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 12 Hang Li, Chunmei Su* Plugging it into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3) yields ∂tw = iAw − i 4eit∂2 xB(e−it∂2 xw + eit∂2 xw)2 − i(at + b)eit∂2 xB(e−it∂2 xw + eit∂2 xw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5) Applying Duhamel’s formula of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5), we obtain w(tn + σ) = eiσAw(tn) − iB 4 � σ 0 ei(σ−s)Aei(tn+s)∂2 x(g1(w(tn), s))2ds − iB � σ 0 ei(σ−s)Aei(tn+s)∂2 x[a(tn + s) + b]g1(w(tn), s)ds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6) where A and B are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7), and g1(w(tn), s) = e−i(tn+s)∂2 xw(tn + s) + ei(tn+s)∂2 xw(tn + s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Based on this, a first-order approximation can be easily derived [31] ∥w(tn + σ) − w(tn)∥r ≤ Cσ, r > 1/2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7) where C only depends on sup 0≤s≤σ ∥u(tn + s)∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Setting σ = τ and approximating w(tn + s) by w(tn) in the integral of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6), applying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7) and Lemma 4 (i), we get w(tn + τ) = eiτAw(tn) − i 4BeiτA � τ 0 ei(tn+s)∂2 x(g2(w(tn), s))2ds − iBeiτA � τ 0 ei(tn+s)∂2 x(atn + b)g2(w(tn), s)ds + R0(τ 2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8) where g2(w(tn), s) = e−i(tn+s)∂2 xw(tn) + ei(tn+s)∂2 xw(tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Twisting the variable back, we obtain an approximation of u(tn + τ) with a local error of order two u(tn + τ) = eiτ⟨∂2 x⟩u(tn) − i 4Beiτ⟨∂2 x⟩ � τ 0 eis∂2 x(e−is∂2 xu(tn) + eis∂2 xu(tn))2ds − i(atn + b)Beiτ⟨∂2 x⟩ � τ 0 eis∂2 x(e−is∂2 xu(tn) + eis∂2 xu(tn))ds + R0(τ 2) = eiτ⟨∂2 x⟩u(tn) − i 4Bτ� Iτ 0 (u(tn)) + Iτ 1 (u(tn)) + 2Iτ 2 (u(tn)) � − iτ(atn + b)Bτ� u(tn) + ψ1(2iτ∂2 x)u(tn) � + R0(τ 2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='9) where ψ1 is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5) and Bτ(f) = Beiτ⟨∂2 x⟩f = ⟨∂2 x⟩−1∂2 xeiτ⟨∂2 x⟩f, Iτ 0 (f) = � τ 0 eis∂2 x� eis∂2 xf �2ds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='10) Iτ 1 (f) = � τ 0 eis∂2 x� e−is∂2 xf �2ds, Iτ 2 (f) = � τ 0 eis∂2 x��e−is∂2 xf ��2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='11) FIRST-ORDER LREI FOR GB 13 Now we calculate the terms Iτ j in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='9) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Firstly for f satisfying F0(f) = 0, as was shown in [31], Iτ 1 (f) and Iτ 2 (f) can be calculated exactly as Iτ 1 (f) = � k � k1+k2=k � τ 0 e−is(k2−k2 1−k2 2)ds �fk1 �fk2eikx = � � k � k1+k2=k k1̸=0,k2̸=0 e−2iτk1k2 − 1 −2ik1k2 + � k � k1+k2=k k1=0 or k2=0 � τ 0 ds � �fk1 �fk2eikx = � k � k1+k2=k k1̸=0,k2̸=0 e−2iτk1k2 − 1 −2ik1k2 �fk1 �fk2eikx + 2τ �f0 � k∈Z �fkeikx − τ �f 2 0 = i 2 � (∂−1 x f)2 − eiτ∂2 x(e−iτ∂2 x∂−1 x f)2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='12) Iτ 2 (f) = � k1,k2∈Z � τ 0 eis(k2 1−k2 2−(k1−k2)2)ds �fk1 �fk2ei(k1−k2)x = − i 2eiτ∂2 x∂−1 x � (e−iτ∂2 xf)(eiτ∂2 x∂−1 x f) � + i 2∂−1 x � f(∂−1 x f) � + τ∥f∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='13) It remains to calculate Iτ 0 (f) which reads as Iτ 0 (f) = � k∈Z � k1+k2=k � τ 0 e−isΦds�fk1�f k2eikx with Φ = k2 + k2 1 + k2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='14) Different from the above two terms Iτ 1 (f) and Iτ 2 (f), in which the obtained integration is a function with separable variables k, k1 and k2 that enables us to compute the obtained convolution efficiently in physical space or Fourier space, it is impossible to compute the exact integral of Iτ 0 efficiently in any space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' To overcome this difficulty, in [22], we proposed an approximation of Iτ 0 by applying the identity 1 = k1+k2 k and an appropriate approximation which can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' In this paper, we utilize similar idea based on the identity 1 = k2 1 + k2 2 + 2k1k2 k2 , Φ = 2k2 2 + 2kk1 = 2k2 1 + 2kk2 = 2k2 − 2k1k2, and approximate the corresponding integrals in a proper way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2 The first-order exponential-type integrator Ψ τ 1 Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' When k = 0, the mean value of Iτ 0 (f) can be computed exactly and efficiently by F0 (Iτ 0 (f)) = i 2F0 �� eiτ∂2 x∂−1 x f �2� − i 2F0 �� ∂−1 x f �2� + τ ��f0 �2 = T τ 0 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='15) Case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' When k ̸= 0, in order to balance the power of k1 and k2 as much as possible in the following estimations, we use different forms of the phase 14 Hang Li, Chunmei Su* function Φ as 2k2 2 + 2kk1, 2k2 1 + 2kk2, and 2k2 − 2k1k2 for coefficients k2 1 k2 , k2 2 k2 , and 2k1k2 k2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Specifically, � k̸=0 Fk (Iτ 0 (f)) eikx = � k̸=0 � k1+k2=k � τ 0 e−is(k2+k2 1+k2 2)ds�fk1 �f k2eikx = � k̸=0 � k1+k2=k k2 1 k2 � τ 0 e−2is(k2 2+kk1)ds�f k1�f k2eikx + � k̸=0 � k1+k2=k k2 2 k2 � τ 0 e−2is(k2 1+kk2)ds�fk1 �f k2eikx + � k̸=0 � k1+k2=k 2k1k2 k2 � τ 0 e−2is(k2−k1k2)ds�f k1 �f k2eikx = T τ 1 (f) + T τ 2 (f) + T τ 3 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='16) By symmetry, obviously we have T τ 1 (f) = T τ 2 (f) and it suffices to approximate T τ 1 (f) and T τ 3 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' To begin with, we decompose T τ 1 (f) as T τ 1 (f) = � k̸=0 � k1+k2=k k2 1 k2 � τ 0 e−2is(k2 2+kk1)ds�f k1�f k2eikx = � k̸=0 � k1+k2=k k2 1 k2 � τ 0 e−2isk2 2ds�fk1 �f k2eikx + � k̸=0 � k1+k2=k k2 1 k2 � τ 0 � e−2iskk1 − 1 � ds�fk1 �f k2eikx + � k̸=0 � k1+k2=k k2 1 k2 � τ 0 � e−2isk2 2 − 1 � � e−2iskk1 − 1 � ds�fk1 �f k2eikx = Lτ 1(f) + Lτ 2(f) + P τ 1 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='17) For f with F0(f) = 0, similarly Lτ 1(f) and Lτ 2(f) can be integrated exactly as Lτ 1(f) = � k̸=0 � k1+k2=k k2=0 k2 1 k2 � τ 0 e−2isk2 2ds�fk1 �f k2eikx + � k̸=0 � k1+k2=k k2̸=0 k2 1 k2 � τ 0 e−2isk2 2ds�f k1 �fk2eikx = − i 2∂−2 x �� e2iτ∂2 x∂−2 x f � � ∂2 xf �� + i 2∂−2 x �� ∂2 xf � � ∂−2 x f �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='18) Lτ 2(f) = − i 2eiτ∂2 x∂−3 x �� eiτ∂2 x∂xf � � e−iτ∂2 xf �� + i 2∂−3 x �� ∂xf � f � − τ∂−2 x �� ∂2 xf � f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='19) FIRST-ORDER LREI FOR GB 15 The remainder term P τ 1 (f) will be thrown away in the scheme and the estimate is postponed to the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Similarly T τ 3 (f) can be decomposed as T τ 3 (f) = � k̸=0 � k1+k2=k 2k1k2 k2 � τ 0 e−2is(k2−k1k2)ds�f k1 �fk2eikx = � k̸=0 � k1+k2=k 2k1k2 k2 � τ 0 e−2isk2ds�f k1 �f k2eikx + � k̸=0 � k1+k2=k 2k1k2 k2 � τ 0 � e2isk1k2 − 1 � ds�f k1 �f k2eikx + � k̸=0 � k1+k2=k 2k1k2 k2 � τ 0 � e−2isk2 − 1 � � e2isk1k2 − 1 � ds�f k1 �f k2eikx = Lτ 3(f) + Lτ 4(f) + P τ 2 (f), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='20) where Lτ 3(f) and Lτ 4(f) can be integrated exactly as Lτ 3(f) = −i∂−4 x � e2iτ∂2 x − 1 � � ∂xf �2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='21) Lτ 4(f) = i∂−2 x e−iτ∂2 x � eiτ∂2 xf �2 − i∂−2 x � f �2 − 2τ∂−2 x � ∂xf �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='22) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='9), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='20), we obtain u(tn + τ) = Ψ τ 1 (u(tn)) − i 2BτP τ 1 (u(tn)) − i 4BτP τ 2 (u(tn)) + R0(τ 2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='23) where Ψ τ 1 (f) = eiτ⟨∂2 x⟩f − i 4Bτ� T τ 0 (f) + 2Lτ 1(f) + 2Lτ 2(f) + Lτ 3(f) + Lτ 4(f) + Iτ 1 (f) + 2Iτ 2 (f) � − iτ(atn + b)Bτ� f + ψ1(2iτ∂2 x)f � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='24) with operators Bτ, Iτ 1 , Iτ 2 , T τ 0 , Lτ 1, Lτ 2, Lτ 3, Lτ 4 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='15), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='18), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='19), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='21), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='22) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Furthermore, notic- ing BτT τ 0 (f) = 0, we can rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='24) simply as Ψ τ 1 (f) = eiτ⟨∂2 x⟩f − i 4Bτ� 2Lτ 1(f) + 2Lτ 2(f) + Lτ 3(f) + Lτ 4(f) + Iτ 1 (f) + 2Iτ 2 (f) � − iτ(atn + b)Bτ� f + ψ1(2iτ∂2 x)f � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='25) which is exactly (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8) when Iτ j and Lτ j are plugged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4) and z = F0(z) + ˇz, now we are able to propose the scheme zn = 1 2(un + un) + atn + b, zn t = i 2⟨∂2 x⟩(un − un) + a, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='26) where un+1 = Ψ τ 1 (un), n ≥ 0, u0 = u(0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='27) The proposed scheme is fully explicit in time and it is easy to implement efficiently if pseudospectral method is used for spatial discretization thanks to FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 16 Hang Li, Chunmei Su* 4 Error estimates In this section, we will establish the global error estimate concerning the first- order scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 Local error estimate In this part, we give the local error estimate of the scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Inspired by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='23) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6), it remains to estimate P τ 1 (f) and P τ 2 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Lemma 6 For r ≥ 1, f ∈ Hr+p(r), it holds ∥P τ 1 (f)∥r ≲ τ2∥f∥2 r+p(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Proof Firstly the k-th Fourier coefficient of P τ 1 (f) can be bounded as |Fk (P τ 1 (f))| ≲ � k1+k2=k |k|−2|k1|2 ���� � τ 0 � e−2isk2 2 − 1 � � e−2iskk1 − 1 � ds ���� ����f k1 ��� ����fk2 ��� ≲ τ � k1+k2=k |k|−2|k1|2 sup 0≤s≤τ ���� e−2isk2 2 − 1 ��� ��� e−2iskk1 − 1 ��� � ����fk1 ��� ����f k2 ��� ≲ τ 1+α+β � k1+k2=k |k|−2|k1|2 |k2|2α |k|β |k1|β ����f k1 ��� ����f k2 ��� ≲ τ 1+α+β|k|−2+β � k1+k2=k |k1|2+β |k2|2α ����fk1 ��� ����f k2 ��� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1) where α, β ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' This gives ∥P τ 1 (f)∥r ≲ ���τ 1+α+β � k̸=0 |k|−2+β � k1+k2=k |k1|2+β |k2|2α ����fk1 ��� ����f k2 ��� eikx��� r ≲ τ1+α+β ���|∂x|−2+β �� |∂x|2+β �f � � |∂x|2α �f ����� r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2) Now we give several estimates for P τ 1 (f) which might be valid in different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (1) For r ≥ 1, setting α = 1 and β = 0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2), applying the inequalities in Lemma 5 yields ∥P τ 1 (f)∥r ≲ τ 2 ���|∂x|−2�� |∂x|2 �f �� |∂x|2 �f ����� r ≲ τ2�� |∂x| �f ��2 r = τ 2∥f∥2 r+1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3) which implies a second-order local error by requiring one additional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2) By applying Lemma 4 (iv), for r + β − 1 > 1/2, we get ∥P τ 1 (f)∥r ≲ τ1+α+β ���|∂x|−1 �� |∂x|2+β �f � � |∂x|2α �f ����� r+β−1 FIRST-ORDER LREI FOR GB 17 ≲ τ1+α+β ��� � |∂x|1+β �f ���� r+β−1 ��� � |∂x|2α �f ���� r+β−1 ≲ τ1+α+β∥f∥r+2β∥f∥r+2α+β−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' To get a local error bound of order two, setting α + β = 1, one obtains ∥P τ 1 (f)∥r ≲ τ 2∥f∥2 r+2β, with β ∈ [1/3, 1/2], r > 3/2 − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4) We clearly see that compared to the estimate in (1), this decreases the addi- tional regularity required when r > 1 and the least order of additional regular- ity can be decreased to 2/3 which is valid when r > 7/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' On the other hand, w observe that it is possible to require less additional regularity by choosing smaller β, however, we have to pay extra price that the error itself is estimated in a much more regular space by noticing the constraint r > 3/2 − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (3) When r +β −2 ∈ [0, 1/2), by applying Lemma 2 and H¨older inequality, one gets ∥P τ 1 (f)∥r ≲ τ 1+α+β ��� � |∂x|2+β �f � � |∂x|2α �f ���� r+β−2 ≲ τ 1+α+β ��� � |∂x|2+β �f � � |∂x|2α �f ���� L 2 5−2r−2β ≲ τ 1+α+β ��� � |∂x|2+β �f ���� L 4 5−2r−2β ��� � |∂x|2α �f ���� L 4 5−2r−2β ≲ τ 1+α+β ∥f∥ 2r+2β−3 4 +2+β ∥f∥ 2r+2β−3 4 +2α ≲ τ 1+α+β ∥f∥2 r 2 + 3 2 β+ 5 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Similarly setting α + β = 1, one derives ∥P τ 1 (f)∥r ≲ τ 2 ∥f∥2 r 2 + 3 2 β+ 5 4 , r ∈ (3/2 − β, 2 − β], β ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5) (4) On the other hand, we can estimate P τ 1 (f) by employing the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4) in Lemma 1, by setting β = 0, α = 1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1), ∥P τ 1 (f)∥r ≲ τ 2 ���Jr−2 � |∂x|2 �f � � |∂x|2 �f ���� ≲ τ 2 ��� � Jr�f ���� Lp1 ��� � |∂x|2 �f ���� Lp2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6) where 2 ≤ p1 < ∞, 2 < p2 ≤ ∞ and 1 p1 + 1 p2 = 1 2, when r > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Applying the Sobolev embedding theorem in Lemma 2, we get ∥P τ 1 (f)∥r ≲ τ 2∥f∥r− 1 p1 + 1 2 ∥f∥ 5 2 − 1 p2 , 2 < p1, p2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' To obtain a lower spatial regularity requirement, it is natural to choose r− 1 p1 + 1 2 = 5 2 − 1 p2 for 1 p1 + 1 p2 = 1 2 with p1 ∈ (2, ∞) and p2 ∈ (2, ∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', p1 = 1 r 2 − 3 4 and p2 = 1 5 4 − r 2 with r ∈ (2, 5/2), which yields the local error estimate as ∥P τ 1 (f)∥r ≲ τ 2∥f∥2 r 2 + 5 4 , r ∈ (2, 5/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7) 18 Hang Li, Chunmei Su* (5) Finally, using the bilinear inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3) in Lemma 1, for α + β = 1, one has ∥P τ 1 (f)∥r ≲ τ 1+α+β ��� � |∂x|2+β �f � � |∂x|2α �f ���� r+β−2 ≲ τ 2 ��� � |∂x|2+β �f ���� r+β−2 ��� � |∂x|2α �f ���� r+β−2 ≲ τ 2∥f∥r+2β∥f∥r+2α+β−2 ≲ τ 2∥f∥2 r+2β, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8) for r > 5/2−β with 0 ≤ β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' This implies an error without loss of regularity when r > 5/2 by choosing β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Taking β = 0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5), one gets ∥P τ 1 (f)∥r ≲ τ 2 ∥f∥2 r 2 + 5 4 , r ∈ (3/2, 2], which combines with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8) gives ∥P τ 1 (f)∥r ≲ τ 2 ∥f∥2 r 2 + 5 4 , r ∈ (3/2, 5/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' ∥P τ 1 (f)∥r ≲ τ 2 ∥f∥2 r+ , r = 5/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' ∥P τ 1 (f)∥r ≲ τ 2 ∥f∥2 r , r > 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='9) For r ≤ 3 2, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5), we have to choose β = ( 3 2 − r)+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', β = 3 2 − r + ε with any sufficiently small ε > 0 which reads as ∥P τ 1 (f)∥r ≲ τ2 ∥f∥2 r 2 + 5 4 + 3 2 ( 3 2 −r)+ = ∥f∥2 ( 7 2 −r)+ , r ∈ [1, 3/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='10) Similarly (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4) equivalents to the estimate ∥P τ 1 (f)∥r ≲ τ2 ∥f∥2 (3−r)+ , r ∈ (1, 7 6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' ∥P τ 1 (f)∥r ≲ τ2 ∥f∥2 r+ 2 3 , r > 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='11) Lemma 6 is concluded by taking the minimum of the required order of regu- larity for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='9), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Concerning the term P τ 2 (f), we have the following estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Lemma 7 For r ≥ 1, we have ∥P τ 2 (f)∥r ≲ τ2∥f∥2 r+q(r), where q(r) = � 5/4 − r/2, 1 ≤ r ≤ 5/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 0, r > 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' FIRST-ORDER LREI FOR GB 19 Proof It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='20) that |Fk (P τ 2 (f))| ≲ � k1+k2=k |k|−2|k1||k2| ���� � τ 0 � e−2isk2 − 1 � � e2isk1k2 − 1 � ds ���� ����f k1 ��� ����fk2 ��� ≲ τ � k1+k2=k |k|−2|k1||k2| sup 0≤s≤τ ���� e−2isk2 − 1 ��� ��� e2isk1k2 − 1 ��� � ����f k1 ��� ����fk2 ��� ≲ τ 1+α+β � k1+k2=k |k|−2|k1||k2| |k|2α |k1|β |k2|β ����f k1 ��� ����f k2 ��� ≲ τ 2|k|−2+2α � k1+k2=k |k1|1+β |k2|1+β ����fk1 ��� ����f k2 ��� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='12) for α, β ∈ [0, 1] satisfying α + β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Using similar approach applied in the proof of Lemma 6, we establish several estimates by applying various tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (1) When r + 2α − 2 ∈ (−1/2, 0], by applying Lemma 2 and H¨older in- equality, one gets ∥P τ 2 (f)∥r ≲ τ2 ��� � |∂x|1+β �f � � |∂x|1+β �f ���� r+2α−2 ≲ τ2 ��� � |∂x|1+β �f � � |∂x|1+β �f ���� L 2 5−2r−4α ≲ τ2 ��� � |∂x|1+β �f ���� 2 L 4 5−2r−4α ≲ τ2 ∥f∥2 r 2 + 5 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Noticing the constraint r ∈ (3/2 − 2α, 2 − 2α] and α ∈ [0, 1], we immediately get ∥P τ 2 (f)∥r ≲ τ 2 ∥f∥2 r 2 + 5 4 , r ∈ [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='13) (2) Setting α = 0 and β = 1, one gets ∥P τ 2 (f)∥r ≲ τ 2 ���|∂x|−2 � |∂x|2 �f � � |∂x|2 �f ���� r ≲ τ2 ���Jr−2 � |∂x|2 �f � � |∂x|2 �f ���� , which is exactly the same as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='7) also holds for P τ 2 (f), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', ∥P τ 2 (f)∥r ≲ τ 2∥f∥2 r 2 + 5 4 , r ∈ (2, 5/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='14) (3) It remains to give a bound of ∥P τ 2 (f)∥r for r > 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Employing the bilinear estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3), one easily gets ∥P τ 2 (f)∥r ≲ τ 2 ��� � |∂x|1+β �f � � |∂x|1+β �f ���� r+2α−2 ≲ τ 2 ��� � |∂x|1+β �f ���� 2 r+2α−2 ≲ τ 2∥f∥2 r+α, for r > 5/2 − 2α, 20 Hang Li, Chunmei Su* which implies ∥P τ 2 (f)∥r ≲ τ2∥f∥2 r, for r > 5/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' ∥P τ 2 (f)∥r ≲ τ2∥f∥2 ( r 2 + 5 4 )+, r ∈ [1, 5/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' This together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='14) concludes Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' It is easy to see q(r) ≤ p(r) by direct computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' In spirit of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='23), Lemma 6 and Lemma 7, we obtain the local error of the scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Lemma 8 Suppose r ≥ 1, u ∈ L∞(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Hr+p(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Then we have ∥u(tn + τ) − Ψ τ 1 (u(tn))∥r ≤ Lτ 2, where L depends on ∥u∥L∞(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='Hr+p(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2 Stability Lemma 9 Suppose r ≥ 1 and f, g ∈ Hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Then for τ > 0, we have ∥Ψ τ 1 (f) − Ψ τ 1 (g)∥r ≤ (1 + Mτ)∥f − g∥r, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='15) where M depends on r and ∥f∥r + ∥g∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Proof To begin with, by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='11), one can easily check that ∥Iτ 1 (f) − Iτ 1 (g)∥r ≤ Cτ sup τ≥s≥0 ���eis∂2 x � (e−is∂2 xf)2 − (e−is∂2 xg)2���� r ≤ Cτ sup τ≥s≥0 ∥e−is∂2 x(f + g)∥r∥e−is∂2 x(f − g)∥r ≤ Cτ(∥f∥r + ∥g∥r)∥f − g∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='16) Similar discussions for Iτ 1 (f) and Iτ 2 (f) yield that ∥Iτ 2 (f) − Iτ 2 (g)∥r ≤ Cτ(∥f∥r + ∥g∥r)∥f − g∥r, ∥Iτ 0 (f) − Iτ 0 (g)∥r ≤ Cτ(∥f∥r + ∥g∥r)∥f − g∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='17) Noticing the decomposition of T τ i (f) (i = 1, 2, 3) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='15), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='20), we have Iτ 0 (f) = T τ 0 (f) + 2Lτ 1(f) + 2Lτ 2(f) + 2P τ 1 (f) + Lτ 3(f) + Lτ 4(f) + P τ 2 (f), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='18) which yields ∥W τ(f) − W τ(g)∥r = ∥Iτ 0 (f) − Iτ 0 (g) − 2P τ 1 (f) + 2P τ 1 (g) − P τ 2 (f) + P τ 2 (g)∥r ≤ ∥Iτ 0 (f) − Iτ 0 (g)∥r + 2∥P τ 1 (f) − P τ 1 (g)∥r + ∥P τ 2 (f) − P τ 2 (g)∥r, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='19) where W τ(f) := T τ 0 (f) + 2Lτ 1(f) + 2Lτ 2(f) + Lτ 3(f) + Lτ 4(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' FIRST-ORDER LREI FOR GB 21 It remains to deal with the terms P τ 1 (f) and P τ 2 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' According to Lemmas 1, 4, 5 and the definition of P τ 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' for r ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='∥P τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 (f) − P τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 (g)∥r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='≲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k̸=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k1+k2=k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e−2isk2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='e−2iskk1 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='��f k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='�fk2 − �gk1�gk2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='eikx��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='≲ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k̸=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k1+k2=k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k−2k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='����f k1�fk2 − �f k1�gk2 + �fk1�gk2 − �gk1�gk2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='��� eikx��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='≲ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k̸=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k1+k2=k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='k−2k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='����f k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='����fk2 − �gk2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='��� + ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='����∂−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x( � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f − g)�g ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='(∂x�f)( � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f − g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='− ∂−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='(∂x�f)∂x( � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f − g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='����∂−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='∂x( � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f − g)�g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='− ∂−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='∂x( � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='f − g)∂x�g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='≲ τ(∥f∥r + ∥g∥r)∥f − g∥r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='20) where we have used the modified version of Newton-Leibniz formula ∂−2 x [(∂2 xf)g] = ∂−1 x [(∂xf)g] − ∂−2 x [(∂xf)(∂xg)], which can be obviously derived by the decomposition k−2k2 1 = k−2k1(k − k2) = k−1k1 − k−2k1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Recalling the definition of P τ 2 (f) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='20), we can establish ∥P τ 2 (f) − P τ 2 (g)∥r ≲ τ(∥f∥r + ∥g∥r)∥f − g∥r, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='21) by employing similar arguments as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='16)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='21), applying Lemma 1 and Lemma 4, for r ≥ 1, we have ∥Ψ τ 1 (f) − Ψ τ 1 (g)∥r = ����eiτ⟨∂2 x⟩(f − g) − i 4Bτ� W τ(f) − W τ(g) + Iτ 1 (f) − Iτ 1 (g) + 2 (Iτ 2 (f) − Iτ 2 (g)) � − iτ(atn + b)Bτ� f − g + ψ1(2iτ∂2 x)(f − g) ����� r ≤ ∥f − g∥r + ���W τ(f) − W τ(g) �� r + ∥Iτ 1 (f) − Iτ 1 (g)∥r + 2 ∥Iτ 2 (f) − Iτ 2 (g)∥r � + Crτ(∥f − g∥r + ∥f − g∥r) ≤ (1 + Mτ)∥f − g∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='22) where M depends on r and ∥f∥r + ∥g∥r and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 22 Hang Li, Chunmei Su* 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='3 Proof of Theorem 1 Proof In spirit of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='27), it suffices to show ∥u(tn) − un∥r ≤ Cτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Combining the local error estimate in Lemma 8 and stability inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='15), we are led to ∥u(tn+1) − un+1∥r ≤ ∥u(tn+1) − Ψ τ 1 (u(tn))∥r + ∥Ψ τ 1 (u(tn)) − Ψ τ 1 (un)∥r ≤ Lτ 2 + ∥Ψ τ 1 (u(tn)) − Ψ τ 1 (un)∥r ≤ Lτ 2 + eτM∥u(tn) − un∥r, where L depends on ∥u∥L∞(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='Hr+p(r)) (or equivalently ∥z∥L∞(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='Hr+p(r)) + ∥zt∥L∞(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='Hr+p(r)−2)), and M depends on ∥u(tn)∥r and ∥un∥r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Then the as- sertion follows by a standard induction argument [18,30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 5 Numerical experiments In this section, we present some numerical experiments of the newly proposed first-order LREI Ψ τ 1 to justify our theoretical convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The numer- ical investigations of convergence of the first-order LREIs in [22,31] will be provided as comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Furthermore, the Fourier pseudospectral method is used for spatial discretization so that each iteration can be calculated by FFT via O(MlogM) operations, where M represents the number of grid points in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' We choose the spatial mesh size ∆x = 1/26 for soliton solutions and ∆x = π/215 for rough solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Moreover, we define the error in Hr as ∥z(tn) − zn∥r + ∥∂tz(tn) − zn t ∥r−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1 Soliton solutions In the first experiment, we numerically verify the temporal convergence of the first-order LREI Ψ τ 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='27) for the soliton solution [24] of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1) z(x, t) = −Asech2[(ω/2)(x − vt + ζ0)], (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1) where ζ0 ∈ R, 0 < ω ≤ 1, and the relations among the amplitude A, velocity v and frequency ω are given as follows A = 3ω2/2, v = ±(1 − ω2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' It can be clearly observed that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1) decays exponentially at far field, which enables us to impose the periodic boundary conditions on a bounded domain [−x0, x0] when x0 is chosen large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 2 shows the error in H2, for the numerical solution obtained by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='27) at tn = 1, where x0 = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 2 we can see that the scheme converges at the first order in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' FIRST-ORDER LREI FOR GB 23 10-3 10-2 10-1 10-6 10-5 10-4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 2 Linear convergence of the LREI scheme Ψτ 1 for the soliton solution with ζ0 = 0, ω = 1/2, v = √ 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Here we select the spatial mesh size as ∆x = 1/26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2 Rough solutions In the second experiment, we apply the first-order LREI Ψ τ 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='27) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='1) under nonsmooth initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' We numerically compare the results with those of the first-order LREIs in [22] and [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Following the construction method of the initial data with desired regular- ity in [31], we choose the spatial mesh size ∆x = 2π/M with M = 216 and the grid points xj = −π + j∆x, 0 ≤ j < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Moreover, a uniformly distributed random vector rand(1, M) can be taken in the computer, which is denoted by Z = (z0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' , zM−1) = rand(1, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Then we define the inverse derivative operator |∂x,M|−θ as a truncation of the operator |∂x|−θ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2), which maps a function f ∈ L2(T) to Hθ(T) |∂x,M|−θf = M/2−1 � k=−M/2,k̸=0 |k|−θ �fkeikx, θ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Then we define Z0 = Z1 + c ∗ ∥Z1∥∞ ∥Z1 + c ∗ ∥Z1∥∞∥ , where c := rand(1) is a random number and Z1 := |∂x,M|−θZ, x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Finally, we get Z0 ∈ Hθ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' For instance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 3 displays the initial data obtained as above for θ = 2 and θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 4-6 show the errors of the scheme Ψ τ 1 and those in [22,31] in Hr with various r at the final time tn = T = 1 for different rough initial data, where the reference solution is obtained by the first-order LREI Ψ τ 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='27) with a tiny time step τ = 3 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Specifically, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 4–6 display the temporal errors of the three schemes when the given initial data has additional order of regularity 0, 1/4, 1/2, 2/3 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' From the numerical results shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 4 to 6, it can be clearly observed that: 24 Hang Li, Chunmei Su* 3 2 1 0 1 2 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 3 Left: initial value z0 ∈ H2 with θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Right: initial value z(0, x) ∈ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5 with θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (1) The newly developed first-order LREI scheme in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='27) has first-order convergence in all cases, which demonstrates the theoretical results pre- sented in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' (2) Compared to the other two schemes in [22,31], the newly proposed scheme behaves most regularly and the oscillations are the weakest while the method in [31] behaves most irregularly and might suffer an order re- duction (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Furthermore, the method presented in this paper is the most accurate when the time step is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' This shows the superiority of the newly proposed method (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 10-3 10-2 10-1 10-3 10-2 10-1 100 10-3 10-2 10-1 10-3 10-2 10-1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 4 Numerical errors in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5 (left) and H2 (right) of three first-order schemes at the final time T = 1 with rough solution in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5 and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' FIRST-ORDER LREI FOR GB 25 10-3 10-2 10-1 10-4 10-3 10-2 10-3 10-2 10-1 10-4 10-3 10-2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 5 Numerical errors in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content='5 and H17/12 of three first-order schemes at the final time T = 1 with rough solution in H2 and H25/12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 10-3 10-2 10-1 10-4 10-3 10-2 10-3 10-2 10-1 10-4 10-3 10-2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 6 Numerical errors in H7/6 and H1 of the three first-order schemes at the final time T = 1 with rough solution in H11/6 and H2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 6 Conclusions In this work, we developed a new first-order low regularity exponential-type integrator for the “good” Boussinesq equation with rough initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The method is based on a twisted variable and the phase space analysis of the nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' By applying the Kato-Ponce inequalities, the Hardy- Littlewood-Sobolev type inequality and Sobolev embedding theorem, we es- tablished the linear convergence in Hr with solutions in Hr+p(r) for r ≥ 1, where p(r) is non-increasing with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Particularly, the first-order ac- curacy can be achieved in Hr for solutions in Hr when r ≥ 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' This is the lowest regularity requirement of the existing methods for the GB equation so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' The analytical result is supported by extensive numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Acknowledgements This work was supported by the NSFC 12201342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 26 Hang Li, Chunmei Su* Data Availability Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Declarations Conflict of interest The author declares no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Adams, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=', Fournier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' J.' metadata={'source': 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second order operator splitting numerical scheme for the “good” Boussinesq equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} +page_content=' 119, 179– 193 (2017)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf'} diff --git a/yNFQT4oBgHgl3EQfxzYK/content/tmp_files/2301.13406v1.pdf.txt b/yNFQT4oBgHgl3EQfxzYK/content/tmp_files/2301.13406v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..57a85c35b18e949d07d180234dabe8bb3efe43da --- /dev/null +++ b/yNFQT4oBgHgl3EQfxzYK/content/tmp_files/2301.13406v1.pdf.txt @@ -0,0 +1,2306 @@ +arXiv:2301.13406v1 [math.LO] 31 Jan 2023 +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +Abstract. We study varieties generated by semi-primal lattice-expansions +by means of category theory. We provide a new proof of the Keimel-Werner +topological duality for such varieties and, using similar methods, establish +its discrete version. We describe multiple adjunctions between the variety of +Boolean algebras and the variety generated by a semi-primal lattice-expansion, +both on the topological side and explicitly algebraic. In particular, we show +that the Boolean skeleton functor has two adjoints, both defined by taking +certain Boolean powers, and we identify properties of these adjunctions which +fully characterize semi-primality of an algebra. Lastly, we give a new charac- +terization of canonical extensions of algebras in semi-primal varieties in terms +of their Boolean skeletons. +1. Introduction +Primality and its variations are classical topics in universal algebra which were +prominently studied during the second half of the 20th century [54, 58, 10]. During +the 1950s, Foster introduced primal algebras in his generalized ‘Boolean’ theory +of universal algebras [24, 25]. +Generalizing functional completeness of the two- +element Boolean algebra, an algebra P is primal if every operation f : P n → P is +term-definable in P. The intuition that a primal algebra P is ‘close to’ the two- +element Boolean algebra 2 was confirmed by Hu’s theorem [36, 37], which states +that a variety V is categorically equivalent to the variety BA of Boolean algebras +(generated by 2) if and only if V is generated by a primal algebra P ∈ V. +In 1964, Foster and Pixley introduced the first variation of primality, which +they called semi-primality [29]. Unlike primal algebras, a semi-primal algebra may +have proper subalgebras. Accordingly, in a semi-primal algebra L, we only require +the operations f : Ln → L which preserve subalgebras to be term-definable in L. +Semi-primal varieties (that is, varieties of the form HSP(L) where L is semi-primal) +are well-understood from the viewpoint of ‘classical’ universal algebraic structure +theory [29, 30, 26, 45] as well as from the viewpoint of duality theory [41, 17]. From +the perspective of category theory, semi-primal varieties were classified up to Morita +equivalence in [7] - however, this is done using purely algebraic tools based on [44]. +In this paper, we further advance the category theoretical study of semi-primality +by putting a semi-primal variety A in relationship with other varieties, in particular +with the primal variety BA. Although Hu’s theorem implies that the varieties are +usually not categorically equivalent, we demonstrate that, nevertheless, there is a +rich relationship between A and BA. In particular we explicate the intuition that +semi-primal algebras are still ‘close to’ the two-element Boolean algebra. +2020 Mathematics Subject Classification. 06E15, 06E75, 08A40, 08C05. +Key words and phrases. semi-primal algebras, primal algebras, ternary discriminator, stone +duality, boolean skeleton, boolean power, canonical extension, universal algebra, category theory. +1 + +2 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +More specifically, we investigate multiple adjunctions between BA and the variety +A generated by a semi-primal algebra L with an underlying bounded lattice (see +Assumption 2.9). For one, this assumption yields a useful characterization of semi- +primality via certain unary terms (see Proposition 2.8) which we prominently use. +Furthermore, since L has no one-element subalgebras, the dual category of A has a +particularly simple description (see Definition 3.1). Apart from these advantages, +the restriction to lattice-based algebras is motivated by the connection to many- +valued logic. +If we consider L as an algebra of propositional truth-degrees, an +underlying bounded lattice is a reasonable assumption. For example, Maruyama +[43] generalized J´onsson-Tarski duality to modal extensions of semi-primal algebras +with bounded lattice reducts. We plan to demonstrate applications of our results +to many-valued (coalgebraic) modal logic in subsequent work. Since, in addition, +there are already plenty of examples of such algebras (see Subsection 2.3), it is +reasonable to stick to this framework. +Although we were mainly motivated by questions arising in logic, we particularly +hope that this paper will be of interest to algebraists interested in category theory as +well as to category theorists interested in universal algebra. Let us point out that, +in this paper, the category theoretical approach to universal algebra is different +from other common ones via Lawvere theories or monads (these are well-exposed +in [38]). Indeed, this paper is not about reformulating and generalizing algebraic +concepts into categorical language, but rather to apply category theory as a tool to +gain new insight into a concrete topic in universal algebra. For example, the fact +that the variety A is the completion of the full subcategory of its finite members +Aω under filtered colimits (i.e., A ≃ Ind(Aω)) can be helpful to make the step from +finite to infinite, for example to extend functors defined on Aω to the full variety A +in a canonical way. Motivated by [40], we furthermore use this fact to give a new +proof of the semi-primal duality [41, 17] by lifting the corresponding finite duality +(see Theorem 3.7 and Theorem 5.5). By replacing Ind(Aω) by Pro(Aω), the closure +under cofiltered limits, we prove the discrete version of the duality (resembling the +duality between Set and the category CABA of complete atomic Boolean algebras) +in a similar manner. +The paper is organized as follows. In Section 2 we recall well-known results about +semi-primal algebras and the varieties they generate. In particular, we discuss semi- +primal bounded-lattice expansions and provide examples thereof. In Section 3 we +describe the topological duality for semi-primal algebras and, as mentioned above, +provide an alternative proof for it. Arguably the most important results of the paper +are exposed in Section 4, where we describe a chain of four adjoint functors between +A and BA (see Figure 3). Most prominently, the adjunction S ⊣ P is described +in detail, first via duality and then explicitly algebraically (see Theorem 4.11). +The Boolean skeleton S: A → BA has, for example, been known for MVn-algebras +[16] and was generalized to arbitrary semi-primal bounded lattice expansions by +Maruyama [43]. Its right-adjoint P: BA → A relies on the construction of a Boolean +power [9], a certain Boolean product [11] which was already introduced for arbitrary +finite algebras in Foster’s original paper on primality [24]. In the case where L is +primal, we retrieve a concrete categorical equivalence witnessing Hu’s theorem (see +Corollary 4.12). We proceed to investigate the subalgebra adjunctions, which exist +for each subalgebra S ≤ L. +We manage to trace them back to the adjunction +S ⊣ P after taking an appropriate inclusion/quotient (see Theorem 4.16). +In + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +3 +particular, we illustrate why the subalgebra adjunction Q ⊣ I corresponding to the +smallest subalgebra of L is of special interest. Indeed, towards the end of Section +4 we also show that the existence of an adjoint situation resembling I ⊣ S ⊣ P +fully characterizes semi-primality of a lattice-based algebra (see Theorem 4.19). +Building on the results of Section 4, in Section 5 we prove the above-mentioned +discrete duality for Pro(Aω). It is well-known that the algebras in this category +correspond to the canonical extensions [33, 21] of algebras in A. Notably, we show +that these canonical extensions may be characterized almost purely in terms of +their Boolean skeletons (see Theorem 5.10). Lastly we connect Sections 4 and 5 +by describing an analogue of the Stone-ˇCech compactification in our setting (see +Proposition 5.11). +We summarize our results schematically in Section 6 (see Figure 6). In addi- +tion to the logical ramifications already mentioned, we believe that there are more +potential ways to follow up our results. In particular, we hope to inspire further +research in universal algebra through the lens of category-theory. Some open ques- +tions directly related to the content of this paper are also collected in Section 6. +2. Semi-primal algebras and the varieties they generate +In the 1950s, Foster introduced the concept of primality in [24, 25], generalizing +functional completeness of the two-element Boolean algebra 2. A finite algebra L +is called primal if, for all n ≥ 1, every function f : Ln → L is term-definable in L. +Besides the two-element Boolean algebra 2, the (n + 1)-element Post chain Pn and +the field of prime order Z/pZ with 0 and 1 as constants are some famous examples +of primal algebras. +Using Stone duality, Hu [36, 37] showed that a variety A is generated by a primal +algebra (in other words, A = HSP(L) for some primal algebra L) if and only if A +is categorically equivalent to the variety of Boolean algebras BA (see also [51] for a +treatment using Lawvere theories). Of course we don’t expect any more meaningful +category theoretical results about the relationship between A and BA in this case. +One purpose of this paper is to demonstrate that, in contrast, such results do arise +as soon as we assume that L is semi-primal. +2.1. Characterizations of semi-primality. Since Foster’s original work, many +variations of primality have been introduced (for an overview see, e.g., [54]). Among +them, intuitively speaking, semi-primality seems to still be rather close to primality +(a central theme of this paper is to show why this intuition is justified). In a slogan: +semi-primal algebras are like primal algebras which allow subalgebras. +Note that a primal algebra L does not have any proper subalgebra S ≨ L. +Otherwise, picking any s ∈ S and ℓ ∈ L\S, no function f : L → L with f(s) = ℓ +can possibly be term-definable. +Semi-primality, introduced by Foster and Pixley in 1964 (see [29]) does not +impose this restriction. Recall that a function f : Ln → L preserves subalgebras +if f(a1, . . . , an) is in the subalgebra generated by {a1, . . . , an} for any choice of +a1, . . . , an ∈ L. Clearly, if a function is term-definable, then it preserves subalge- +bras. In semi-primal algebras, the converse also holds. +Definition 2.1. A finite algebra L is semi-primal if for every n ≥ 1, every function +f : Ln → L which preserves subalgebras is term-definable in L. + +4 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +For example, the field of prime-order Z/pZ with only 0 as constant is semi-primal +but not primal anymore - it now has {0} as proper subalgebra. More interesting +examples are described in detail in Subsection 2.3. In the following we recall two +well-known equivalent characterizations of semi-primality. The first one is based +on the ternary discriminator term and the second one is based on the existence of +a majority term. +First we recall the characterization of semi-primal algebras as special instances of +discriminator algebras. These are the algebras in which the ternary discriminator +t(x, y, z) = +� +z +if x = y +x +if x ̸= y +is term-definable. Finite discriminator algebras are also called quasi-primal. +An internal isomorphism of L is an isomorphism ϕ: S1 → S2 between any two +(not necessarily distinct) subalgebras S1 and S2 of L. For example, if S ≤ L is a +subalgebra, then the identity idS is an internal isomorphism of L. In semi-primal +algebras, there are no other internal isomorphisms. +Proposition 2.2. [50, Theorem 3.2.] A finite algebra L is semi-primal if and only +if it is quasi-primal and the only internal isomorphisms of L are the identities on +subalgebras of L. +Secondly, we recall the characterization of semi-primality based on a majority +term, which can be useful to generate examples (see, for example, [22]). Recall that +a majority term is a ternary term m(x, y, z) satisfying +m(x, x, y) = m(x, y, x) = m(y, x, x) = x. +In particular, every lattice L = (L, ∧, ∨) has a majority term given by the median +m(x, y, z) = (x ∧ y) ∨ (x ∧ z) ∨ (y ∧ z). +Proposition 2.3. [3, Theorem 7.2.] A finite algebra L is semi-primal if and only +if it has a majority term and every subalgebra of L2 is either the product of two +subalgebras or the diagonal of a subalgebra of L. +The structure of semi-primal varieties was already well-studied in the original +work by Foster and Pixley during the 1960s. To stay self-contained, we recall some +results about these varieties which will be of use for us later. +Proposition 2.4. [29, Theorem 4.2] The variety A generated by a semi-primal +algebra L coincides with the quasi-variety generated by L, that is A = ISP(L). +In addition to the characterizations above, there is a nice characterization of +semi-primality of L in terms of A. Recall that a variety is called arithmetical if it +is congruence distributive and congruence permutable. +Proposition 2.5. [30, Theorem 3.1] A finite algebra L is semi-primal if and only +if the variety generated by L is arithmetical, every subalgebra of L is simple, and +the only internal isomorphisms of L are the identities of subalgebras. +Remark 1. Together with Proposition 3.5 this implies that if L is semi-primal, +then the collection of subalgebras S(L) considered as a subcategory of the variety +generated by L, forms a lattice. +■ + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +5 +The finite members of A are particularly well-behaved. For notation, given a +concrete category C, we use Cω to denote the full subcategory of C generated by its +finite members. In particular, if A is a variety, we use Aω to denote the category +of finite algebras in A. +Proposition 2.6. [29, Theorem 7.1] Every finite algebra A ∈ Aω is isomorphic to +a direct product of subalgebras of L. +We add yet another characterization of semi-primality in our particular case +of interest (in which the algebra is based on a bounded lattice) in the following +subsection (see Proposition 2.8). +2.2. Semi-primal bounded lattice expansions. In this subsection we set the +scene for the remainder of this paper. We aim to describe the relationship between +the variety BA of Boolean algebras and the variety generated by a semi-primal +algebra with underlying bounded lattice. +Under the additional assumption that L is based on a bounded lattice, there +is another nice characterization of semi-primality of L which will be particularly +useful for our purposes. It relies on the following unary terms. +Definition 2.7. Let L be an algebra based on a bounded lattice L♭ = (L, ∧, ∨, 0, 1). +For all ℓ ∈ L we define Tℓ : L → L and τℓ : L → L to be the characteristic function +of {ℓ} and {ℓ′ ≥ ℓ}, respectively. That is, +Tℓ(x) = +� +1 +if x = ℓ +0 +if x ̸= ℓ +and +τℓ(x) = +� +1 +if x ≥ ℓ +0 +if x ̸≥ ℓ. +Even though the following result is essentially an instance of the more general +[26, Theorem 4.1], we include an easy direct proof here. +Proposition 2.8. [26, Theorem 4.1] Let L be a finite algebra with an underlying +bounded lattice. Then the following conditions are equivalent: +(1) L is semi-primal. +(2) For every ℓ ∈ L, the function Tℓ is term-definable in L. +(3) T0 is term-definable and for every ℓ ∈ L, the function τℓ is term-definable +in L. +Proof. (1) ⇒ (2): Since every subalgebra of L contains the set {0, 1}, semi-primality +of L implies that all Tℓ are term-definable, since they preserve subalgebras. +(2) ⇒ (1): First we show that the ternary discriminator is term-definable in L. +Consider the term +c(x, y) = +� +ℓ∈L +� +(Tℓ(x) ∧ Tℓ(y) +� +, +which satisfies +c(x, y) = +� +1 +if x = y +0 +if x ̸= y +and d(x, y) := T0(c(x, y)) (note that this is the discrete metric). The term +t(x, y, z) = (d(x, y) ∧ x) ∨ (c(x, y) ∧ z) +yields the ternary discriminator on L. Now we show that the only internal iso- +morphisms of L are the identities of subalgebras. Let ϕ : S1 → S2 be an internal + +6 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +isomorphism of L and let s ∈ S1 be arbitrary. Then +1 = Tϕ(s) +� +ϕ(s) +� += ϕ +� +Tϕ(s)(s) +� +Since ϕ(0) = 0 we necessarily have Tϕ(s)(s) = 1, which is equivalent to ϕ(s) = s. +Altogether, due to Proposition 2.2, we showed that L is semi-primal. +(2) ⇒ (3): If the Tℓ are term-definable we can define +τℓ(x) = +� +ℓ′≥ℓ +Tℓ′(x). +(3) ⇒ (2): If T0 and the τℓ are term-definable we can define +Tℓ(x) = τℓ(x) ∧ +� +ℓ′>ℓ +T0 +� +τℓ′(x) +� +, +which concludes the proof. +□ +Remark 2. In light of this result, we can turn any finite bounded lattice into a semi- +primal algebra by adding Tℓ as unary operation for every element ℓ ∈ L. One might +wonder how this differs from adding a constant symbol (i.e., a nullary operation) +for every element. The difference is that adding a constant imposes the requirement +that every subalgebra needs to contain the element corresponding to this constant. +Thus, the algebra that results after adding all constants does not have any proper +subalgebras. +■ +We now state our main assumption, which from now on holds for the remainder +of this paper. +Assumption 2.9. The finite algebra L is semi-primal and has an under- +lying bounded lattice. +From now on, let A := HSP(L) denote the variety generated by L. In Subsection +2.3 we provide various examples of algebras satisfying Assumption 2.9. +As noted in [43] (where the same assumption on L is made), from the point of +view of many-valued logic, semi-primal algebras make good candidates for algebras +of truth-values. In this context the underlying bounded lattice is a natural minimal +requirement. +2.3. Examples of semi-primal algebras. In this subsection we collect some ex- +amples of semi-primal algebras. All of them are bounded lattice expansions (since +most of them stem from many-valued logic), thus they all fit the scope of this paper +(see Assumption 2.9). For other examples we refer the reader to [10, 58, 45]. +First, we describe several different semi-primal algebras based on finite chains. +To get examples based on lattices which are not necessarily totally ordered, in +Example 2.3.2 (and Appendix A) we discuss semi-primal residuated lattices. In +particular we describe a systematic way to identify them among the FLew-algebras. +Similarly, Example 2.3.3 illustrates how to identify semi-primal algebras which need +not be totally ordered among the pseudo-logics. At the end of this subsection we +recall Murski˘ı’s Theorem which states that, in some sense, almost all finite lattice- +based algebras are semi-primal. + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +7 +2.3.1. Chain-based algebras. We will describe several different ways of turning the +(n + 1)-element chain {0, 1 +n, . . . , n−1 +n , 1} with its usual lattice-order into a semi- +primal algebra. We present the examples ordered decreasingly by the amount of +subalgebras. +First, turning a chain into a semi-primal algebra without any further impositions +may be achieved as follows. +Example 2.10. The n-th general semi-primal chain is given by +Tn = +� +{0, 1 +n, . . . , n−1 +n , 1}, ∧, ∨, 0, 1, (T i +n )n +i=0 +� +, +where the unary operations T i +n are the ones from Definition 2.7. For all n ≥ 1 the +algebra Tn is semi-primal (this immediately follows from Proposition 2.8). Every +subset of Tn which contains the set {0, 1} defines a subalgebra of Tn. +Next we find examples among the �Lukasiewicz-Moisil algebras, which were orig- +inally intended to give algebraic semantics for �Lukasiewicz finitely-valued logic. It +turns out, however, that they encompass a bit more than that (see [15]). The logic +corresponding to these algebras is nowadays named after Moisil. +Example 2.11. The n-th �Lukasiewicz-Moisil chain is given by +Mn = +� +{0, 1 +n, . . . , n−1 +n , 1}, ∧, ∨, ¬, 0, 1, (τ i +n )n +i=1 +� +, +where ¬x = 1 − x and the unary operations τ i +n are the ones from Definition 2.7. +For all n ≥ 1, the algebra Mn is semi-primal. This follows from characterization +(3) of Proposition 2.8 - we only have to check that T0 is term-definable. To see this +note that we can define T1(x) = τ1(x) and T0(x) = T1(¬x). +We proceed with a classical example from many-valued logic among the finite +MV-algebras introduced by Chang (see [13, 14]). They give rise to the algebraic +counterpart of �Lukasiewicz finite-valued logic. +Example 2.12. The n-th �Lukasiewicz chain is given by +�Ln = +� +{0, 1 +n, . . . , n−1 +n , 1}, ∧, ∨, ⊕, ⊙, ¬, 0, 1 +� +, +where x ⊕ y = min(x + y, 1), x ⊙ y = max(x + y − 1, 0) and ¬x = 1 − x. For all +n ≥ 1, the algebra �Ln is semi-primal. The proof of this fact can be found in [48, +Proposition 2.1]. The subalgebras of �Ln correspond to the divisors d of n and are +of the form +�Ld = {0, k +n, . . . , (d−1)k +n +, 1} where n = kd. +Other semi-primal chains are found among the Cornish algebras, which generalize +Ockham algebras (see [18, 19]). +Example 2.13. The n-th semi-primal Cornish chain is given by +COn = +� +{0, 1 +n, . . . , n−1 +n , 1}, ∧, ∨, ¬, f, 0, 1 +� +, +where ¬x = 1 − x, f(0) = 0, f(1) = 1 and f( i +n) = i+1 +n +for 1 ≤ i ≤ n − 1. For all +n ≥ 1, the algebra COn is semi-primal. The proof of this fact can be found in [19, +Example 5.15]. The only proper subalgebra of COn is {0, 1}. +Finally, among the Post-algebras we find the well-known examples of chain-based +algebras which are not only semi-primal, but even primal. + +8 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +Example 2.14. The n-th Post chain is given by +Pn = +� +{0, 1 +n, . . . , n−1 +n , 1}, ∧, ∨,′ , 0, 1 +� +where 1′ = 0 and ( i +n)′ = ( i+1 +n ) for 0 ≤ i < n. For all n ≥ 1, the algebra Pn is +primal (see, e.g., [24, Theorem 35]) +2.3.2. Residuated Lattices. For a general survey of residuated lattices we refer the +reader to [32, 39]. We only consider bounded commutative residuated lattices here, +with a particular focus on FLew-algebras. +Definition 2.15. A (bounded commutative) residuated lattice is an algebra +R = (R, ∧, ∨, 0, 1, ⊙, e, →) +such that (R, ∧, ∨, 0, 1) is a bounded lattice, (R, ⊙, e) is a commutative monoid and +the binary operation → satisfies the residuation condition +x ⊙ y ≤ z ⇔ x ≤ y → z. +We call R a FLew-algebra if, in addition, it satisfies e = 1. +Our main tool to identify semi-primal FLew-algebras is [42, Theorem 3.10], which +implies that a FLew-algebra R is quasi-primal if and only if there is some n ≥ 1 +such that +(1) +x ∨ ¬(xn) = 1 for all x ∈ R, +where, as usual, we define ¬x as x → 0 (and xn refers to the n-th power with respect +to ⊙). For our purposes this theorem has the following practical consequence. +Corollary 2.16. Let R be a finite FLew-algebra. If R does not contain any idem- +potent elements (that is, elements with x ⊙ x = x) other than 0 and 1, then R is +quasi-primal. If R is based on a chain, the converse also holds. +Proof. Let R be a finite FLew-algebra with no other idempotent elements than 0 +and 1. Recall that, for any a ∈ R, we have ¬a = a → 0 = �{b ∈ R | a ⊙ b ≤ 0}. +Let a ∈ R\{0, 1}. We show that there is some na such that ana = 0. Since a is +not idempotent we have a2 < a. Either a2 = 0 and we are done or a2 is again not +idempotent. In this case we have a4 < a2 and we repeat the argument. Since R is +finite, continuing this process we eventually need to find a2k = 0. Now R satisfies +equation (1) for n = �{na | a ∈ R\{0, 1}}, since we always have +a ∨ ¬(an) = a ∨ ¬0 = a ∨ 1 = 1. +Thus R is quasi-primal. +Now suppose that R is based on a chain. If a ∈ R\{0, 1} is idempotent, then +¬a < a since for all b ≥ a we have a ⊙ b ≥ a ⊙ a = a. Therefore, for all n ≥ 1 we +have a ∨ ¬(an) = a ∨ ¬a = a ̸= 1. Thus, R does not satisfy equation (1) and is not +quasi-primal. +□ +Remark 3. The second part of the argument really requires R to be based on a +chain. +For example, consider the 4-element diamond lattice 0 ≤ a, b ≤ 1 with +a ∧ b = 0 and a ∨ b = 1. We can define a FLew-algebra based on this lattice by +stipulating a2 = a, b2 = b and a ⊙ b = 0. Even though a and b are idempotent, we +have a ∨ ¬a = a ∨ b = 1 and b ∨ ¬b = b ∨ a = 1. Therefore, this algebra is quasi- +primal (it is, however, not semi-primal, since it has the non-trivial automorphism +swapping a and b). +■ + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +9 +In [31] Galatos and Jipsen provide a list of all finite residuated lattices of size +up to 6. Corollary 2.16 enables us to find quasi-primal FLew-algebras among them +and thus, using Proposition 2.2, we can identify the semi-primal ones by ruling out +the existence of non-trivial internal isomorphisms. For example, there is a total of +six quasi-primal FLew-chains with 5 elements (R5,1 +1,17, R5,1 +1,18 . . . R5,1 +1,22 in [31]), five of +which are semi-primal (all except R5,1 +1,17). Examples of semi-primal FLew-algebras +not based on a chain are, e.g., R6,2 +1,11 and R6,3 +1,9 in [31]. The algebras in question are +depicted in Appendix A, where we also provide detailed proofs of these claims. +While until now we discussed how to identify semi-primal FLew-algebras, we +end this subsection with two examples of semi-primal algebras based on residuated +lattices where 1 ̸= e. +Specifically, we consider the bounded De Morgan monoids C01 +4 and D01 +4 depicted +in Figure 1. +0 +e +a +1 = a2 +0 +1 = a2 +e +a +Figure 1. The (semi-)primal bounded De Morgan monoids C01 +4 +and D01 +4 . +They are bounded commutative residuated lattices with an additional involution +∼ which, in both examples, is defined by ∼e = a and ∼0 = 1. Our names for these +algebras are inspired by [46], where C4 and D4 are used for the corresponding De +Morgan monoids with the bounds 0 and 1 excluded from the signature (in [46] it is +shown that each of these two algebras generates a minimal subvariety of the variety +of all De Morgan monoids). +Proposition 2.17. The algebras C01 +4 and D4 +01 are primal. Their reducts obtained +by removing the neutral element e from the signature, are semi-primal. +Proof. Starting with C01 +4 , we directly verify that it satisfies characterization (3) of +Proposition 2.8. First we define T1 and, therefore, T0(x) = T1(∼x). As in [22], we +do this by, for all ℓ ∈ {0, e, a}, defining unary terms uℓ satisfying +uℓ(x) = + + + + + + + +1 +if x = 1 +0 +if x = ℓ +∗ +otherwise, +were ∗ indicates that any value is allowed. For instance, we can define such terms +by +u0(x) = x ∧ 1, ue(x) = ∼ +� +(∼x)2� +and ua(x) = ∼ +� +(∼x) ⊙ 1 +� +. + +10 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +Through these terms we can clearly define T1(x) = u0(x) ∧ ue(x) ∧ ua(x). Lastly, +we need to define τℓ for ℓ ∈ {e, a}. Again, it suffices to find terms τ ∗ +ℓ which satisfy +τ ∗ +ℓ (x) = +� +1 +if x ≥ ℓ +̸= 1 +if x ̸≥ ℓ, +since then we get τℓ = T1(τ ∗ +ℓ ). Our desired terms are given by +τ ∗ +e (x) = +� +(∼x)2 ⊙ x +� +∨ x2 and τ ∗ +a(x) = x2. +This concludes the proof for C01 +4 . The proof for D01 +4 +is completely analogous, +except that we use τ ∗ +e (x) = +� +(∼x)2 ⊙x +� +∨x instead. Thus we showed that these two +algebras are semi-primal, and since they don’t have any proper subalgebras they +are primal. Since we never relied on the constant e in the above, the last part of +the statement follows. Note that in both cases, if we exclude e from the signature +then {0, 1} becomes a proper subalgebra. +□ +2.3.3. Pseudo-logics. We illustrate how to generate more examples of semi-primal +algebras which are based on a bounded lattice which is not necessarily a chain. The +results and terminology are due to [17, 22]. A pseudo-logic +L = (L, ∧, ∨,′ , 0, 1) +is a bounded lattice with an additional unary operation ′ which satisfies 0′ = 1 and +1′ = 0. In [22] it is shown that every subalgebra of L2 which is not the graph of +an internal isomorphism is a product of subalgebras if the following two properties +are satisfied: +(1) There is no a ∈ L\{0} with a′ = 1, +(2) For all a ∈ L there exists an n ≥ 1 with a ∧ a(2n) = 0 (where a(k) denotes +the k-fold iteration of ′ on a). +Using this and the characterization of Proposition 2.3, we can find more examples +of semi-primal algebras. Here, we only need to assure that the above mentioned +conditions are satisfied and that there are no non-trivial internal isomorphisms. +For example, the three algebras depicted in Figure 2 are semi-primal (the pseudo- +negation ′ is indicated by dotted arrows). +0 +1 +a +b +c +0 +1 +a +b +c +d +0 +1 +a +b +c +d +e +Figure 2. Some semi-primal pseudo-logics ([22, 17]). +2.3.4. Murski˘ı’s Theorem. While semi-primal algebras may seem rare, quite the +opposite is suggested by the following. +In 1975, Murski˘ı proved his surprising +theorem about the proportion of semi-primal algebras of a fixed signature under +increasing order. The original paper [47] is in Russian, the version we recall here is +due to [6, Section 6.2]. + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +11 +Theorem 2.18. [47] Let σ be an algebraic type which contains at least one operation +symbol which is at least binary. Let Aσ,n be the number of algebras of type σ and +size n and let SPσ,n be the number of such algebras which are semi-primal. Then +lim +n→∞ +SPσ,n +Aσ,n += 1. +3. Semi-primal duality +One of the nice features of the variety of Boolean algebras BA is the famous +Stone duality [56]. Categorically speaking, it asserts that there is a dual equivalence +between BA and the category Stone of Stone spaces (that is, compact, Hausdorff +and zero-dimensional topological spaces) with continuous maps: +Stone +Π +� BA +Σ +� +The functor Σ assigns to a Boolean algebra B its collection of ultrafilters and the +functor Π assigns to a Stone space X the Boolean algebra of its clopen subsets +with the usual set-theoretical Boolean operations. Note that these functors can be +defined on objects by +Σ(B) = BA(B, 2) and Π(X) = Stone(X, 2), +where in the latter equation 2 denotes the two-element discrete space. +Stone duality has been extended to quasi-primal algebras by Keimel and Werner +in [41]. This duality fits the general framework of Natural Dualities. For us, the +Semi-primal Strong Duality Theorem [17, Theorem 3.3.14] is of high importance. +However, we present it self-contained and in a way which particularly suits our +purpose. Furthermore, we will use categorical constructions to provide a new proof +of this duality. Such a proof has, to the best of our knowledge, not appeared in the +literature yet. +First we introduce the dual category of A. In the following, we always consider +S(L) as a complete lattice in its usual ordering. +Definition 3.1. The category StoneL has objects (X, v) where X ∈ Stone and +v: X → S(L) +assigns to every point x ∈ X a subalgebra v(x) ≤ L, such that for every subalgebra +S ≤ L the preimage v−1(S↓) is closed. A morphism m: (X, v) → (Y, w) in StoneL +is a continuous map X → Y which, for all x ∈ X, satisfies +w(m(x)) ≤ v(x). +Remark 4. In the framework of natural dualities [17], the dual category of A is +defined slightly differently, using Stone spaces with unary relations (i.e., subsets). +Let X be the category with objects (X, {RS | S ≤ L}), where X ∈ Stone and +RS is a closed subset of X for each subalgebra S ≤ L, satisfying RL = X and +RS1 ∩ RS2 = RS1∩S2 for all S1, S2 ≤ L. A morphism m: (X, {RS | S ≤ L}) → +(X′, {R′ +S | S ≤ L}) in X is a continuous relation-preserving map X → X′, i.e., it +satisfies x ∈ RS ⇒ m(x) ∈ R′ +S for all x ∈ X and S ∈ S(L). + +12 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +The categories X and StoneL are isomorphic, as witnessed by the following mu- +tually inverse functors φ and ψ. The functor φ : X → StoneL is given on objects +by (X, {RS | S ≤ L}) �→ (X, v), where +v(x) = +� +{S | x ∈ RS}. +The functor ψ: StoneL → X is given on objects by (X, v) �→ (X, {RS | S ≤ L}) +where +RS = {x ∈ X | v(x) ≤ S}. +Both φ and ψ map every morphism to itself. +■ +We now describe the two contravariant functors ΣL and ΠL which give rise to +the duality between A and StoneL: +StoneL +ΠL +� A +ΣL +� +On objects A ∈ A, let the functor ΣL be defined by +ΣL(A) = +� +A(A, L), im +� +where im assigns to a homomorphism h: A → L its image im(h) = h(A) ∈ S(L). +A clopen subbasis for the topology on A(A, L) is given by the collection of sets of +the following form with a ∈ A and ℓ ∈ L: +[a : ℓ] = {h ∈ A(A, L) | h(a) = ℓ}. +On morphisms f ∈ A(A1, A2) the functor acts via composition +ΣLf : A(A2, L) → A(A1, L) +h �→ h ◦ f. +Note that this is a morphism in StoneL since im(h ◦ f) ≤ im(h). +Before we define the functor ΠL, we describe the canonical way to consider L as +a member of StoneL. Simply endow L with the discrete topology and +⟨·⟩: L → S(L) +assigning to an element ℓ ∈ L the subalgebra ⟨ℓ⟩ ≤ L it generates. +Now, as +expected, we can define the functor ΠL on objects (X, v) ∈ StoneL by +ΠL(X, v) = StoneL +� +(X, v), (L, ⟨·⟩) +� +with the algebraic operations defined pointwise. This means that the carrier-set +of ΠL(X, v) is the set of continuous maps g : X → L which respect v in the sense +that, for all x ∈ X, they satisfy +g(x) ∈ v(x). +Again, on morphisms m: (X, v) → (Y, w) the functor is defined via composition +ΠLm: StoneL +� +(Y, w), (L, ⟨·⟩) +� +→ StoneL +� +(X, v), (L, ⟨·⟩) +� +g �→ g ◦ m. +This is well-defined due to the condition on morphisms in StoneL: +(g ◦ m)(x) = g(m(x)) ∈ w(m(x)) ⊆ v(x). +It is also clearly a homomorphism since the operations are defined pointwise. + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +13 +Theorem 3.2. [41, 17] The functors ΠL and ΣL are fully faithful and establish a +dual equivalence between A and StoneL. +The remainder of this section is dedicated to an alternative proof of this theorem. +The idea is to directly prove the duality on the finite level, and then lift it to the +infinite level using the following categorical constructions. +Definition 3.3. For a finitely complete and cocomplete category C, its comple- +tion under filtered colimits is denoted by Ind(C) and, dually, its completion under +cofiltered limits is denoted by Pro(C). +For example, Ind(BAω) ≃ BA and Pro(Setω) ≃ Stone. More material about these +completions can be found in Johnstone’s book [40, Chapter VI] (in particular, a +more rigorous definition of the Ind-completion is given in VI.1.2). We only recite +the following, which allows us to lift dualities between small categories (following +Johnstone, dualities arising this way are called Stone type dualities). +Lemma 3.4. [40, Lemma VI 3.1] Let C and D be small categories which are dually +equivalent. Then Ind(C) is dually equivalent to Pro(C). +Our argument to prove Theorem 3.2 now has the following outline. The role of +C will be played by Aω. Since A is locally finite (see, e.g., [17, Lemma 1.3.2]), it +is well-known that Ind(Aω) ≃ A (see, e.g., [40, Corollary VI 2.2]). The role of D +will be played by Stoneω +L. Since the topology doesn’t matter here (because it is +always discrete), we will denote this category by Setω +L instead. To get the finite +dual equivalence, we make the following observation +Proposition 3.5. Let S1, . . . , Sn be subalgebras of L. Then the set of homomor- +phisms A(� +i≤n Si, L) consists exactly of the projections followed by inclusions +pri : +� +i≤n +Si → Si ֒→ L +in each component i ≤ n. +Proof. Our proof is similar to that of [12, Theorem 2.5]. Let h: � +i≤n Si → L +be a homomorphism. Since A is congruence distributive (Proposition 2.5), it has +the Fraser-Horn property, meaning that the congruence θ := ker(h) is a product of +congruences θi on Si. By the isomorphism theorem we find +( +� +i≤n +Si)/θ ∼= +� +i≤n +(Si/θi) ∼= im(h). +Since im(h) is a subalgebra of L and thus directly irreducible, at most one factor of +� +i≤n(Si/θi) can be non-trivial. Since im(h) contains at least two elements (that is, +0 and 1), precisely one factor, say Sj/θj, is non-trivial. Since Sj is itself semi-primal, +it is simple, so Sj/θj ∼= Sj. So h induces an internal isomorphism Sj ∼= im(h), but +by Proposition 2.2 this can only be the identity on Sj, thus h coincides with prj. +□ +Corollary 3.6. The (restrictions of the) functors ΠL and ΣL establish a dual +equivalence between the small categories SetL +ω and Aω. +Proof. Let (X, v) ∈ Setω +L. Then +ΣLΠL(X, v) = +� +A +� � +x∈X +v(x), L +� +, im +� +. + +14 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +By Proposition 3.5 this is equal to ({prx | x ∈ X}, im), which is clearly isomorphic +to (X, v). +On the other hand, starting with A ∈ Aω, we know by Proposition 2.6 that it is +a product of subalgebras A = � +i≤n Si. Now, again due to Proposition 3.5, we get +ΣL(A) = ({pri | i ≤ n}, im), and thus +ΠLΣL(A) ∼= +� +i≤n +im(pri) ∼= +� +i≤n +Si. +To see that ΠL and ΣL form a dual adjunction we note that for A = � +i≤n Si ∈ +Aω and (X, v) ∈ Setω +L we have +Aω� +ΠL(X, v), A +� ∼= +� +i≤n +Aω� +ΠL(X, v), Si +� +and +Setω +L +� +ΣL(A), (X, v) +� ∼= Setω +L( +� +i≤n +({pri}, im), (X, v)) ∼= +� +i≤n +Setω +L +� +({pri}, im), (X, v) +� +where the coproduct in Setω +L is the obvious disjoint union. So we only need to show +that +Aω� +ΠL(X, v), Si +� ∼= Setω +L +� +({pri}, im), (X, v) +� +. +But this is obvious since the elements of the left-hand side are exactly the projec- +tions with image contained in Si, which are in bijective correspondence with the +points of X with v(x) ≤ Si, that is, with elements of the right-hand side. +□ +In order to successfully apply Lemma 3.4, it remains to show the following. +Theorem 3.7. Pro(Setω +L) is categorically equivalent to StoneL. +Proof. First we show that the category StoneL is complete. For an index set I +(which we often omit), we claim that the product is computed as +� +i∈I +(Xi, vi) = ( +� +i∈I +Xi, +� +vi), +where � vi(p) = �(vi(pi)) for all p ∈ � Xi. It follows from +( +� +vi)−1(S↓) = +� +vi +−1(S↓) +that this defines a member of StoneL. Note that the projections are morphisms in +StoneL since +vi(πi(p)) = vi(pi) ≤ +� +j∈I +vj(pj) = ( +� +vj)(p). +If (γi : (Y, w) → (Xi, vi) | i ∈ I) is another cone, there is a unique continuous map +f : Y → � Xi with πi ◦ f = γi. This map is a morphism in StoneL since +( +� +vi)(f(y)) = +� +vi +� +πi(f(y)) +� += +� +vi +� +γi(f(y)) +� +≤ w(y), +where the last inequality follows from vi(γi)(y) ≤ w(y) which is true since the γi +are morphisms in StoneL. The equalizer of f, g : (X, v) → (Y, w) is simply given by +(Eq, v|Eq) where Eq ⊆ X is the corresponding equalizer in Stone. It follows that +StoneL has all limits. In particular, StoneL has all cofiltered limits, so the natural +inclusion functor ι: Setω +L ֒→ StoneL has a unique cofinitary (that is, cofiltered limit +preserving) extension +ˆι: Pro(Setω +L) ֒→ StoneL. + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +15 +Since ι is fully faithful, to conclude that the functor ˆι is fully faithful as well it +suffices to show that ι maps all objects to finitely copresentable objects in StoneL +(this is due to the analogue of [40, Theorem VI.1.8] for the Pro-completion). So +we need to show that any (C, w) ∈ StoneL where C is a finite discrete space is +finitely copresentable. In other words, we need to show that, whenever (X, v) ∼= +limi∈I(Xi, vi) is a cofiltered limit of a diagram (fij : (Xj, vj) → (Xi, vi) | i ≤ j) +in StoneL with limit morphisms pi : (X, v) → (Xi, vi), any morphism f : (X, v) → +(C, w) factors essentially uniquely through one of the pi. For this we can employ +an argument similar to the one in the proof of [55, Lemma 1.1.16(b)]. +On the +underlying level of Stone, where finite discrete spaces are finitely copresentable, +the continuous map f factors essentially uniquely through some pi, say via the +continuous map gi : Xi → C. However, gi is not necessarily a morphism in StoneL. +Consider J = {j ≥ i} and for each j ∈ J define gj = fij ◦ gi. Define the continuous +maps µ: X → S(L)2 and µj : Xj → S(L)2 for all j ∈ J by +µ(x) = +� +w(f(x)), v(x) +� +and µj(x) = +� +w(gj(x)), vj(x) +� +. +Since µ(X) = limj∈J µj(Xj) = � +j≥i µj(Xj) is contained in the finite set S(L)2 and +J is directed, there is some k ∈ J such that +µ(X) = µk(Xk). +But now, since f is a morphism in StoneL, we have that µ(X) ⊆ {(S, T) | S ≤ T}, +and thus the same holds for µk(Xk). Thus gk is a morphism in StoneL which has +the desired properties. +To finish the proof we show that ˆι is essentially surjective, in other words, we show +that every element (X, v) of StoneL is isomorphic to a cofiltered limit of elements of +Setω +L. We do this in a manner similar to [55, Theorem 1.1.12]. Let R consist of all +finite partitions of X into clopen sets. Together with the order R ≤ R′ if and only +if R′ refines R this forms a codirected set and in [55, Theorem 1.1.12] it is shown +that X ∼= limR∈R R. We now turn every R ∈ R into a member of Setω +L by endowing +it with an appropriate vR : R → S(L) and show that (X, v) = limR∈R(R, vR). For +R ∈ R, say R = {Ω1, . . . , Ωk}, we define +v−1 +R (S↓) = {Ωi | Ωi ∩ v−1(S↓) ̸= ∅}. +The map pR : X → R defined by pR(x) = Ωi ⇔ x ∈ Ωi is a morphism in +StoneL since v(x) = S and x ∈ Ωi implies vR(pR(x)) ∈ v−1 +R (S↓). Is is easy to +see that this defines a cone over the diagram (R, vR)R∈R, so there is a unique +f : (X, v) → limR∈R(R, vR) in StoneL. As in Stone, the map f is a homeomor- +phism. To complete the proof it suffices to show that f −1 is a morhpism in StoneL +as well. Say limR∈R(R, vR) = (Y, w) and let πR : (Y, w) → (R, vR) denote the limit +morphisms. Assuming w(y) = S we want to show f −1(y) ∈ v−1(S↓). Let Ω ⊆ X +be an arbitrary clopen set containing f −1(y). Then R = {Ω, X\Ω} ∈ R and +Ω = pR(f −1(y)) = πR(y) ∈ v−1 +R (S↓). +By definition this means that Ω ∩ v−1(S↓) ̸= ∅. +Since this holds for every Ω +containing f −1(y), this implies that f −1(y) is in the closure v−1(S↓). However, +this closure coincides with v−1(S↓), since by definition of StoneL this is a closed +set already. +□ + +16 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +As discussed before, this yields our alternative proof of Theorem 3.2. In Section +5 we investigate the other dual equivalence which can be obtained from the finite +dual equivalence of Corollary 3.6. More specifically, there we describe Ind(Setω +L) +and its dual, the category of profinite algebras Pro(Aω). This is the ‘semi-primal +version’ of the duality between Ind(Setω) ≃ Set and Pro(BAω) ≃ CABA. +Before that, in the following section we investigate the relationship between +StoneL and Stone and, more interestingly, between A and BA. +4. A chain of adjuntions +In this section we explore the relationship between Stone duality and the semi- +primal duality discussed in the previous section. +We start with the connection +between StoneL and Stone, which will be expressed in terms of a chain of four adjoint +functors (similar to one in [57]). Then we look at the duals of these functors and give +them purely algebraic descriptions to gain insight into the structure of A relative to +that of BA. The entire situation is summarized in Figure 3, which we will have fully +described at the end of this section (note that left-adjoints on the topological side +correspond to right-adjoints on the algebraic side and vice-versa, since the functors +ΠL, ΣL and Σ, Π which establish the two dualities are contravariant). +StoneL +ΠL +� +� +⊣ +V⊤ +U +⊣ +� +� +V⊥ +⊣ C +� +A +ΣL +� +� +⊢ +P +S +⊢ +� +� +I +⊢ Q +� +Stone +Π +� BA +Σ +� +Figure 3. The chain of adjunctions on the topological and the +algebraic side. +4.1. Four functors on the topological side. Let U: StoneL → Stone be the +obvious forgetful functor. This functor has a left-adjoint and a right-adjoint V⊤ ⊣ +U ⊣ V⊥. The two functors V⊤, V⊥ : StoneL → Stone are given on objects by +V ⊤(X) = (X, v⊤) where ∀x ∈ X : v⊤(x) = L, +V ⊥(X) = (X, v⊥) where ∀x ∈ X : v⊥(x) = ⟨0, 1⟩ +and both assign every morphism to itself. Here ⟨0, 1⟩ is the subalgebra generated +by {0, 1}, that is, the (unique) smallest subalgebra of L. +To see V ⊤ ⊣ U note that by definition we have +m ∈ StoneL +� +(X, v⊤), (Y, w) +� +⇔ m ∈ Stone(X, Y ) ∧ ∀x ∈ X : w(m(x)) ≤ v⊤(x), +and w(m(x)) ≤ v⊤(x) = L is trivially satisfied for every m ∈ Stone(X, Y ). + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +17 +Similarly we see U ⊣ V⊥, since every m ∈ Stone(X, Y ) automatically satisfies +v⊥(m(x)) ≤ w(x) and, therefore, m ∈ StoneL +� +(X, w), (Y, v⊥) +� +. +The functor V⊥ also has a right-adjoint C : StoneL → Stone defined by +C(X, v) = {x ∈ X | v(x) = ⟨0, 1⟩} +on objects. On morphisms m: (X, v) → (Y, w) it acts via restriction m �→ m|C(X,v), +which is well-defined since m ∈ StoneL +� +(X, v), (Y, w) +� +and x ∈ C(X, v) means +w(m(x)) ≤ v(x) = ⟨0, 1⟩ +which is equivalent to m(x) ∈ C(W, w). Again V⊥ ⊣ C is easy to see since +m ∈ StoneL +� +(X, v⊥), (Y, w) +� +⇔ ∀x : w(m(x)) ≤ ⟨0, 1⟩ ⇔ m ∈ Stone +� +X, C(Y, w) +� +. +The functor V⊤ preserves almost all limits, however, there is one important +exception. The terminal object (that is, the limit of the empty diagram) in StoneL is +given by ({∗}, v⊥), implying that V⊤ does not preserve terminal objects. Therefore, +contrary to a claim made in [57], no further left-adjoint of V⊤ exists. +It is obvious that both the unit idStone ⇒ U ◦ V⊤ of the adjunction V⊤ ⊣ U and +the counit U ◦ V⊥ ⇒ idStone of the adjunction U ⊣ V⊥ are natural isomorphisms. +We hold on to this fact, which will also be interesting on the algebraic side. +Proposition 4.1. The category Stone is categorically equivalent to +(i) a coreflective subcategory of StoneL, witnessed by the fully faithful functor V⊤. +(ii) a reflective and coreflective subcategory of StoneL, witnessed by the fully faith- +ful functor V⊥. +The functors described in this subsection can be carried through the dualities, +resulting in a a corresponding chain of adjunctions between A and BA. For example, +the dual of U is given by ΠUΣL : A → BA. In the next subsection we show that this +functor can be understood algebraically as the Boolean skeleton. Throughout the +subsections that follow, we will give similar algebraic descriptions for all of these +functors between A and BA in Figure 3. +4.2. The Boolean skeleton functor. In the theory of MVn-algebras (that is, the +case where L = �Ln), the Boolean skeleton is a well-known and useful tool (see, for +example, [16]). An appropriate generalization of this concept to our setting was +made by Maruyama in [43] (where it is called the Boolean core). +Due to Proposition 2.8 and [43, Lemma 3.11], the following definition is justified. +Definition 4.2. Let A ∈ A. The Boolean skeleton of A is the Boolean algebra +S(A) = (S(A), ∧, ∨, T0, 0, 1) on the carrier set +S(A) = {a ∈ A | T1(a) = a}, +where the lattice operations ∧ and ∨ are inherited from A and the unary operations +T0 and T1 correspond to the ones from Definition 2.7 (which by Proposition 2.8 are +term-definable in L), interpreted in A. +For example, for each A ∈ A, a ∈ A and ℓ ∈ L we have Tℓ(a) ∈ S(A). This +holds since the equation T1(Tℓ(x)) ≈ Tℓ(x) holds in L, and therefore also in A. +Remark 5. For A ∈ A, suppose that A′ ⊆ A is a subset such that (A′, ∧, ∨, T0, 0, 1) +forms a Boolean algebra. Then, for all a′ ∈ A′, we have T1(a′) = T1(T0(T0(a′))) = +T0(T0(a′)) = a′ and thus a′ ∈ S(A) (the second equation always holds since A |= + +18 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +T1(T0(x)) ≈ T0(x), which is easily checked in L). Therefore, S(A) is the largest +such subset. +■ +To extend the construction of the Boolean skeleton to a functor S: A → BA, on +homomorphisms f ∈ A(A1, A2) we define Sf to be the restriction f|S(A1). This +is well-defined since +a ∈ S(A1) ⇔ T1(a) = a ⇒ T1(f(a)) = f(T1(a)) = f(a) ⇔ f(a) ∈ S(A2). +The following is arguably the most important property of the Boolean skeleton. +Proposition 4.3. For all A ∈ A, there is a homeomorphism between UΣL(A) = +A(A, L) and ΣS(A) = BA(S(A), 2) given by h �→ h|S(A). +Proof. First we show that the map is a bijection. For injectivity, suppose that g +and h satisfy g|S(A) = h|S(A). Take an arbitrary element a ∈ A and let g(a) = ℓ. +Using that Tℓ(a) ∈ S(A) we get +1 = Tℓ(g(a)) = g(Tℓ(a)) = h(Tℓ(a)) = Tℓ(h(a)), +which implies h(a) = ℓ and, since a was arbitrary, that g = h. For surjectivity, +let h ∈ BA(S(A), 2) be arbitrary. Due to [43, Lemma 3.12] the following yields a +well-defined homomorphism ¯h ∈ A(A, L): +¯h(a) = ℓ ⇔ h(Tℓ(a)) = 1. +Since for a ∈ S(A) we have +h(T1(a)) = 1 ⇔ h(a) = 1 and +h(T0(a)) = 1 ⇔ T0(h(a)) = 1 ⇔ h(a) = 0, +we conclude that ¯h|S(A) = h. +We now have a bijection between two Stone spaces, so it only remains to show +that it is continuous. But this is easy to see since the preimage of an open subbasis +element [a : i] ⊆ BA(S(A), 2) is the open subbasis element [a : i] ⊆ A(A, L). +□ +Corollary 4.4. There is a natural isomorphism between the functor S and the +dual ΠUΣL of the forgetful functor U. +Proof. By Proposition 4.3, for every A ∈ A, setting +φA : UΣL(A) → ΣS(A) +h �→ h|S(A) +defines a natural isomorphism φ: UΣL ⇒ ΣS (naturality is easy to check using the +definitions of Σ, ΣL and S on morphisms). Applying Π and using the fact that ΠΣ is +naturally isomorphic to idBA, we get the natural isomorphism Πφ: S ⇒ ΠUΣL. +□ +In the next subsection we explain the right-adjoint of the Boolean skeleton func- +tor. +4.3. The Boolean power functor. In this subsection we give an algebraic de- +scription of a functor naturally isomorphic to the dual ΠLV⊤Σ of the functor V⊤. +This functor, which we call P, turns out to be an instance of the the well-known +Boolean power (or Boolean extension), which was introduced for arbitrary finite +algebras in Foster’s first paper on primal algebras [24]. Boolean powers are special +instances of Boolean products (see, e.g., [11, Chapter IV]), but for our purposes it +is more convenient to work with the following equivalent definition found in [9]. + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +19 +Definition 4.5. Given a Boolean algebra B ∈ BA and a finite algebra M, the +Boolean power M[B] is defined on the carrier set +M[B] ⊆ BM +consisting of all maps ξ : M → B which satisfy +(1) If ℓ and ℓ′ are distinct elements of M, then ξ(ℓ) ∧ ξ(ℓ′) = 0, +(2) �{ξ(ℓ) | ℓ ∈ M} = 1. +If oL : M k → M is a k-ary operation of M, we define a corresponding operation +oM[B] : M[B] → M[B] by +oM[B](ξ1, . . . , ξk)(ℓ) = +� +oM(ℓ1,...,ℓk)=ℓ +(ξ1(ℓ1) ∧ · · · ∧ ξk(ℓk)). +The resulting algebra M[B] = (M[B], oM[B]) is a member of the variety HSP(M) +generated by M (since it satisfies the same equations as M). +There is a straightforward way to extend this construction to a functor. +Definition 4.6. Given a finite algebra M, we define the functor PM : BA → +HSP(M) as follows. On objects B ∈ BA we define +PM(B) = M[B]. +For a Boolean homomorphism ϕ: B → B′, the homomorphism PMϕ: M[B] → +M[B′] is defined via composition ξ �→ ϕ ◦ ξ (this is a homomorphism because +operations in M[B] are defined by Boolean expressions, which commute with ϕ). +In particular, we will use the shorthand notation P for PL. In the remainder of +this subsection we aim to show that P is indeed the right-adjoint of the Boolean +skeleton functor S. For this, we need the following well-known properties of the +Boolean power. +Lemma 4.7. [9, Proposition 2.1] The functor PM has the following properties: +(i) PM(2) ∼= M, +(ii) PM preserves products. +In particular, PM(2κ) ∼= Mκ holds for all index sets κ. +In the next proposition we describe the interplay between the functors S and P. +Again, the terms Tℓ from Proposition 2.8 play an important role. +Proposition 4.8. For every A ∈ A there is an embedding T(·) : A ֒→ P(S(A)) +given by a �→ Ta where +Ta(ℓ) = Tℓ(a). +The restriction to S(A) yields an isomorphism S(A) ∼= S +� +P(S(A)) +� +. +Proof. The map is well-defined, that is, Ta is in P(S(A)), since the equations +Tℓ(x) ∧ Tℓ′(x) ≈ 0 (for distinct ℓ, ℓ′) and �{Tℓ(x) | ℓ ∈ L} ≈ 1 hold in L. +We now fix an embedding A ֒→ LI. It is easy to see that T(·) is injective since, +for distinct a, a′ ∈ A, there is some component i ∈ I with a(i) = ℓ ̸= a′(i), thus +Ta(ℓ) ̸= Ta′(ℓ). To conclude that T(·) is an embedding we need to show that it is a +homomorphism, that is we want to show that for any k-ary operation o: Lk → L +of L we have +ToA(a1,...,ak) = oL[B(A)](Ta1, . . . Tak). + +20 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +By definition the i-th component of the left-hand side is given by +ToA(a1,...,ak)(ℓ)(i) = Tℓ +� +oL(a1(i), . . . , ak(i)) +� += +� +1 +if oL(a1(i), . . . , ak(i)) = ℓ +0 +otherwise. +The right-hand side is given by +oL[B(A)](Ta1, . . . Tak)(ℓ) = +� +oL(ℓ1,...,ℓk)=ℓ +(Ta1(ℓ1) ∧ · · · ∧ Tak(ℓk)). +In its i-th component this again corresponds to +� +oL(ℓ1,...,ℓk)=ℓ +� +Tℓ1(a1(i)) ∧ · · · ∧ Tℓk(ak(i)) +� += +� +1 +if oL(a1(i), . . . , ak(i)) = ℓ +0 +otherwise. +Thus T(·) is an embedding, which concludes the proof of the first statement. +For the second statement, note that, since S preserves injectivity of homomor- +phisms, it suffices to show that the restriction of T(·) to S(A) is a surjection onto +S +� +P(S(A)) +� +. So consider an element ξ ∈ S +� +P(S(A)) +� +, that is ξ ∈ P(S(A)) and +T L[S(A)] +1 +(ξ) = ξ. The latter by definition means +T L[S(A)] +1 +(ξ)(1) = ξ(1), +T L[S(A)] +1 +(ξ)(0) = +� +{ξ(ℓ) | ℓ ∈ L, ℓ ̸= 1} = ξ(0) and +T L[S(A)] +1 +(ξ)(ℓ) = +� +∅ = 0 = ξ(ℓ) for all ℓ ∈ L\{0, 1}. +We claim that ξ = Tξ(1). Indeed, we know that ξ(1) ∈ S(A) so ξ(1) = T1(ξ(1)). +Furthermore, in the component i ∈ I, we have ξ(0)(i) = 1 if and only if ξ(1)(i) = 0, +so T0(ξ(1)) = T1(ξ(0)) = ξ(0) since ξ(0) ∈ S(A). Finally, for ℓ ̸∈ {0, 1} we have +Tℓ(ξ(1)) = 0 since for all i ∈ I we have ξ(1)(i) ∈ {0, 1}. +This concludes the +proof. +□ +Since S is dual to the essentially surjective functor U, we know that every B ∈ +BA is isomorphic to S(A) for some A ∈ A. Therefore, the following is a direct +consequence of the second part of Proposition 4.8. +Corollary 4.9. Every Boolean algebra B ∈ BA is isomorphic to S(P(B)). +Another immediate consequence of Proposition 4.8 is the following. +Corollary 4.10. For every Boolean algebra B ∈ BA, the algebra P(B) is the largest +algebra in A which has B as Boolean skeleton. That is, for every algebra A ∈ A +with S(A) ∼= B there exists an embedding A ֒→ P(B). +We now have everything at hand to prove the main theorem of this subsection. +Theorem 4.11. P is naturally isomorphic to the dual of V⊤ and, therefore, +S ⊣ P. +Proof. First we prove the statement on the finite level. In other words, we want to +show that, in StoneL, +ΣLP(B) ∼= V⊤Σ(B) +holds for every finite Boolean algebra B. More explicitly, after spelling out the +definition of the functors involved we want to show +(2) +� +A(P(B), L), im +� ∼= +� +BA(B, 2), v⊤� + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +21 +for every finite Boolean algebra B. First, since B is finite there is some positive +integer k such that B ∼= 2k. We combine the following isomorphisms in Stone. Due +to Proposition 4.3 we know +A +� +P(B), L +� ∼= BA +� +S(P(B)), 2 +� +, +And due to Corollary 4.9 we know +S(P(B)) ∼= B. +Putting these together, we get +A(P(B), L) ∼= BA(B, 2). +In fact, this even yields an isomorphism in StoneL as desired in (2), because +� +A(P(B), L), im +� ∼= +� +A(Lk, L), im +� ∼= +� +A(Lk, L), v⊤� +where the last equation holds due to Proposition 3.5. +So we know that the restriction of P to the category of finite Boolean algebras +Pω : BAω → A is dual to the restriction (V⊤)ω of V⊤ to the category Setω +L. There +is a unique (up to natural iso) finitary (i.e., filtered colimit preserving) extension +of Pω to Ind(BAω) ≃ BA, and this extension is naturally isomorphic to the dual of +V⊤ (since V⊤ preserves all limits except for the terminal object, it is the unique +cofinitary extension of (V⊤)ω). To show that P coincides with this unique extension +(up to natural isomorphism), it suffices to show that P is finitary as well. Since P +preserves monomorphisms (it is easy to see by definition that if ϕ ∈ BA(B1, B2) is +injective, then Pϕ is injective as well), we can apply [2, Theorem 3.4], which states +that P is finitary if and only if the following holds. +For every Boolean algebra B ∈ BA, for every finite subalgebra A ֒→ P(B) the +inclusion factors through the image of the inclusion of some finite subalgebra B′ ֒→ +B under P. +To see this write A ∼= +� +i≤n Si as product of finite subalgebras of L. Then, by +Corollary 4.9, we know that S(A) ∼= 2n embeds into B. Now by Lemma 4.7 we +have P(2n) ∼= Ln and the natural inclusion � +i≤n Si ֒→ Ln yields our factorization +A +P(B) +P(2n) +This concludes the proof. +□ +In particular, if L is primal, we get an explicit categorical equivalence witnessing +Hu’s theorem. +Corollary 4.12. [36] If L is primal, then S ⊣ P yields a categorical equivalence +between A and BA. +We also get an algebraic analogue of Proposition 4.1(i). +Corollary 4.13. The functor P is fully faithful and identifies BA with a reflective +subcategory of A. +By now we found detailed descriptions of most of the functors appearing in Figure +3. We are still missing is an algebraic understanding of the adjunction Q ⊣ I. This + +22 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +gap is filled in the next subsection. As we will see, it is closely connected to the +adjunction S ⊣ P. +4.4. The subalgebra adjunctions. For every subalgebra S ≤ L, there is an +adjunction +(3) +Stone +VS +⊥ +� StoneL +CS +� +which we explore in this subsection. +The functor VS : Stone → StoneL is given on objects by +VS(X) = (X, vS) where ∀x ∈ X : vS(x) = S, +and assigns every morphism to itself. +The functor CS : StoneL → Stone is given on objects by +CS(X, v) = {x ∈ X | v(x) ≤ S}. +On morphisms it acts via restriction, that is, given a morphism m: (X, v) → (Y, w), +define m |CS(X) : CS(X) → CS(Y ). This is well-defined since +x ∈ CS(X, v) ⇔ v(x) ≤ S ⇔ w(m(x)) ≤ v(x) ≤ S ⇔ m(x) ∈ CS(Y, w). +Comparing this with Subsection 4.1, the reader may easily verify V S ⊣ CS. +Indeed, the adjunction V S ⊣ CS generalizes the following adjunctions in Figure 3: +• V⊤ ⊣ U in the case where S = L is the largest subalgebra of L, +• V⊥ ⊣ C in the case where S = ⟨0, 1⟩ is the smallest subalgebra of L. +What is special about these two extreme cases is the additional adjunction U ⊣ V⊤, +which ‘glues’ the two adjunctions into the chain described in Subsection 4.1. +To better understand the adjunction corresponding to a subalgebra S ≤ L, we +dissect it into two parts as follows. +Stone +V⊤ +⊥ +� StoneS +U +� +ιS +⊥ +� StoneL +(CS,−) +� +Here, ιS is the natural inclusion and the functor (CS, −) is defined by +(X, v) �→ (CS(X), v|CS(X)) +on objects and, exactly like CS, acts via restriction on morphisms. It is easy to see +that this really is a decomposition of the adjunction (3), that is, +VS = ιS ◦ V⊤ and CS = U ◦ (CS, −). +As before, we want to carry everything over to the algebraic side, where the dissec- +tion takes place through the subvariety +AS := HSP(S). +We illustrate the entire situation in Figure 4. Note that S ≤ L is itself semi-primal, +so the semi-primal duality given by ΣS and ΠS as well as the adjunction S ⊣ PS +make sense in this context. + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +23 +StoneL +ΠL +� +� +ιS +⊣ +(CS,−) +� +A +ΣL +� +� +ιS +QS +⊢ +� +StoneS +ΠS +� +� +V⊤ +⊣ +U +� +AS +ΣS +� +� +PS +S +⊢ +� +Stone +Π +� BA +Σ +� +Figure 4. Dissecting the subalgebra adjunction of S ≤ L. +Again, ιS denotes the natural inclusion. Although it may seem obvious, it is not +immediate that ιS really is the dual of ιS. To prove it, we make use of the following +unary term, which will play an important role for the remainder of the subsection: +χS(x) = +� +s∈S +Ts(x). +On L, this simply corresponds to the characteristic function of S ⊆ L. +It is, +furthermore, characteristic for the subvariety AS in the following sense. +Lemma 4.14. An algebra in A is a member of AS if and only if it satisfies the +equation χS(x) ≈ 1. +Proof. Clearly every member of AS satisfies the equation since S satisfies it. For +the other direction, let A ∈ A satisfy χS(a) = 1 for all a ∈ A. We know that A can +be embedded into some LI and for each a ∈ A and i ∈ I, we have χS(πi(a)) = 1 +which implies that πi(a) ∈ S. Therefore, A can be embedded into SI. +□ +Now, let A ∈ AS and let h ∈ A(ιS(A), L) be a homomorphism. Since h preserves +equations, for every a ∈ A we get +χS(a) = 1 ⇒ χS(h(a)) = 1 +and, therefore, h ∈ A(A, S). So we showed A(A, L) = AS(A, S) for A ∈ AS, which +immediately implies the following. +Corollary 4.15. The inclusion functor ιS is the dual of the inclusion functor ιS. +To complete the picture, we only need to describe the functor QS from Figure 4. +Let α: idA ⇒ ιS ◦ QS be the unit of the adjunction QS ⊣ ιS. For any A ∈ A, the +algebra QS(A) is universal for AS in the following sense: +For every B ∈ AS and every homomorphism f : A → B, there is a unique +ˆf : QS(A) → B such that ˆf ◦ αA = f. +A +QS(A) +B +f +πA +∃ ˆ +f + +24 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +Therefore, the functor QS may be understood as a quotient. +There is a well- +known connection between quotients and equations introduced by Banaschewski +and Herrlich in [4]. Not surprisingly, the equation corresponding to QS is given +by χS(x) ≈ 1, which is an easy consequence of the above discussion together with +Lemma 4.14. We summarize the results of this subsection as follows. +Theorem 4.16. For every subalgebra S ≤ L, there is an adjunction +BA +IS +⊤ +� A +KS +� +which can be dissected as +BA +PS +⊤ +� AS +S +� +ιS +⊤ +� A +QS +� +where ιS is the natural inclusion functor of the subvariety HSP(S) ֒→ HSP(L) and +QS is the quotient functor corresponding to the equation χS(x) ≈ 1. +In particular, in the case where S is the smallest subalgebra of L, we can recover +the functors I = ιS ◦ PS and Q from Figure 3 . +Corollary 4.17. The functor I: BA → A is, up to categorical equivalence, an +inclusion. The functor Q: A → BA is, up to categorical equivalence, the quotient +by the equation +χE(x) ≈ 1, +where E = ⟨0, 1⟩ is the smallest subalgebra of L. +Proof. Being the smallest subalgebra of a semi-primal algebra, E is primal. There- +fore, by Corollary 4.12, the adjunction S ⊣ PE is an equivalence of categories. The +statement follows from Theorem 4.16. +□ +Clearly Corollary 4.13 holds not only for P, but for all the functors IS. Among +them, I is special since it also has a right-adjoint. This yields the following algebraic +version of Proposition 4.1(ii). +Corollary 4.18. The functor I is fully faithful and identifies BA with a reflective +and coreflective subcategory of A. +We showed that, if a finite lattice-based algebra M is semi-primal, then there +is an adjunction PE ⊣ S ⊣ PM, where E is the smallest subalgebra of M. In the +next subsection we show that, conversely, the existence of an adjunction resembling +this one fully characterizes semi-primality of a finite lattice-based algebra M. +4.5. Characterizing semi-primality via adjunctions. The aim of this subsec- +tion is to find sufficient conditions for an adjoint of PM to imply semi-primality +of the algebra M. +We will then show that, in particular, these conditions are +consequences of U and S from Figure 3 being (essentially) topological functors. +Recall that, in Definition 4.6, the Boolean power functor PM : BA → HSP(M) +was defined for arbitrary finite algebras M. Of course, if S is a subalgebra of M, +then PS can also be seen as a functor into HSP(M), and in the following there +is no need to distinguish between these two functors in our notation. The functor + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +25 +PM is faithful (unless M is trivial), but it is usually not full. In fact, it is easy to +see that PM can only be full if M does not have any non-trivial automorphisms. +In the main theorem of this subsection we show that, if PM is full and has a +left-adjoint resembling S, then a lattice-based algebra M is semi-primal. +Theorem 4.19. Let M be a finite lattice-based algebra. Then M is semi-primal +if and only if PM is full and there is a faithful functor s: HSP(M) → BA which +satisfies +PE ⊣ s ⊣ PM, +where E = ⟨0, 1⟩ is the smallest subalgebra of M. +Proof. If M is semi-primal, then PM is full since it is dual to the full functor V⊤, the +functor s = S is faithful since it is dual to the faithful functor U and PE ⊣ S ⊣ PM +was shown in the last two subsections. +Now for the converse, assume that PM is full and there is a faithful functor +s: HSP(M) → BA with PE ⊣ s ⊣ PM. For abbreviation we write V for HSP(M). +We will make use of the following properties of s: +(i) The unit η: idV ⇒ PM ◦ s is a monomorphism in each component, +(ii) s preserves monomorphisms and finite products. +Condition (i) follows from s being faithful and (ii) follows from s being a right- +adjoint. +Our first goal is to prove the equivalence +(4) +s(A) ∼= 2 ⇔ ∃S ∈ S(M) : A ∼= S. +If s(A) ∼= 2, use that by (i) there is an embedding A ֒→ PM(s(A)). +Since +PM(s(A)) ∼= M, it follows that A is isomorphic to a subalgebra of M. +Con- +versely, first note that s(M) ∼= 2 since, using that PM is full and s ⊣ PM, we +have +1 = |BA(2, 2)| = |V(M, M)| = |V +� +M, PM(2) +� +| = |BA +� +s(M), 2) +� +|, +which is only possible for s(M) ∼= 2. Now if A ∼= S ∈ S(M) then, due to (ii), +the natural embedding S ֒→ M induces an embedding s(S) ֒→ s(M). Therefore +s(S) ∼= 2 since s(M) ∼= 2 does not have any proper subalgebras. +Next we show that M does not have any non-trivial internal isomorphisms. +For every subalgebra S ∈ S(M), there is a bijection between the set of Boolean +homomorphisms s(S) → 2 and the set of homomorphisms S → PM(2). Due to +(4) we have s(S) ∼= 2, so the former only has one element. Since PM(2) ∼= M this +means that there is only one homomorphism S → M, namely the identity on S. +Every non-trivial internal isomorphism with domain S would define another such +homomorphism, resulting in a contradiction. +We now show that M is semi-primal, using the characterization of semi-primality +in Proposition 2.3. That is, we want to show that M has a majority term and every +subalgebra of M2 is either a product of subalgebras or the diagonal of a subalgebra +of M. +Since M is based on a lattice, a majority term is given by the median +(see the paragraph before Proposition 2.3). +Let A ≤ M2 be a subalgebra and +let ι: A ֒→ M be its natural embedding. +Due to (ii), this embedding induces +an embedding s(A) ֒→ s(M2) into s(M2) ∼= 22. Therefore, either s(A) ∼= 22 or +s(A) ∼= 2. +Let p1 : A → M and p2 : A → M be ι followed by the respective +projections M2 → M. + +26 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +First assume that p1 and p2 coincide. +Then clearly A embeds into M, and +therefore it is isomorphic to some subalgebra S of M. Since M has no non-trivial +internal isomorphisms, A needs to coincide with the diagonal of S. +If p1 and p2 are distinct then, using that s is faithful, the morphisms sp1 : s(A) → +2 and sp2 : s(A) → 2 are distinct as well. This implies that s(A) ∼= 22. Using the +adjunction PE ⊣ s we get +4 = |BA(22, s(A))| = |V(E2, A)| and 4 = |BA(22, s(M2))| = |V(E2, M2)|. +So there are exactly four distinct homomorphisms E2 → A and, since ι is a +monomorphism, their compositions with ι are also four distinct homomorphisms +E2 → M2. Therefore every of the former homomorphisms arises in such a way. +In particular, the natural embedding E2 ֒→ M2 arises in this way, which implies +(0, 1) ∈ A and (1, 0) ∈ A. As noted in [22], this leads to A = p1(A) × p2(A), since +whenever (a, b), (c, d) ∈ A we also have +(a, d) = +� +(a, b) ∧ (1, 0) +� +∨ +� +(c, d) ∧ (0, 1) +� +∈ A. +This concludes the proof. +□ +In the remainder of this subsection we show how this theorem relates to the +theory of topological functors (see, e.g., [1, Chapter VI.21] or [8, Chapter 7]). In- +tuitively speaking, topological functors behave similarly to the forgetful functor +Top → Set out of the category of all topological spaces. Still, the definitions in- +volved are rather technical and the reader not familiar with this topic may skip this +part. +Definition 4.20. We call a functor F: C → D +(1) topological if it is faithful and every F-structured source has an initial lift, +(2) essentially topological if it is topological up to categorical equivalence of C +and D. +The need for this distinction arises because certain properties of topological +functors, e.g. amnesticity [1, Definition 3.27], are not preserved under categorical +equivalence (this issue is addressed in [49]). +The following is our key observation for the last part of this subsection. +Proposition 4.21. The forgetful functor U: StoneL → Stone is topological and the +Boolean skeleton functor S: A → BA is essentially topological. +Proof. We only need to show that U is topological, which immediately implies that +S is essentially topological due to [1, Theorem 21.9] together with the fact that S +is naturally isomorphic to the dual of U. +Of course U is faithful since it is the identity on morphisms. +Now let X ∈ +Stone and let (fi : X → U(Xi, vi))i∈I be a U-structured source (i.e., a collection of +continuous maps) indexed by a class I. We define v: X → S(L) by +v(x) = +� +i∈I +vi(fi(x)). +Note that this is well-defined, since S(L) is finite and that (X, v) is a member +of StoneL, since v−1(S↓) = � +i∈I f −1 +i +(v−1 +i +(S↓)) is closed. Every fi is now also a +morphism in StoneL, which defines a lift of the source. To show that it is initial, +assume there are StoneL-morphisms (gi : (Y, w) → (Xi, vi))i∈I and a continuous + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +27 +map g : Y → X with fi ◦ g = gi. All we need to show is that g defines a StoneL- +morphism (Y, w) → (X, v). To see this simply note that +v(g(y)) = +� +i∈I +vi +� +fi(g(y)) +� += +� +i∈I +vi(gi(y)) ≤ w(y), +which concludes the proof. +□ +We can now easily show the following. +Corollary 4.22. Let M be a finite lattice-based algebra. Then M is semi-primal +if and only if there is an essentially topological functor s: HSP(M) → BA which +satisfies +PE ⊣ s ⊣ PM, +where E = ⟨0, 1⟩ is the smallest subalgebra of M. +Proof. In the previous proposition we showed that if M is semi-primal, then S is +essentially topological. +Conversely, if such an essentially topological s exists, it is faithful by definition +and both its adjoints PM and PE are full by [1, Proposition 21.12]. Therefore, due +to Theorem 4.19, M is semi-primal. +□ +In this section we gained an algebraic understanding of all the functors between A +and BA appearing on the right-hand side of Figure 3. Furthermore, we now showed +how properties of the Boolean skeleton functor S characterize semi-primality. In +the next section we investigate how canonical extensions of algebras in A behave +under these functors. One of the main results is that the Boolean skeleton functor +S may be used to identify canonical extensions of algebras in A. +5. Discrete duality and canonical extensions +In this section we describe a semi-primal discrete duality similar to the well- +known discrete duality between Set and CABA, the category of complete atomic +Boolean algebras with complete homomorphisms. +It can be obtained from the +finite duality in a similar way to the one of Section 3, except that now we lift it +to the level of Ind(Setω +L) and Pro(Aω). The members of the latter category are +known to be precisely the canonical extensions [33] of members of A (see [21]), +and we will provide two new characterizations of this category (Corollary 5.8 and +Theorem 5.10). Lastly we show that, as in the primal case L = 2, the topological +duality from Section 3 can be connected to its discrete version via an analogue of +the Stone- ˇCech compactification. +Our first goal is to identify Ind(Setω +L). Although it may not be surprising, it will +still take some work to prove that it can be identified with the following category. +Definition 5.1. The category SetL has objects of the form (X, v) where X ∈ Set +and v: X → S(L) is an arbitrary map. A morphism m: (X, v) → (Y, w) is a map +X → Y which always satisfies +w(m(x)) ≤ v(x). +Remark 6. In the context of fuzzy sets, Goguen [34, 35] initiated the study of such +categories. This research was continued, e.g., in [5, 57]. In this remark we stick +to the notation of [35]. Given a complete lattice V, the category Set(V) of V-fuzzy +sets has objects (X, A) where A: X → V. Morphisms (X, A) → (X′, A′) are maps + +28 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +m: X → Y which satisfy A′(m(x)) ≥ A(x) for all x ∈ X. In the context of fuzzy +set theory, people were mainly interested in the case where V = [0, 1]. However, we +retrieve SetL in the case where V is the order-dual of S(L). +■ +Since we are interested in the Ind-completion of Setω +L, we will first discuss (fil- +tered) colimits in this category. +Lemma 5.2. The category SetL is cocomplete. The colimit colimi∈I(Xi, vi) of a +filtered diagram +� +fij : (Xi, vi) → (Xj, vj) | i ≤ j +� +is realized by +� +(� +i∈I Xi)/∼, ¯v +� +. +Here, for xi ∈ Xi and xj ∈ Xj, +xi ∼ xj ⇐⇒ ∃k ≥ i, j : fik(xi) = fjk(xj) +and +¯v([xi]) = +� +xi∼xj∈Xj +vj(xj). +Proof. The proof that SetL is cocomplete is completely analogous to the one in [57]. +For filtered colimits, on the underlying level of Set we know that X := � +i∈I(Xi)/∼ +with the canonical inclusions ρi : Xi → X is the colimit of the diagram. To see that +all the ρi are morphisms in SetL note +¯v(ρi(xi)) = +� +xi∼xj∈Xj +vj(xj) ≤ vi(xi). +Given another cocone γi : (Xi, vi) → (Z, u), the unique map g : X → Z is a mor- +phism in SetL since, for xi ∈ Xi and xi ∼ xj ∈ Xj we have u +� +g(ρj(xj)) +� += +u(γj(xj)) ≤ vj(xj) and thus +u +� +g([xi]) +� +≤ +� +xi∼xj∈Xj +vj(xj) = ¯v([xi]), +which concludes the proof. +□ +We will also make use of the following general result. +Lemma 5.3. Let F: C → D be a functor between categories C and D which both +admit filtered colimits. If F has a right-adjoint G which preserves filtered colimits, +then F preserves finitely presentable objects. +Proof. Let C ∈ C be finitely presentable. We want to show that F(C) is finitely +presentable in D. Let colimiDi be a filtered colimit in D. Then +D +� +F(C), colimiDi +� ∼= colimiC +� +C, G(Di) +� ∼= colimiD +� +F(C), Di +� +, +where the first isomorphism comes from the fact that G preserves filtered colimits +and C is finitely presentable. +□ +Corollary 5.4. If X is a finite set, then (X, v) is finitely presentable in SetL for +every v: X → S(L). +Proof. Let X = {x1, . . . , xn} and let v(xi) = Si. Then we can clearly identify +(X, v) ∼= +� +1≤i≤n +({xi}, vSi). +Since filtered colimits commute with finite limits in Set, it now suffices to show that +all ({xi}, vSi) are finitely presentable. Just like in Subsection 4.4 we can define the +adjunction VS ⊣ CS between SetL and Set for every subalgebra S ≤ L. By Lemma + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +29 +5.3 it now suffices to show that CS preserves filtered colimits. +So let (X, ¯v) be +a filtered colimit as in Lemma 5.2. We know that CS(X) = {[xi] | ∃xi ∼ xj ∈ +Xj, vj(xj) ≤ S}. Therefore, for all [xi] ∈ CS we can choose representatives with +xi ∈ CS(Xi, vi). This yields a bijection between CS(X) and colimCS(Xi, vi). +□ +We now have everything at hand to easily prove the following. +Theorem 5.5. Ind(Setω +L) is categorically equivalent to SetL. +Proof. Since SetL is cocomplete, the inclusion ι: Setω +L → SetL has a unique finitary +extension ˆι: Ind(Setω +L) → SetL. Since ι is fully faithful and, by the above corollary, +maps all objects to finitely presentable objects in SetL, this extension is also fully +faithful. To see that it is essentially surjective note that, just like in Set, every +member of SetL is the filtered colimit of its finite subsets. +□ +We now take a closer look at the category Pro(Aω). It is well-known that it +consists of the canonical extensions [33] of algebras in A. In [21] a description of +these canonical extensions as topological algebras can be found. But, as in the +case of complete atomic Boolean algebras CABA ≃ IP(2), this need not be the +only description. In the following we apply results of Section 4 to find two easy +alternatives. The first one is in terms of (arbitrary) products of subalgebras of L +with complete homomorphisms. +Definition 5.6. Let ˆ +A be the category with algebras from IPS(L) as objects and +complete homomorphisms as morphisms. +We can essentially repeat our proof of the finite duality from Corollary 3.6, once +we prove the following result analogous to Proposition 3.5. +Proposition 5.7. Let A = � +i∈I Si ∈ +ˆ +A. +Then the complete homomorphisms +A → L are precisely the projections (followed by inclusions) in each component. +Proof. By Proposition 4.3 there is a bijection between A(A, L) and BA(S(A), 2) +given by h �→ h|S(A). In particular, if h is complete, then so is its restriction. Since +S(A) = 2I, the only complete homomorphisms S(A) → 2 are the projections, and +they are the restrictions of the respective projections A → L. +□ +Corollary 5.8. Pro(Aω) is categorically equivalent to ˆ +A +Proof. By Theorem 5.5 it suffices to show that SetL is dually equivalent to ˆ +A. This +is done completely analogous to the proof of Corollary 3.6. +□ +The second description of Pro(Aω) is in terms of the Boolean skeleton. +Definition 5.9. The category CAA has as objects algebras A ∈ A which have a +complete lattice-reduct and which satisfy S(A) ∈ CABA. The morphisms in CAA +are the complete homomorphisms. +Theorem 5.10. Pro(Aω) is categorically equivalent to CAA. +Proof. Using Corollary 5.8 we show that CAA is categorically equivalent to +ˆ +A. +Clearly there is a fully faithful inclusion functor ˆ +A ֒→ CAA. So it suffices to show +that this functor is essentially surjective. In other words, we want to show that +every object of CAA is isomorphic to a product of subalgebras of L. + +30 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +So consider A ∈ CAA. Since the adjunction S ⊣ P restricts to CABA and CAA, +we can use Corollary 4.10 to get a complete embedding ηA : A ֒→ P(S(A)). Since +S(A) is in CABA it is isomorphic to 2I for some index set I. Thus P(S(A)) ∼= +P(2I) ∼= LI. We show that A is isomorphic to the direct product of subalgebras +� +i∈I pri(ηA(A)). For this it suffices to show that the injective homomorphism ηA +maps onto it. So let α be an element of this product. For each i ∈ I choose ai ∈ A +such that pri(ηA(ai)) = α(i). Since 2I ∼= S(A) ⊆ A all atoms bi ∈ 2I (defined by +bi(j) = 1 iff j = i) can be considered as members of A. Now define +a = +� +{ai ∧ bi | i ∈ I}. +Since A is complete, we have a ∈ A. And since ηA is a complete homomorphism +we have ηA(a) = α (because pri(ηA(a)) = ηA(ai) = α(i)). +□ +With the results from this section thus far, it is clear that the chains of adjunc- +tions from Section 4 (summarized in Figure 3) have their discrete counterparts, +equally defined, between SetL and Set and CAA and CABA, respectively. To make +the connection between Figure 3 and its discrete counterpart, we finish this section +by connecting the respective dualities as indicated in Figure 5. +StoneL +� +� +βL +⊣ +(−)♭ +� +A +� +� +ιc +(−)δ +⊢ +� +SetL +� CAA +� +Figure 5. Compactification and canonical extension. +Here (−)♭ : StoneL → SetL is the forgetful functor with respect to topology and +ιc : CAA → A is the obvious inclusion functor (note that both these functors are not +full). The functor (−)δ takes an algebra to its canonical extension. In the primal +case L = 2, it is well-known that β2 =: β is the Stone- ˇCech compactification (see, +e.g., [40, Section IV.2]). This has been generalized to the Bohr compactification in +a (much broader) framework which includes ours in [20]. However, since things are +particularly simple in our setting, we directly show how to define βL. +Given (X, v) ∈ SetL, there is a natural way to extend v to the Stone-ˇCech +compactification β(X) of X. Indeed, since v: X → S(L) can be thought of as +a continuous map between discrete spaces, by the universal property of β it has +a unique continuous extension ˜v: β(X) → S(L). Here, ˜v−1(S↓) is given by the +topological closure of v−1(S↓) in β(X). Thus, for every morphism f : (X, v) → +(Y, w) in SetL, the continuous map βf defines a morphism (β(X), ˜v) → (β(Y ), ˜w) +in StoneL. This is due to the observation that whenever x ∈ ˜v−1(S↓) = v−1(S↓), +by continuity of βf and the morphism property of f, we have βf(x) ∈ w−1(S↓) = +˜w−1(S↓). +Proposition 5.11. The functor βL : SetL → StoneL defined on objects by +βL(X, v) = (β(X), ˜v) + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +31 +and by f �→ βf on morphisms is the dual of the canonical extension functor +(−)δ : A → CAA. +Proof. It suffices to show that βL satisfies the following universal property. Given +(Y, w) ∈ StoneL, every SetL-morphism f : (X, v) → (Y, w) extends uniquely to a +StoneL-morphism ˜f : (β(X), ˜v) → (Y, w). On the levels of Set and Stone we get +a unique continuous extension ˜f. To show it is a StoneL-morphism, similarly to +before, note that if x ∈ v−1(S↓), then by continuity +˜f(x) ∈ f +� +v−1(S↓) +� +⊆ w−1(S↓). +Since w−1(S↓) is closed it equals its own closure. This concludes the proof. +□ +This nicely wraps up this paper by connecting all of its main sections. In the +last section we give a quick summary and discuss some possible directions of future +research along similar lines. +6. Concluding Remarks and Further Research +We explored semi-primality by means of category theory, showing how a variety +generated by a semi-primal lattice expansion relates to the variety of Boolean alge- +bras. Various adjunctions provide insight into the many similarities between these +varieties. A schematic summary of our results can be found in Figure 6, which also +emphasizes once more how close BA and A really are. +Setω +L +Aω +SetL +CAA +StoneL +A +Set +CABA +Stone +BA +Setω +BAω +Pro +Ind +Pro +Ind +Ind +Pro +Ind +Pro +Figure 6. Summary of our results. +We plan to follow up this research by developing a coalgebraic framework for +modal extensions of the many-valued logic corresponding to a semi-primal variety. +As mentioned before, from this point of view it is reasonable to assume that L is +based on a lattice. However, it seems plausible that our results generalize to the +slightly more general case of semi-primal algebras which possess a coupling in the +sense of [26], essentially since Proposition 2.8 and Theorem 3.2 still apply to this +case. +We will now sketch some more potential ways to follow up this research. In +general, we hope to have set an example in exploring concepts in universal algebra +through the lens of (mostly elementary) category theory. + +32 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +For example, other variants of primality could be investigated in a similar man- +ner. +Definition 6.1. A finite algebra M is called +(1) demi-semi-primal if it is quasi-primal and every internal isomorphism of M +can be extended to an automorphism of M (see [53]). +(2) demi-primal if it is quasi-primal and has no proper subalgebras (see [53]). +(3) infra-primal if it is demi-semi primal and every internal isomorphism is an +automorphism on its domain (see [27]). +(4) hemi-primal if every operation on M which preserves congruences is term- +definable in M (see [28]). +Question 1. What is the categorical relationship between BA and the variety gener- +ated by an algebra which is quasi-primal or which satisfies one of the properties of +Definition 6.1? What about the relationship between distinct variations of primality +to each other? +For quasi-primal algebras (and thus, in particular, for algebras satisfying (1), +(2) or (3)), there is the duality theorem by Keimel-Werner [41] (which is also a +natural duality [17]), possibly a good starting point to a discussion similar to the +one presented here. +Hemi-primality seems to have received less attention. To the best of the authors +knowledge, no duality for varieties generated by hemi-primal algebras is known thus +far. +Question 2. Is it possible to obtain a duality for hemi-primal varieties, for example +one which stems from a finite dual equivalence using methods similar to our proof +of semi-primal duality in Section 3? +The Boolean power functor PM: BA → HSP(M) was defined for an arbitrary +finite algebra M. In the light of our results from Section 4, the following question +arises. +Question 3. Under which circumstances does the functor PM have a left-adjoint? +Which information about M can be retrieved from properties of the functors of the +form PS with S ≤ M? +If we consider this work as not only comparing varieties but comparing dualities, +another range of questions appears. +Question 4. What is the category theoretical relationship between different dual +equivalences? +For example, one could consider Priestley duality [52] or Esakia +duality [23]. +Lastly, another category theoretical approach to universal algebra, which has +not been discussed in this paper, is given by Lawvere theories. For example, Hu’s +theorem has been analyzed from this angle in [51]. Of course, one could also try to +find out more about other variants of primality in this context. +Question 5. How can semi-primality and other variants of primality be expressed +in terms of Lawvere theories? + +NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES +33 +Appendix A. Some semi-primal FLewalgebras +Here we go into more detail in some claims made in Subsection 2.3.2. We provide +examples of semi-primal FLew-algebras, both chain-based and non chain-based, in- +cluding the proof of semi-primality for each one of them. +All of the examples +and their labels are taken from the list [31] by Galatos and Jipsen. For simplicity +we only discus FLew-algebras without any idempotent elements other than 0 and +1. Due to Corollary 2.16 they are all quasi-primal. To prove semi-primality, by +Proposition 2.2, it suffices to describe all subalgebras and argue why there can’t be +any non-trivial isomorphisms between then. +We begin with the quasi-primal FLew-chains of five elements R5,1 +1,17 to R5,1 +1,22 in +[31, p.2, row 2] depicted in Figure 7. +1 +a +b +c +a2 +R1,5 +1,17 +1 +a +b +a2 +ab +R1,5 +1,18 +1 +a +a2 +c +ab +R1,5 +1,19 +1 +a +b +a2 = ab +b2 = ac +R1,5 +1,20 +1 +a +a2 +ab +b2 = ac +R1,5 +1,21 +1 +a +b +a2 = b2 +ac +R1,5 +1,22 +Figure 7. The quasi-primal FLew-chains of order five. +Claim 1. Except for the first one, all algebras depicted in Figure 7 are semi-primal. +Proof. R1,5 +1,17 is not semi-primal because it has isomorphic subalgebras {0, 1, a, c} +and {0, 1, a, d}. +In the following we show why the other ones are semi-primal by describing the +subalgebras other than the obvious ones {0, 1} and {0, 1, a, b, c}. +Since isomor- +phisms need to be order-preserving, it suffices to note that there are never two +subalgebras of the same size in the examples below. +R1,5 +1,18: There are no other subalgebras since {¬a, a2} = {b, c} ⊆ ⟨a⟩ and ¬b = +¬c = a, thus a ∈ ⟨b⟩ and a ∈ ⟨c⟩. +R1,5 +1,19: There is the subalgebra ⟨a⟩ = ⟨b⟩ = {0, 1, a, b} since a → b = a, ¬a = b and +¬b = a. Since a = ¬c we have a ∈ ⟨c⟩, so c generates the entire algebra. +R1,5 +1,20: There are two different sized subalgebras ⟨a⟩ = ⟨c⟩ = {0, 1, a, c} (since +¬a = c, ¬c = a and a → c = a) and ⟨b⟩ = {0, 1, b} (since ¬b = b → b = b) +R1,5 +1,21: Note that this algebra corresponds to the �Lukasiewicz-chain �L4. As thus +expected, there is the subalgebra ⟨b⟩ = {0, 1, b}, while b ∈ ⟨a⟩∩⟨c⟩ since a = ¬c, c = +¬a and b = a2. +R1,5 +1,22: There is the subalgebra ⟨a⟩ = ⟨c⟩ = {0, 1, a, c} (since ¬a = c, ¬c = a +and a → c = a). Since ¬b = c and ¬c = a we find that b generates the entire +algebra. +□ +To also provide non chain-based examples, we examine the FLew-algebras R6,2 +1,11 +([31, p.18, row 4]) and R6,3 +1,9 ([31, p.20, row 1]) depicted in Figure 8. + +34 +ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX +1 +a +b +c = a2 +d +ab +R6,2 +1,11 +1 +a +b +c +d = a2 = c2 +ab = bc +R6,3 +1,9 +Figure 8. Two semi-primal FLew-algebras of order six. +Claim 2. The two FLew-algebras depicted in Figure 8 are semi-primal. +Proof. R6,2 +1,11: The only possible candidate for an automorphism of this algebra is +the bijection f exchanging c and d (since it needs to be order-preserving). This +map, however, is not a homomorphism, as witnessed by the fact that f(a2) = +f(c) = d while f(a)2 = a2 = c. The only other subalgebra other than {0, 1} is +⟨a⟩ = {0, 1, a, b, c} since we have ¬a = b, a2 = c, ¬c = a, a → b = a, a → c = a +and b → c = a. Since this subalgebra is a chain, it does not have any non-trivial +isomorphisms. Since ¬d = a we know that d generates the entire algebra, so there +are no more subalgebras to consider. +R6,3 +1,9: Again, there is only one possible candidate for an automorphism of this +algebra, namely the bijection g exchanging b and c. This is not a homomorphism +because g(b2) = g(0) = 0 while g(b)2 = c2 = d. 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Studien zur Algebra und ihre Anwendungen, vol. 6. +Akademie Verlag, 1978. +Fowler School of Engineering, Chapman University, 1 University Drive, 92866 Or- +ange, California, USA +Email address: akurz@chapman.edu +Department of Mathematics, FSTM, University of Luxembourg, 6 Avenue de la Fonte, +L-4364 Esch-sur-Alzette, Luxembourg +Email address: wolfgang.poiger@uni.lu, bruno.teheux@uni.lu + diff --git a/yNFQT4oBgHgl3EQfxzYK/content/tmp_files/load_file.txt b/yNFQT4oBgHgl3EQfxzYK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..809f9a35d8346c35f3ed6c8b0ee3205ad43528fb --- /dev/null +++ b/yNFQT4oBgHgl3EQfxzYK/content/tmp_files/load_file.txt @@ -0,0 +1,1465 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf,len=1464 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='13406v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='LO] 31 Jan 2023 NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We study varieties generated by semi-primal lattice-expansions by means of category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We provide a new proof of the Keimel-Werner topological duality for such varieties and, using similar methods, establish its discrete version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We describe multiple adjunctions between the variety of Boolean algebras and the variety generated by a semi-primal lattice-expansion, both on the topological side and explicitly algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, we show that the Boolean skeleton functor has two adjoints, both defined by taking certain Boolean powers, and we identify properties of these adjunctions which fully characterize semi-primality of an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lastly, we give a new charac- terization of canonical extensions of algebras in semi-primal varieties in terms of their Boolean skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Introduction Primality and its variations are classical topics in universal algebra which were prominently studied during the second half of the 20th century [54, 58, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' During the 1950s, Foster introduced primal algebras in his generalized ‘Boolean’ theory of universal algebras [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Generalizing functional completeness of the two- element Boolean algebra, an algebra P is primal if every operation f : P n → P is term-definable in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The intuition that a primal algebra P is ‘close to’ the two- element Boolean algebra 2 was confirmed by Hu’s theorem [36, 37], which states that a variety V is categorically equivalent to the variety BA of Boolean algebras (generated by 2) if and only if V is generated by a primal algebra P ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In 1964, Foster and Pixley introduced the first variation of primality, which they called semi-primality [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Unlike primal algebras, a semi-primal algebra may have proper subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Accordingly, in a semi-primal algebra L, we only require the operations f : Ln → L which preserve subalgebras to be term-definable in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Semi-primal varieties (that is, varieties of the form HSP(L) where L is semi-primal) are well-understood from the viewpoint of ‘classical’ universal algebraic structure theory [29, 30, 26, 45] as well as from the viewpoint of duality theory [41, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' From the perspective of category theory, semi-primal varieties were classified up to Morita equivalence in [7] - however, this is done using purely algebraic tools based on [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In this paper, we further advance the category theoretical study of semi-primality by putting a semi-primal variety A in relationship with other varieties, in particular with the primal variety BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Although Hu’s theorem implies that the varieties are usually not categorically equivalent, we demonstrate that, nevertheless, there is a rich relationship between A and BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular we explicate the intuition that semi-primal algebras are still ‘close to’ the two-element Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 06E15, 06E75, 08A40, 08C05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' semi-primal algebras, primal algebras, ternary discriminator, stone duality, boolean skeleton, boolean power, canonical extension, universal algebra, category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 1 2 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX More specifically, we investigate multiple adjunctions between BA and the variety A generated by a semi-primal algebra L with an underlying bounded lattice (see Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For one, this assumption yields a useful characterization of semi- primality via certain unary terms (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8) which we prominently use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Furthermore, since L has no one-element subalgebras, the dual category of A has a particularly simple description (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Apart from these advantages, the restriction to lattice-based algebras is motivated by the connection to many- valued logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If we consider L as an algebra of propositional truth-degrees, an underlying bounded lattice is a reasonable assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, Maruyama [43] generalized J´onsson-Tarski duality to modal extensions of semi-primal algebras with bounded lattice reducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We plan to demonstrate applications of our results to many-valued (coalgebraic) modal logic in subsequent work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since, in addition, there are already plenty of examples of such algebras (see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3), it is reasonable to stick to this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Although we were mainly motivated by questions arising in logic, we particularly hope that this paper will be of interest to algebraists interested in category theory as well as to category theorists interested in universal algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let us point out that, in this paper, the category theoretical approach to universal algebra is different from other common ones via Lawvere theories or monads (these are well-exposed in [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Indeed, this paper is not about reformulating and generalizing algebraic concepts into categorical language, but rather to apply category theory as a tool to gain new insight into a concrete topic in universal algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, the fact that the variety A is the completion of the full subcategory of its finite members Aω under filtered colimits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', A ≃ Ind(Aω)) can be helpful to make the step from finite to infinite, for example to extend functors defined on Aω to the full variety A in a canonical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Motivated by [40], we furthermore use this fact to give a new proof of the semi-primal duality [41, 17] by lifting the corresponding finite duality (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='7 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' By replacing Ind(Aω) by Pro(Aω), the closure under cofiltered limits, we prove the discrete version of the duality (resembling the duality between Set and the category CABA of complete atomic Boolean algebras) in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In Section 2 we recall well-known results about semi-primal algebras and the varieties they generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, we discuss semi- primal bounded-lattice expansions and provide examples thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In Section 3 we describe the topological duality for semi-primal algebras and, as mentioned above, provide an alternative proof for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Arguably the most important results of the paper are exposed in Section 4, where we describe a chain of four adjoint functors between A and BA (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Most prominently, the adjunction S ⊣ P is described in detail, first via duality and then explicitly algebraically (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The Boolean skeleton S: A → BA has, for example, been known for MVn-algebras [16] and was generalized to arbitrary semi-primal bounded lattice expansions by Maruyama [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Its right-adjoint P: BA → A relies on the construction of a Boolean power [9], a certain Boolean product [11] which was already introduced for arbitrary finite algebras in Foster’s original paper on primality [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the case where L is primal, we retrieve a concrete categorical equivalence witnessing Hu’s theorem (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We proceed to investigate the subalgebra adjunctions, which exist for each subalgebra S ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We manage to trace them back to the adjunction S ⊣ P after taking an appropriate inclusion/quotient (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 3 particular, we illustrate why the subalgebra adjunction Q ⊣ I corresponding to the smallest subalgebra of L is of special interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Indeed, towards the end of Section 4 we also show that the existence of an adjoint situation resembling I ⊣ S ⊣ P fully characterizes semi-primality of a lattice-based algebra (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Building on the results of Section 4, in Section 5 we prove the above-mentioned discrete duality for Pro(Aω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It is well-known that the algebras in this category correspond to the canonical extensions [33, 21] of algebras in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Notably, we show that these canonical extensions may be characterized almost purely in terms of their Boolean skeletons (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lastly we connect Sections 4 and 5 by describing an analogue of the Stone-ˇCech compactification in our setting (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We summarize our results schematically in Section 6 (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In addi- tion to the logical ramifications already mentioned, we believe that there are more potential ways to follow up our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, we hope to inspire further research in universal algebra through the lens of category-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Some open ques- tions directly related to the content of this paper are also collected in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Semi-primal algebras and the varieties they generate In the 1950s, Foster introduced the concept of primality in [24, 25], generalizing functional completeness of the two-element Boolean algebra 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A finite algebra L is called primal if, for all n ≥ 1, every function f : Ln → L is term-definable in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Besides the two-element Boolean algebra 2, the (n + 1)-element Post chain Pn and the field of prime order Z/pZ with 0 and 1 as constants are some famous examples of primal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Using Stone duality, Hu [36, 37] showed that a variety A is generated by a primal algebra (in other words, A = HSP(L) for some primal algebra L) if and only if A is categorically equivalent to the variety of Boolean algebras BA (see also [51] for a treatment using Lawvere theories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Of course we don’t expect any more meaningful category theoretical results about the relationship between A and BA in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' One purpose of this paper is to demonstrate that, in contrast, such results do arise as soon as we assume that L is semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Characterizations of semi-primality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since Foster’s original work, many variations of primality have been introduced (for an overview see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Among them, intuitively speaking, semi-primality seems to still be rather close to primality (a central theme of this paper is to show why this intuition is justified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In a slogan: semi-primal algebras are like primal algebras which allow subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Note that a primal algebra L does not have any proper subalgebra S ≨ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Otherwise, picking any s ∈ S and ℓ ∈ L\\S, no function f : L → L with f(s) = ℓ can possibly be term-definable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Semi-primality, introduced by Foster and Pixley in 1964 (see [29]) does not impose this restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Recall that a function f : Ln → L preserves subalgebras if f(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , an) is in the subalgebra generated by {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , an} for any choice of a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , an ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Clearly, if a function is term-definable, then it preserves subalge- bras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In semi-primal algebras, the converse also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A finite algebra L is semi-primal if for every n ≥ 1, every function f : Ln → L which preserves subalgebras is term-definable in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 4 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX For example, the field of prime-order Z/pZ with only 0 as constant is semi-primal but not primal anymore - it now has {0} as proper subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' More interesting examples are described in detail in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the following we recall two well-known equivalent characterizations of semi-primality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The first one is based on the ternary discriminator term and the second one is based on the existence of a majority term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' First we recall the characterization of semi-primal algebras as special instances of discriminator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' These are the algebras in which the ternary discriminator t(x, y, z) = � z if x = y x if x ̸= y is term-definable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Finite discriminator algebras are also called quasi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' An internal isomorphism of L is an isomorphism ϕ: S1 → S2 between any two (not necessarily distinct) subalgebras S1 and S2 of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, if S ≤ L is a subalgebra, then the identity idS is an internal isomorphism of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In semi-primal algebras, there are no other internal isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [50, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='] A finite algebra L is semi-primal if and only if it is quasi-primal and the only internal isomorphisms of L are the identities on subalgebras of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Secondly, we recall the characterization of semi-primality based on a majority term, which can be useful to generate examples (see, for example, [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Recall that a majority term is a ternary term m(x, y, z) satisfying m(x, x, y) = m(x, y, x) = m(y, x, x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, every lattice L = (L, ∧, ∨) has a majority term given by the median m(x, y, z) = (x ∧ y) ∨ (x ∧ z) ∨ (y ∧ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [3, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='] A finite algebra L is semi-primal if and only if it has a majority term and every subalgebra of L2 is either the product of two subalgebras or the diagonal of a subalgebra of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The structure of semi-primal varieties was already well-studied in the original work by Foster and Pixley during the 1960s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To stay self-contained, we recall some results about these varieties which will be of use for us later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [29, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2] The variety A generated by a semi-primal algebra L coincides with the quasi-variety generated by L, that is A = ISP(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In addition to the characterizations above, there is a nice characterization of semi-primality of L in terms of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Recall that a variety is called arithmetical if it is congruence distributive and congruence permutable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [30, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1] A finite algebra L is semi-primal if and only if the variety generated by L is arithmetical, every subalgebra of L is simple, and the only internal isomorphisms of L are the identities of subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Together with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5 this implies that if L is semi-primal, then the collection of subalgebras S(L) considered as a subcategory of the variety generated by L, forms a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' ■ NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 5 The finite members of A are particularly well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For notation, given a concrete category C, we use Cω to denote the full subcategory of C generated by its finite members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, if A is a variety, we use Aω to denote the category of finite algebras in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [29, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1] Every finite algebra A ∈ Aω is isomorphic to a direct product of subalgebras of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We add yet another characterization of semi-primality in our particular case of interest (in which the algebra is based on a bounded lattice) in the following subsection (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Semi-primal bounded lattice expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In this subsection we set the scene for the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We aim to describe the relationship between the variety BA of Boolean algebras and the variety generated by a semi-primal algebra with underlying bounded lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Under the additional assumption that L is based on a bounded lattice, there is another nice characterization of semi-primality of L which will be particularly useful for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It relies on the following unary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let L be an algebra based on a bounded lattice L♭ = (L, ∧, ∨, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For all ℓ ∈ L we define Tℓ : L → L and τℓ : L → L to be the characteristic function of {ℓ} and {ℓ′ ≥ ℓ}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' That is, Tℓ(x) = � 1 if x = ℓ 0 if x ̸= ℓ and τℓ(x) = � 1 if x ≥ ℓ 0 if x ̸≥ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Even though the following result is essentially an instance of the more general [26, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1], we include an easy direct proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [26, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1] Let L be a finite algebra with an underlying bounded lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then the following conditions are equivalent: (1) L is semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (2) For every ℓ ∈ L, the function Tℓ is term-definable in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (3) T0 is term-definable and for every ℓ ∈ L, the function τℓ is term-definable in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (1) ⇒ (2): Since every subalgebra of L contains the set {0, 1}, semi-primality of L implies that all Tℓ are term-definable, since they preserve subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (2) ⇒ (1): First we show that the ternary discriminator is term-definable in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Consider the term c(x, y) = � ℓ∈L � (Tℓ(x) ∧ Tℓ(y) � , which satisfies c(x, y) = � 1 if x = y 0 if x ̸= y and d(x, y) := T0(c(x, y)) (note that this is the discrete metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The term t(x, y, z) = (d(x, y) ∧ x) ∨ (c(x, y) ∧ z) yields the ternary discriminator on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now we show that the only internal iso- morphisms of L are the identities of subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let ϕ : S1 → S2 be an internal 6 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX isomorphism of L and let s ∈ S1 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then 1 = Tϕ(s) � ϕ(s) � = ϕ � Tϕ(s)(s) � Since ϕ(0) = 0 we necessarily have Tϕ(s)(s) = 1, which is equivalent to ϕ(s) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Altogether, due to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2, we showed that L is semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (2) ⇒ (3): If the Tℓ are term-definable we can define τℓ(x) = � ℓ′≥ℓ Tℓ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (3) ⇒ (2): If T0 and the τℓ are term-definable we can define Tℓ(x) = τℓ(x) ∧ � ℓ′>ℓ T0 � τℓ′(x) � , which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In light of this result, we can turn any finite bounded lattice into a semi- primal algebra by adding Tℓ as unary operation for every element ℓ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' One might wonder how this differs from adding a constant symbol (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', a nullary operation) for every element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The difference is that adding a constant imposes the requirement that every subalgebra needs to contain the element corresponding to this constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Thus, the algebra that results after adding all constants does not have any proper subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' ■ We now state our main assumption, which from now on holds for the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The finite algebra L is semi-primal and has an under- lying bounded lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' From now on, let A := HSP(L) denote the variety generated by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3 we provide various examples of algebras satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' As noted in [43] (where the same assumption on L is made), from the point of view of many-valued logic, semi-primal algebras make good candidates for algebras of truth-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In this context the underlying bounded lattice is a natural minimal requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Examples of semi-primal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In this subsection we collect some ex- amples of semi-primal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' All of them are bounded lattice expansions (since most of them stem from many-valued logic), thus they all fit the scope of this paper (see Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For other examples we refer the reader to [10, 58, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' First, we describe several different semi-primal algebras based on finite chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To get examples based on lattices which are not necessarily totally ordered, in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2 (and Appendix A) we discuss semi-primal residuated lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular we describe a systematic way to identify them among the FLew-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Similarly, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3 illustrates how to identify semi-primal algebras which need not be totally ordered among the pseudo-logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' At the end of this subsection we recall Murski˘ı’s Theorem which states that, in some sense, almost all finite lattice- based algebras are semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Chain-based algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We will describe several different ways of turning the (n + 1)-element chain {0, 1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , n−1 n , 1} with its usual lattice-order into a semi- primal algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We present the examples ordered decreasingly by the amount of subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' First, turning a chain into a semi-primal algebra without any further impositions may be achieved as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The n-th general semi-primal chain is given by Tn = � {0, 1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , n−1 n , 1}, ∧, ∨, 0, 1, (T i n )n i=0 � , where the unary operations T i n are the ones from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For all n ≥ 1 the algebra Tn is semi-primal (this immediately follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Every subset of Tn which contains the set {0, 1} defines a subalgebra of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Next we find examples among the �Lukasiewicz-Moisil algebras, which were orig- inally intended to give algebraic semantics for �Lukasiewicz finitely-valued logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It turns out, however, that they encompass a bit more than that (see [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The logic corresponding to these algebras is nowadays named after Moisil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The n-th �Lukasiewicz-Moisil chain is given by Mn = � {0, 1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , n−1 n , 1}, ∧, ∨, ¬, 0, 1, (τ i n )n i=1 � , where ¬x = 1 − x and the unary operations τ i n are the ones from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For all n ≥ 1, the algebra Mn is semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This follows from characterization (3) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8 - we only have to check that T0 is term-definable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To see this note that we can define T1(x) = τ1(x) and T0(x) = T1(¬x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We proceed with a classical example from many-valued logic among the finite MV-algebras introduced by Chang (see [13, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' They give rise to the algebraic counterpart of �Lukasiewicz finite-valued logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The n-th �Lukasiewicz chain is given by �Ln = � {0, 1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , n−1 n , 1}, ∧, ∨, ⊕, ⊙, ¬, 0, 1 � , where x ⊕ y = min(x + y, 1), x ⊙ y = max(x + y − 1, 0) and ¬x = 1 − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For all n ≥ 1, the algebra �Ln is semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The proof of this fact can be found in [48, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The subalgebras of �Ln correspond to the divisors d of n and are of the form �Ld = {0, k n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , (d−1)k n , 1} where n = kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Other semi-primal chains are found among the Cornish algebras, which generalize Ockham algebras (see [18, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The n-th semi-primal Cornish chain is given by COn = � {0, 1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , n−1 n , 1}, ∧, ∨, ¬, f, 0, 1 � , where ¬x = 1 − x, f(0) = 0, f(1) = 1 and f( i n) = i+1 n for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For all n ≥ 1, the algebra COn is semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The proof of this fact can be found in [19, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The only proper subalgebra of COn is {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Finally, among the Post-algebras we find the well-known examples of chain-based algebras which are not only semi-primal, but even primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 8 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The n-th Post chain is given by Pn = � {0, 1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , n−1 n , 1}, ∧, ∨,′ , 0, 1 � where 1′ = 0 and ( i n)′ = ( i+1 n ) for 0 ≤ i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For all n ≥ 1, the algebra Pn is primal (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', [24, Theorem 35]) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Residuated Lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For a general survey of residuated lattices we refer the reader to [32, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We only consider bounded commutative residuated lattices here, with a particular focus on FLew-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A (bounded commutative) residuated lattice is an algebra R = (R, ∧, ∨, 0, 1, ⊙, e, →) such that (R, ∧, ∨, 0, 1) is a bounded lattice, (R, ⊙, e) is a commutative monoid and the binary operation → satisfies the residuation condition x ⊙ y ≤ z ⇔ x ≤ y → z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We call R a FLew-algebra if, in addition, it satisfies e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Our main tool to identify semi-primal FLew-algebras is [42, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='10], which implies that a FLew-algebra R is quasi-primal if and only if there is some n ≥ 1 such that (1) x ∨ ¬(xn) = 1 for all x ∈ R, where, as usual, we define ¬x as x → 0 (and xn refers to the n-th power with respect to ⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For our purposes this theorem has the following practical consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let R be a finite FLew-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If R does not contain any idem- potent elements (that is, elements with x ⊙ x = x) other than 0 and 1, then R is quasi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If R is based on a chain, the converse also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let R be a finite FLew-algebra with no other idempotent elements than 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Recall that, for any a ∈ R, we have ¬a = a → 0 = �{b ∈ R | a ⊙ b ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let a ∈ R\\{0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We show that there is some na such that ana = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since a is not idempotent we have a2 < a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Either a2 = 0 and we are done or a2 is again not idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In this case we have a4 < a2 and we repeat the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since R is finite, continuing this process we eventually need to find a2k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now R satisfies equation (1) for n = �{na | a ∈ R\\{0, 1}}, since we always have a ∨ ¬(an) = a ∨ ¬0 = a ∨ 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Thus R is quasi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now suppose that R is based on a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If a ∈ R\\{0, 1} is idempotent, then ¬a < a since for all b ≥ a we have a ⊙ b ≥ a ⊙ a = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore, for all n ≥ 1 we have a ∨ ¬(an) = a ∨ ¬a = a ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Thus, R does not satisfy equation (1) and is not quasi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The second part of the argument really requires R to be based on a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, consider the 4-element diamond lattice 0 ≤ a, b ≤ 1 with a ∧ b = 0 and a ∨ b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We can define a FLew-algebra based on this lattice by stipulating a2 = a, b2 = b and a ⊙ b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Even though a and b are idempotent, we have a ∨ ¬a = a ∨ b = 1 and b ∨ ¬b = b ∨ a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore, this algebra is quasi- primal (it is, however, not semi-primal, since it has the non-trivial automorphism swapping a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' ■ NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 9 In [31] Galatos and Jipsen provide a list of all finite residuated lattices of size up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='16 enables us to find quasi-primal FLew-algebras among them and thus, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2, we can identify the semi-primal ones by ruling out the existence of non-trivial internal isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, there is a total of six quasi-primal FLew-chains with 5 elements (R5,1 1,17, R5,1 1,18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' R5,1 1,22 in [31]), five of which are semi-primal (all except R5,1 1,17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Examples of semi-primal FLew-algebras not based on a chain are, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', R6,2 1,11 and R6,3 1,9 in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The algebras in question are depicted in Appendix A, where we also provide detailed proofs of these claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' While until now we discussed how to identify semi-primal FLew-algebras, we end this subsection with two examples of semi-primal algebras based on residuated lattices where 1 ̸= e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Specifically, we consider the bounded De Morgan monoids C01 4 and D01 4 depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 0 e a 1 = a2 0 1 = a2 e a Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The (semi-)primal bounded De Morgan monoids C01 4 and D01 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' They are bounded commutative residuated lattices with an additional involution ∼ which, in both examples, is defined by ∼e = a and ∼0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Our names for these algebras are inspired by [46], where C4 and D4 are used for the corresponding De Morgan monoids with the bounds 0 and 1 excluded from the signature (in [46] it is shown that each of these two algebras generates a minimal subvariety of the variety of all De Morgan monoids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The algebras C01 4 and D4 01 are primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Their reducts obtained by removing the neutral element e from the signature, are semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Starting with C01 4 , we directly verify that it satisfies characterization (3) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' First we define T1 and, therefore, T0(x) = T1(∼x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' As in [22], we do this by, for all ℓ ∈ {0, e, a}, defining unary terms uℓ satisfying uℓ(x) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 1 if x = 1 0 if x = ℓ ∗ otherwise, were ∗ indicates that any value is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For instance, we can define such terms by u0(x) = x ∧ 1, ue(x) = ∼ � (∼x)2� and ua(x) = ∼ � (∼x) ⊙ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 10 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX Through these terms we can clearly define T1(x) = u0(x) ∧ ue(x) ∧ ua(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lastly, we need to define τℓ for ℓ ∈ {e, a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Again, it suffices to find terms τ ∗ ℓ which satisfy τ ∗ ℓ (x) = � 1 if x ≥ ℓ ̸= 1 if x ̸≥ ℓ, since then we get τℓ = T1(τ ∗ ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Our desired terms are given by τ ∗ e (x) = � (∼x)2 ⊙ x � ∨ x2 and τ ∗ a(x) = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This concludes the proof for C01 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The proof for D01 4 is completely analogous, except that we use τ ∗ e (x) = � (∼x)2 ⊙x � ∨x instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Thus we showed that these two algebras are semi-primal, and since they don’t have any proper subalgebras they are primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since we never relied on the constant e in the above, the last part of the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Note that in both cases, if we exclude e from the signature then {0, 1} becomes a proper subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Pseudo-logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We illustrate how to generate more examples of semi-primal algebras which are based on a bounded lattice which is not necessarily a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The results and terminology are due to [17, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A pseudo-logic L = (L, ∧, ∨,′ , 0, 1) is a bounded lattice with an additional unary operation ′ which satisfies 0′ = 1 and 1′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In [22] it is shown that every subalgebra of L2 which is not the graph of an internal isomorphism is a product of subalgebras if the following two properties are satisfied: (1) There is no a ∈ L\\{0} with a′ = 1, (2) For all a ∈ L there exists an n ≥ 1 with a ∧ a(2n) = 0 (where a(k) denotes the k-fold iteration of ′ on a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Using this and the characterization of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3, we can find more examples of semi-primal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Here, we only need to assure that the above mentioned conditions are satisfied and that there are no non-trivial internal isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, the three algebras depicted in Figure 2 are semi-primal (the pseudo- negation ′ is indicated by dotted arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 0 1 a b c 0 1 a b c d 0 1 a b c d e Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Some semi-primal pseudo-logics ([22, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Murski˘ı’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' While semi-primal algebras may seem rare, quite the opposite is suggested by the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In 1975, Murski˘ı proved his surprising theorem about the proportion of semi-primal algebras of a fixed signature under increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The original paper [47] is in Russian, the version we recall here is due to [6, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 11 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [47] Let σ be an algebraic type which contains at least one operation symbol which is at least binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let Aσ,n be the number of algebras of type σ and size n and let SPσ,n be the number of such algebras which are semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then lim n→∞ SPσ,n Aσ,n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Semi-primal duality One of the nice features of the variety of Boolean algebras BA is the famous Stone duality [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Categorically speaking, it asserts that there is a dual equivalence between BA and the category Stone of Stone spaces (that is, compact, Hausdorff and zero-dimensional topological spaces) with continuous maps: Stone Π � BA Σ � The functor Σ assigns to a Boolean algebra B its collection of ultrafilters and the functor Π assigns to a Stone space X the Boolean algebra of its clopen subsets with the usual set-theoretical Boolean operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Note that these functors can be defined on objects by Σ(B) = BA(B, 2) and Π(X) = Stone(X, 2), where in the latter equation 2 denotes the two-element discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Stone duality has been extended to quasi-primal algebras by Keimel and Werner in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This duality fits the general framework of Natural Dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For us, the Semi-primal Strong Duality Theorem [17, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='14] is of high importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' However, we present it self-contained and in a way which particularly suits our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Furthermore, we will use categorical constructions to provide a new proof of this duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Such a proof has, to the best of our knowledge, not appeared in the literature yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' First we introduce the dual category of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the following, we always consider S(L) as a complete lattice in its usual ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The category StoneL has objects (X, v) where X ∈ Stone and v: X → S(L) assigns to every point x ∈ X a subalgebra v(x) ≤ L, such that for every subalgebra S ≤ L the preimage v−1(S↓) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A morphism m: (X, v) → (Y, w) in StoneL is a continuous map X → Y which, for all x ∈ X, satisfies w(m(x)) ≤ v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the framework of natural dualities [17], the dual category of A is defined slightly differently, using Stone spaces with unary relations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', subsets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let X be the category with objects (X, {RS | S ≤ L}), where X ∈ Stone and RS is a closed subset of X for each subalgebra S ≤ L, satisfying RL = X and RS1 ∩ RS2 = RS1∩S2 for all S1, S2 ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A morphism m: (X, {RS | S ≤ L}) → (X′, {R′ S | S ≤ L}) in X is a continuous relation-preserving map X → X′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', it satisfies x ∈ RS ⇒ m(x) ∈ R′ S for all x ∈ X and S ∈ S(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 12 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX The categories X and StoneL are isomorphic, as witnessed by the following mu- tually inverse functors φ and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor φ : X → StoneL is given on objects by (X, {RS | S ≤ L}) �→ (X, v), where v(x) = � {S | x ∈ RS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor ψ: StoneL → X is given on objects by (X, v) �→ (X, {RS | S ≤ L}) where RS = {x ∈ X | v(x) ≤ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Both φ and ψ map every morphism to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' ■ We now describe the two contravariant functors ΣL and ΠL which give rise to the duality between A and StoneL: StoneL ΠL � A ΣL � On objects A ∈ A, let the functor ΣL be defined by ΣL(A) = � A(A, L), im � where im assigns to a homomorphism h: A → L its image im(h) = h(A) ∈ S(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A clopen subbasis for the topology on A(A, L) is given by the collection of sets of the following form with a ∈ A and ℓ ∈ L: [a : ℓ] = {h ∈ A(A, L) | h(a) = ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' On morphisms f ∈ A(A1, A2) the functor acts via composition ΣLf : A(A2, L) → A(A1, L) h �→ h ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Note that this is a morphism in StoneL since im(h ◦ f) ≤ im(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Before we define the functor ΠL, we describe the canonical way to consider L as a member of StoneL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Simply endow L with the discrete topology and ⟨·⟩: L → S(L) assigning to an element ℓ ∈ L the subalgebra ⟨ℓ⟩ ≤ L it generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now, as expected, we can define the functor ΠL on objects (X, v) ∈ StoneL by ΠL(X, v) = StoneL � (X, v), (L, ⟨·⟩) � with the algebraic operations defined pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This means that the carrier-set of ΠL(X, v) is the set of continuous maps g : X → L which respect v in the sense that, for all x ∈ X, they satisfy g(x) ∈ v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Again, on morphisms m: (X, v) → (Y, w) the functor is defined via composition ΠLm: StoneL � (Y, w), (L, ⟨·⟩) � → StoneL � (X, v), (L, ⟨·⟩) � g �→ g ◦ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This is well-defined due to the condition on morphisms in StoneL: (g ◦ m)(x) = g(m(x)) ∈ w(m(x)) ⊆ v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It is also clearly a homomorphism since the operations are defined pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 13 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [41, 17] The functors ΠL and ΣL are fully faithful and establish a dual equivalence between A and StoneL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The remainder of this section is dedicated to an alternative proof of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The idea is to directly prove the duality on the finite level, and then lift it to the infinite level using the following categorical constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For a finitely complete and cocomplete category C, its comple- tion under filtered colimits is denoted by Ind(C) and, dually, its completion under cofiltered limits is denoted by Pro(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, Ind(BAω) ≃ BA and Pro(Setω) ≃ Stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' More material about these completions can be found in Johnstone’s book [40, Chapter VI] (in particular, a more rigorous definition of the Ind-completion is given in VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We only recite the following, which allows us to lift dualities between small categories (following Johnstone, dualities arising this way are called Stone type dualities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [40, Lemma VI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1] Let C and D be small categories which are dually equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then Ind(C) is dually equivalent to Pro(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Our argument to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2 now has the following outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The role of C will be played by Aω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since A is locally finite (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', [17, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2]), it is well-known that Ind(Aω) ≃ A (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', [40, Corollary VI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The role of D will be played by Stoneω L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since the topology doesn’t matter here (because it is always discrete), we will denote this category by Setω L instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To get the finite dual equivalence, we make the following observation Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , Sn be subalgebras of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then the set of homomor- phisms A(� i≤n Si, L) consists exactly of the projections followed by inclusions pri : � i≤n Si → Si ֒→ L in each component i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Our proof is similar to that of [12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let h: � i≤n Si → L be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since A is congruence distributive (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5), it has the Fraser-Horn property, meaning that the congruence θ := ker(h) is a product of congruences θi on Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' By the isomorphism theorem we find ( � i≤n Si)/θ ∼= � i≤n (Si/θi) ∼= im(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since im(h) is a subalgebra of L and thus directly irreducible, at most one factor of � i≤n(Si/θi) can be non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since im(h) contains at least two elements (that is, 0 and 1), precisely one factor, say Sj/θj, is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since Sj is itself semi-primal, it is simple, so Sj/θj ∼= Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So h induces an internal isomorphism Sj ∼= im(h), but by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2 this can only be the identity on Sj, thus h coincides with prj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The (restrictions of the) functors ΠL and ΣL establish a dual equivalence between the small categories SetL ω and Aω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let (X, v) ∈ Setω L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then ΣLΠL(X, v) = � A � � x∈X v(x), L � , im � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 14 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5 this is equal to ({prx | x ∈ X}, im), which is clearly isomorphic to (X, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' On the other hand, starting with A ∈ Aω, we know by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='6 that it is a product of subalgebras A = � i≤n Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now, again due to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5, we get ΣL(A) = ({pri | i ≤ n}, im), and thus ΠLΣL(A) ∼= � i≤n im(pri) ∼= � i≤n Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To see that ΠL and ΣL form a dual adjunction we note that for A = � i≤n Si ∈ Aω and (X, v) ∈ Setω L we have Aω� ΠL(X, v), A � ∼= � i≤n Aω� ΠL(X, v), Si � and Setω L � ΣL(A), (X, v) � ∼= Setω L( � i≤n ({pri}, im), (X, v)) ∼= � i≤n Setω L � ({pri}, im), (X, v) � where the coproduct in Setω L is the obvious disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So we only need to show that Aω� ΠL(X, v), Si � ∼= Setω L � ({pri}, im), (X, v) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' But this is obvious since the elements of the left-hand side are exactly the projec- tions with image contained in Si, which are in bijective correspondence with the points of X with v(x) ≤ Si, that is, with elements of the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ In order to successfully apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='4, it remains to show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Pro(Setω L) is categorically equivalent to StoneL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' First we show that the category StoneL is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For an index set I (which we often omit), we claim that the product is computed as � i∈I (Xi, vi) = ( � i∈I Xi, � vi), where � vi(p) = �(vi(pi)) for all p ∈ � Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It follows from ( � vi)−1(S↓) = � vi −1(S↓) that this defines a member of StoneL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Note that the projections are morphisms in StoneL since vi(πi(p)) = vi(pi) ≤ � j∈I vj(pj) = ( � vj)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If (γi : (Y, w) → (Xi, vi) | i ∈ I) is another cone, there is a unique continuous map f : Y → � Xi with πi ◦ f = γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This map is a morphism in StoneL since ( � vi)(f(y)) = � vi � πi(f(y)) � = � vi � γi(f(y)) � ≤ w(y), where the last inequality follows from vi(γi)(y) ≤ w(y) which is true since the γi are morphisms in StoneL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The equalizer of f, g : (X, v) → (Y, w) is simply given by (Eq, v|Eq) where Eq ⊆ X is the corresponding equalizer in Stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It follows that StoneL has all limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, StoneL has all cofiltered limits, so the natural inclusion functor ι: Setω L ֒→ StoneL has a unique cofinitary (that is, cofiltered limit preserving) extension ˆι: Pro(Setω L) ֒→ StoneL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 15 Since ι is fully faithful, to conclude that the functor ˆι is fully faithful as well it suffices to show that ι maps all objects to finitely copresentable objects in StoneL (this is due to the analogue of [40, Theorem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8] for the Pro-completion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So we need to show that any (C, w) ∈ StoneL where C is a finite discrete space is finitely copresentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In other words, we need to show that, whenever (X, v) ∼= limi∈I(Xi, vi) is a cofiltered limit of a diagram (fij : (Xj, vj) → (Xi, vi) | i ≤ j) in StoneL with limit morphisms pi : (X, v) → (Xi, vi), any morphism f : (X, v) → (C, w) factors essentially uniquely through one of the pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For this we can employ an argument similar to the one in the proof of [55, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='16(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' On the underlying level of Stone, where finite discrete spaces are finitely copresentable, the continuous map f factors essentially uniquely through some pi, say via the continuous map gi : Xi → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' However, gi is not necessarily a morphism in StoneL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Consider J = {j ≥ i} and for each j ∈ J define gj = fij ◦ gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Define the continuous maps µ: X → S(L)2 and µj : Xj → S(L)2 for all j ∈ J by µ(x) = � w(f(x)), v(x) � and µj(x) = � w(gj(x)), vj(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since µ(X) = limj∈J µj(Xj) = � j≥i µj(Xj) is contained in the finite set S(L)2 and J is directed, there is some k ∈ J such that µ(X) = µk(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' But now, since f is a morphism in StoneL, we have that µ(X) ⊆ {(S, T) | S ≤ T}, and thus the same holds for µk(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Thus gk is a morphism in StoneL which has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To finish the proof we show that ˆι is essentially surjective, in other words, we show that every element (X, v) of StoneL is isomorphic to a cofiltered limit of elements of Setω L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We do this in a manner similar to [55, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let R consist of all finite partitions of X into clopen sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Together with the order R ≤ R′ if and only if R′ refines R this forms a codirected set and in [55, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='12] it is shown that X ∼= limR∈R R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We now turn every R ∈ R into a member of Setω L by endowing it with an appropriate vR : R → S(L) and show that (X, v) = limR∈R(R, vR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For R ∈ R, say R = {Ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , Ωk}, we define v−1 R (S↓) = {Ωi | Ωi ∩ v−1(S↓) ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The map pR : X → R defined by pR(x) = Ωi ⇔ x ∈ Ωi is a morphism in StoneL since v(x) = S and x ∈ Ωi implies vR(pR(x)) ∈ v−1 R (S↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Is is easy to see that this defines a cone over the diagram (R, vR)R∈R, so there is a unique f : (X, v) → limR∈R(R, vR) in StoneL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' As in Stone, the map f is a homeomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To complete the proof it suffices to show that f −1 is a morhpism in StoneL as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Say limR∈R(R, vR) = (Y, w) and let πR : (Y, w) → (R, vR) denote the limit morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Assuming w(y) = S we want to show f −1(y) ∈ v−1(S↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let Ω ⊆ X be an arbitrary clopen set containing f −1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then R = {Ω, X\\Ω} ∈ R and Ω = pR(f −1(y)) = πR(y) ∈ v−1 R (S↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' By definition this means that Ω ∩ v−1(S↓) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since this holds for every Ω containing f −1(y), this implies that f −1(y) is in the closure v−1(S↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' However, this closure coincides with v−1(S↓), since by definition of StoneL this is a closed set already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ 16 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX As discussed before, this yields our alternative proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In Section 5 we investigate the other dual equivalence which can be obtained from the finite dual equivalence of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' More specifically, there we describe Ind(Setω L) and its dual, the category of profinite algebras Pro(Aω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This is the ‘semi-primal version’ of the duality between Ind(Setω) ≃ Set and Pro(BAω) ≃ CABA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Before that, in the following section we investigate the relationship between StoneL and Stone and, more interestingly, between A and BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A chain of adjuntions In this section we explore the relationship between Stone duality and the semi- primal duality discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We start with the connection between StoneL and Stone, which will be expressed in terms of a chain of four adjoint functors (similar to one in [57]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then we look at the duals of these functors and give them purely algebraic descriptions to gain insight into the structure of A relative to that of BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The entire situation is summarized in Figure 3, which we will have fully described at the end of this section (note that left-adjoints on the topological side correspond to right-adjoints on the algebraic side and vice-versa, since the functors ΠL, ΣL and Σ, Π which establish the two dualities are contravariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' StoneL ΠL � � ⊣ V⊤ U ⊣ � � V⊥ ⊣ C � A ΣL � � ⊢ P S ⊢ � � I ⊢ Q � Stone Π � BA Σ � Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The chain of adjunctions on the topological and the algebraic side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Four functors on the topological side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let U: StoneL → Stone be the obvious forgetful functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This functor has a left-adjoint and a right-adjoint V⊤ ⊣ U ⊣ V⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The two functors V⊤, V⊥ : StoneL → Stone are given on objects by V ⊤(X) = (X, v⊤) where ∀x ∈ X : v⊤(x) = L, V ⊥(X) = (X, v⊥) where ∀x ∈ X : v⊥(x) = ⟨0, 1⟩ and both assign every morphism to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Here ⟨0, 1⟩ is the subalgebra generated by {0, 1}, that is, the (unique) smallest subalgebra of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To see V ⊤ ⊣ U note that by definition we have m ∈ StoneL � (X, v⊤), (Y, w) � ⇔ m ∈ Stone(X, Y ) ∧ ∀x ∈ X : w(m(x)) ≤ v⊤(x), and w(m(x)) ≤ v⊤(x) = L is trivially satisfied for every m ∈ Stone(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 17 Similarly we see U ⊣ V⊥, since every m ∈ Stone(X, Y ) automatically satisfies v⊥(m(x)) ≤ w(x) and, therefore, m ∈ StoneL � (X, w), (Y, v⊥) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor V⊥ also has a right-adjoint C : StoneL → Stone defined by C(X, v) = {x ∈ X | v(x) = ⟨0, 1⟩} on objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' On morphisms m: (X, v) → (Y, w) it acts via restriction m �→ m|C(X,v), which is well-defined since m ∈ StoneL � (X, v), (Y, w) � and x ∈ C(X, v) means w(m(x)) ≤ v(x) = ⟨0, 1⟩ which is equivalent to m(x) ∈ C(W, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Again V⊥ ⊣ C is easy to see since m ∈ StoneL � (X, v⊥), (Y, w) � ⇔ ∀x : w(m(x)) ≤ ⟨0, 1⟩ ⇔ m ∈ Stone � X, C(Y, w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor V⊤ preserves almost all limits, however, there is one important exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The terminal object (that is, the limit of the empty diagram) in StoneL is given by ({∗}, v⊥), implying that V⊤ does not preserve terminal objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore, contrary to a claim made in [57], no further left-adjoint of V⊤ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It is obvious that both the unit idStone ⇒ U ◦ V⊤ of the adjunction V⊤ ⊣ U and the counit U ◦ V⊥ ⇒ idStone of the adjunction U ⊣ V⊥ are natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We hold on to this fact, which will also be interesting on the algebraic side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The category Stone is categorically equivalent to (i) a coreflective subcategory of StoneL, witnessed by the fully faithful functor V⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (ii) a reflective and coreflective subcategory of StoneL, witnessed by the fully faith- ful functor V⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functors described in this subsection can be carried through the dualities, resulting in a a corresponding chain of adjunctions between A and BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, the dual of U is given by ΠUΣL : A → BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the next subsection we show that this functor can be understood algebraically as the Boolean skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Throughout the subsections that follow, we will give similar algebraic descriptions for all of these functors between A and BA in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The Boolean skeleton functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the theory of MVn-algebras (that is, the case where L = �Ln), the Boolean skeleton is a well-known and useful tool (see, for example, [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' An appropriate generalization of this concept to our setting was made by Maruyama in [43] (where it is called the Boolean core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Due to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8 and [43, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='11], the following definition is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let A ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The Boolean skeleton of A is the Boolean algebra S(A) = (S(A), ∧, ∨, T0, 0, 1) on the carrier set S(A) = {a ∈ A | T1(a) = a}, where the lattice operations ∧ and ∨ are inherited from A and the unary operations T0 and T1 correspond to the ones from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='7 (which by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8 are term-definable in L), interpreted in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, for each A ∈ A, a ∈ A and ℓ ∈ L we have Tℓ(a) ∈ S(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This holds since the equation T1(Tℓ(x)) ≈ Tℓ(x) holds in L, and therefore also in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For A ∈ A, suppose that A′ ⊆ A is a subset such that (A′, ∧, ∨, T0, 0, 1) forms a Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then, for all a′ ∈ A′, we have T1(a′) = T1(T0(T0(a′))) = T0(T0(a′)) = a′ and thus a′ ∈ S(A) (the second equation always holds since A |= 18 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX T1(T0(x)) ≈ T0(x), which is easily checked in L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore, S(A) is the largest such subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' ■ To extend the construction of the Boolean skeleton to a functor S: A → BA, on homomorphisms f ∈ A(A1, A2) we define Sf to be the restriction f|S(A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This is well-defined since a ∈ S(A1) ⇔ T1(a) = a ⇒ T1(f(a)) = f(T1(a)) = f(a) ⇔ f(a) ∈ S(A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The following is arguably the most important property of the Boolean skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For all A ∈ A, there is a homeomorphism between UΣL(A) = A(A, L) and ΣS(A) = BA(S(A), 2) given by h �→ h|S(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' First we show that the map is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For injectivity, suppose that g and h satisfy g|S(A) = h|S(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Take an arbitrary element a ∈ A and let g(a) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Using that Tℓ(a) ∈ S(A) we get 1 = Tℓ(g(a)) = g(Tℓ(a)) = h(Tℓ(a)) = Tℓ(h(a)), which implies h(a) = ℓ and, since a was arbitrary, that g = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For surjectivity, let h ∈ BA(S(A), 2) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Due to [43, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='12] the following yields a well-defined homomorphism ¯h ∈ A(A, L): ¯h(a) = ℓ ⇔ h(Tℓ(a)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since for a ∈ S(A) we have h(T1(a)) = 1 ⇔ h(a) = 1 and h(T0(a)) = 1 ⇔ T0(h(a)) = 1 ⇔ h(a) = 0, we conclude that ¯h|S(A) = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We now have a bijection between two Stone spaces, so it only remains to show that it is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' But this is easy to see since the preimage of an open subbasis element [a : i] ⊆ BA(S(A), 2) is the open subbasis element [a : i] ⊆ A(A, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' There is a natural isomorphism between the functor S and the dual ΠUΣL of the forgetful functor U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3, for every A ∈ A, setting φA : UΣL(A) → ΣS(A) h �→ h|S(A) defines a natural isomorphism φ: UΣL ⇒ ΣS (naturality is easy to check using the definitions of Σ, ΣL and S on morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Applying Π and using the fact that ΠΣ is naturally isomorphic to idBA, we get the natural isomorphism Πφ: S ⇒ ΠUΣL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ In the next subsection we explain the right-adjoint of the Boolean skeleton func- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The Boolean power functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In this subsection we give an algebraic de- scription of a functor naturally isomorphic to the dual ΠLV⊤Σ of the functor V⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This functor, which we call P, turns out to be an instance of the the well-known Boolean power (or Boolean extension), which was introduced for arbitrary finite algebras in Foster’s first paper on primal algebras [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Boolean powers are special instances of Boolean products (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', [11, Chapter IV]), but for our purposes it is more convenient to work with the following equivalent definition found in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 19 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Given a Boolean algebra B ∈ BA and a finite algebra M, the Boolean power M[B] is defined on the carrier set M[B] ⊆ BM consisting of all maps ξ : M → B which satisfy (1) If ℓ and ℓ′ are distinct elements of M, then ξ(ℓ) ∧ ξ(ℓ′) = 0, (2) �{ξ(ℓ) | ℓ ∈ M} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If oL : M k → M is a k-ary operation of M, we define a corresponding operation oM[B] : M[B] → M[B] by oM[B](ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , ξk)(ℓ) = � oM(ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=',ℓk)=ℓ (ξ1(ℓ1) ∧ · · · ∧ ξk(ℓk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The resulting algebra M[B] = (M[B], oM[B]) is a member of the variety HSP(M) generated by M (since it satisfies the same equations as M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' There is a straightforward way to extend this construction to a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Given a finite algebra M, we define the functor PM : BA → HSP(M) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' On objects B ∈ BA we define PM(B) = M[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For a Boolean homomorphism ϕ: B → B′, the homomorphism PMϕ: M[B] → M[B′] is defined via composition ξ �→ ϕ ◦ ξ (this is a homomorphism because operations in M[B] are defined by Boolean expressions, which commute with ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, we will use the shorthand notation P for PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the remainder of this subsection we aim to show that P is indeed the right-adjoint of the Boolean skeleton functor S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For this, we need the following well-known properties of the Boolean power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1] The functor PM has the following properties: (i) PM(2) ∼= M, (ii) PM preserves products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, PM(2κ) ∼= Mκ holds for all index sets κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the next proposition we describe the interplay between the functors S and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Again, the terms Tℓ from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8 play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For every A ∈ A there is an embedding T(·) : A ֒→ P(S(A)) given by a �→ Ta where Ta(ℓ) = Tℓ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The restriction to S(A) yields an isomorphism S(A) ∼= S � P(S(A)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The map is well-defined, that is, Ta is in P(S(A)), since the equations Tℓ(x) ∧ Tℓ′(x) ≈ 0 (for distinct ℓ, ℓ′) and �{Tℓ(x) | ℓ ∈ L} ≈ 1 hold in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We now fix an embedding A ֒→ LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It is easy to see that T(·) is injective since, for distinct a, a′ ∈ A, there is some component i ∈ I with a(i) = ℓ ̸= a′(i), thus Ta(ℓ) ̸= Ta′(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To conclude that T(·) is an embedding we need to show that it is a homomorphism, that is we want to show that for any k-ary operation o: Lk → L of L we have ToA(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=',ak) = oL[B(A)](Ta1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Tak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 20 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX By definition the i-th component of the left-hand side is given by ToA(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=',ak)(ℓ)(i) = Tℓ � oL(a1(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , ak(i)) � = � 1 if oL(a1(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , ak(i)) = ℓ 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The right-hand side is given by oL[B(A)](Ta1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Tak)(ℓ) = � oL(ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=',ℓk)=ℓ (Ta1(ℓ1) ∧ · · · ∧ Tak(ℓk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In its i-th component this again corresponds to � oL(ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=',ℓk)=ℓ � Tℓ1(a1(i)) ∧ · · · ∧ Tℓk(ak(i)) � = � 1 if oL(a1(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , ak(i)) = ℓ 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Thus T(·) is an embedding, which concludes the proof of the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For the second statement, note that, since S preserves injectivity of homomor- phisms, it suffices to show that the restriction of T(·) to S(A) is a surjection onto S � P(S(A)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So consider an element ξ ∈ S � P(S(A)) � , that is ξ ∈ P(S(A)) and T L[S(A)] 1 (ξ) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The latter by definition means T L[S(A)] 1 (ξ)(1) = ξ(1), T L[S(A)] 1 (ξ)(0) = � {ξ(ℓ) | ℓ ∈ L, ℓ ̸= 1} = ξ(0) and T L[S(A)] 1 (ξ)(ℓ) = � ∅ = 0 = ξ(ℓ) for all ℓ ∈ L\\{0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We claim that ξ = Tξ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Indeed, we know that ξ(1) ∈ S(A) so ξ(1) = T1(ξ(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Furthermore, in the component i ∈ I, we have ξ(0)(i) = 1 if and only if ξ(1)(i) = 0, so T0(ξ(1)) = T1(ξ(0)) = ξ(0) since ξ(0) ∈ S(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Finally, for ℓ ̸∈ {0, 1} we have Tℓ(ξ(1)) = 0 since for all i ∈ I we have ξ(1)(i) ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Since S is dual to the essentially surjective functor U, we know that every B ∈ BA is isomorphic to S(A) for some A ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore, the following is a direct consequence of the second part of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Every Boolean algebra B ∈ BA is isomorphic to S(P(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Another immediate consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8 is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For every Boolean algebra B ∈ BA, the algebra P(B) is the largest algebra in A which has B as Boolean skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' That is, for every algebra A ∈ A with S(A) ∼= B there exists an embedding A ֒→ P(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We now have everything at hand to prove the main theorem of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' P is naturally isomorphic to the dual of V⊤ and, therefore, S ⊣ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' First we prove the statement on the finite level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In other words, we want to show that, in StoneL, ΣLP(B) ∼= V⊤Σ(B) holds for every finite Boolean algebra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' More explicitly, after spelling out the definition of the functors involved we want to show (2) � A(P(B), L), im � ∼= � BA(B, 2), v⊤� NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 21 for every finite Boolean algebra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' First, since B is finite there is some positive integer k such that B ∼= 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We combine the following isomorphisms in Stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Due to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3 we know A � P(B), L � ∼= BA � S(P(B)), 2 � , And due to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='9 we know S(P(B)) ∼= B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Putting these together, we get A(P(B), L) ∼= BA(B, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In fact, this even yields an isomorphism in StoneL as desired in (2), because � A(P(B), L), im � ∼= � A(Lk, L), im � ∼= � A(Lk, L), v⊤� where the last equation holds due to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So we know that the restriction of P to the category of finite Boolean algebras Pω : BAω → A is dual to the restriction (V⊤)ω of V⊤ to the category Setω L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' There is a unique (up to natural iso) finitary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', filtered colimit preserving) extension of Pω to Ind(BAω) ≃ BA, and this extension is naturally isomorphic to the dual of V⊤ (since V⊤ preserves all limits except for the terminal object, it is the unique cofinitary extension of (V⊤)ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To show that P coincides with this unique extension (up to natural isomorphism), it suffices to show that P is finitary as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since P preserves monomorphisms (it is easy to see by definition that if ϕ ∈ BA(B1, B2) is injective, then Pϕ is injective as well), we can apply [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='4], which states that P is finitary if and only if the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For every Boolean algebra B ∈ BA, for every finite subalgebra A ֒→ P(B) the inclusion factors through the image of the inclusion of some finite subalgebra B′ ֒→ B under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To see this write A ∼= � i≤n Si as product of finite subalgebras of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='9, we know that S(A) ∼= 2n embeds into B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='7 we have P(2n) ∼= Ln and the natural inclusion � i≤n Si ֒→ Ln yields our factorization A P(B) P(2n) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ In particular, if L is primal, we get an explicit categorical equivalence witnessing Hu’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [36] If L is primal, then S ⊣ P yields a categorical equivalence between A and BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We also get an algebraic analogue of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor P is fully faithful and identifies BA with a reflective subcategory of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' By now we found detailed descriptions of most of the functors appearing in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We are still missing is an algebraic understanding of the adjunction Q ⊣ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This 22 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX gap is filled in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' As we will see, it is closely connected to the adjunction S ⊣ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The subalgebra adjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For every subalgebra S ≤ L, there is an adjunction (3) Stone VS ⊥ � StoneL CS � which we explore in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor VS : Stone → StoneL is given on objects by VS(X) = (X, vS) where ∀x ∈ X : vS(x) = S, and assigns every morphism to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor CS : StoneL → Stone is given on objects by CS(X, v) = {x ∈ X | v(x) ≤ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' On morphisms it acts via restriction, that is, given a morphism m: (X, v) → (Y, w), define m |CS(X) : CS(X) → CS(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This is well-defined since x ∈ CS(X, v) ⇔ v(x) ≤ S ⇔ w(m(x)) ≤ v(x) ≤ S ⇔ m(x) ∈ CS(Y, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Comparing this with Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1, the reader may easily verify V S ⊣ CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Indeed, the adjunction V S ⊣ CS generalizes the following adjunctions in Figure 3: V⊤ ⊣ U in the case where S = L is the largest subalgebra of L, V⊥ ⊣ C in the case where S = ⟨0, 1⟩ is the smallest subalgebra of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' What is special about these two extreme cases is the additional adjunction U ⊣ V⊤, which ‘glues’ the two adjunctions into the chain described in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To better understand the adjunction corresponding to a subalgebra S ≤ L, we dissect it into two parts as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Stone V⊤ ⊥ � StoneS U � ιS ⊥ � StoneL (CS,−) � Here, ιS is the natural inclusion and the functor (CS, −) is defined by (X, v) �→ (CS(X), v|CS(X)) on objects and, exactly like CS, acts via restriction on morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It is easy to see that this really is a decomposition of the adjunction (3), that is, VS = ιS ◦ V⊤ and CS = U ◦ (CS, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' As before, we want to carry everything over to the algebraic side, where the dissec- tion takes place through the subvariety AS := HSP(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We illustrate the entire situation in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Note that S ≤ L is itself semi-primal, so the semi-primal duality given by ΣS and ΠS as well as the adjunction S ⊣ PS make sense in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 23 StoneL ΠL � � ιS ⊣ (CS,−) � A ΣL � � ιS QS ⊢ � StoneS ΠS � � V⊤ ⊣ U � AS ΣS � � PS S ⊢ � Stone Π � BA Σ � Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Dissecting the subalgebra adjunction of S ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Again, ιS denotes the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Although it may seem obvious, it is not immediate that ιS really is the dual of ιS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To prove it, we make use of the following unary term, which will play an important role for the remainder of the subsection: χS(x) = � s∈S Ts(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' On L, this simply corresponds to the characteristic function of S ⊆ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It is, furthermore, characteristic for the subvariety AS in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' An algebra in A is a member of AS if and only if it satisfies the equation χS(x) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Clearly every member of AS satisfies the equation since S satisfies it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For the other direction, let A ∈ A satisfy χS(a) = 1 for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We know that A can be embedded into some LI and for each a ∈ A and i ∈ I, we have χS(πi(a)) = 1 which implies that πi(a) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore, A can be embedded into SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Now, let A ∈ AS and let h ∈ A(ιS(A), L) be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since h preserves equations, for every a ∈ A we get χS(a) = 1 ⇒ χS(h(a)) = 1 and, therefore, h ∈ A(A, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So we showed A(A, L) = AS(A, S) for A ∈ AS, which immediately implies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The inclusion functor ιS is the dual of the inclusion functor ιS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To complete the picture, we only need to describe the functor QS from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let α: idA ⇒ ιS ◦ QS be the unit of the adjunction QS ⊣ ιS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For any A ∈ A, the algebra QS(A) is universal for AS in the following sense: For every B ∈ AS and every homomorphism f : A → B, there is a unique ˆf : QS(A) → B such that ˆf ◦ αA = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A QS(A) B f πA ∃ ˆ f 24 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX Therefore, the functor QS may be understood as a quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' There is a well- known connection between quotients and equations introduced by Banaschewski and Herrlich in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Not surprisingly, the equation corresponding to QS is given by χS(x) ≈ 1, which is an easy consequence of the above discussion together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We summarize the results of this subsection as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For every subalgebra S ≤ L, there is an adjunction BA IS ⊤ � A KS � which can be dissected as BA PS ⊤ � AS S � ιS ⊤ � A QS � where ιS is the natural inclusion functor of the subvariety HSP(S) ֒→ HSP(L) and QS is the quotient functor corresponding to the equation χS(x) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, in the case where S is the smallest subalgebra of L, we can recover the functors I = ιS ◦ PS and Q from Figure 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor I: BA → A is, up to categorical equivalence, an inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor Q: A → BA is, up to categorical equivalence, the quotient by the equation χE(x) ≈ 1, where E = ⟨0, 1⟩ is the smallest subalgebra of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Being the smallest subalgebra of a semi-primal algebra, E is primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' There- fore, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='12, the adjunction S ⊣ PE is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The statement follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Clearly Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='13 holds not only for P, but for all the functors IS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Among them, I is special since it also has a right-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This yields the following algebraic version of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor I is fully faithful and identifies BA with a reflective and coreflective subcategory of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We showed that, if a finite lattice-based algebra M is semi-primal, then there is an adjunction PE ⊣ S ⊣ PM, where E is the smallest subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the next subsection we show that, conversely, the existence of an adjunction resembling this one fully characterizes semi-primality of a finite lattice-based algebra M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Characterizing semi-primality via adjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The aim of this subsec- tion is to find sufficient conditions for an adjoint of PM to imply semi-primality of the algebra M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We will then show that, in particular, these conditions are consequences of U and S from Figure 3 being (essentially) topological functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Recall that, in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='6, the Boolean power functor PM : BA → HSP(M) was defined for arbitrary finite algebras M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Of course, if S is a subalgebra of M, then PS can also be seen as a functor into HSP(M), and in the following there is no need to distinguish between these two functors in our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 25 PM is faithful (unless M is trivial), but it is usually not full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In fact, it is easy to see that PM can only be full if M does not have any non-trivial automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the main theorem of this subsection we show that, if PM is full and has a left-adjoint resembling S, then a lattice-based algebra M is semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let M be a finite lattice-based algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then M is semi-primal if and only if PM is full and there is a faithful functor s: HSP(M) → BA which satisfies PE ⊣ s ⊣ PM, where E = ⟨0, 1⟩ is the smallest subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If M is semi-primal, then PM is full since it is dual to the full functor V⊤, the functor s = S is faithful since it is dual to the faithful functor U and PE ⊣ S ⊣ PM was shown in the last two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now for the converse, assume that PM is full and there is a faithful functor s: HSP(M) → BA with PE ⊣ s ⊣ PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For abbreviation we write V for HSP(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We will make use of the following properties of s: (i) The unit η: idV ⇒ PM ◦ s is a monomorphism in each component, (ii) s preserves monomorphisms and finite products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Condition (i) follows from s being faithful and (ii) follows from s being a right- adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Our first goal is to prove the equivalence (4) s(A) ∼= 2 ⇔ ∃S ∈ S(M) : A ∼= S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If s(A) ∼= 2, use that by (i) there is an embedding A ֒→ PM(s(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since PM(s(A)) ∼= M, it follows that A is isomorphic to a subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Con- versely, first note that s(M) ∼= 2 since, using that PM is full and s ⊣ PM, we have 1 = |BA(2, 2)| = |V(M, M)| = |V � M, PM(2) � | = |BA � s(M), 2) � |, which is only possible for s(M) ∼= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now if A ∼= S ∈ S(M) then, due to (ii), the natural embedding S ֒→ M induces an embedding s(S) ֒→ s(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore s(S) ∼= 2 since s(M) ∼= 2 does not have any proper subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Next we show that M does not have any non-trivial internal isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For every subalgebra S ∈ S(M), there is a bijection between the set of Boolean homomorphisms s(S) → 2 and the set of homomorphisms S → PM(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Due to (4) we have s(S) ∼= 2, so the former only has one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since PM(2) ∼= M this means that there is only one homomorphism S → M, namely the identity on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Every non-trivial internal isomorphism with domain S would define another such homomorphism, resulting in a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We now show that M is semi-primal, using the characterization of semi-primality in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' That is, we want to show that M has a majority term and every subalgebra of M2 is either a product of subalgebras or the diagonal of a subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since M is based on a lattice, a majority term is given by the median (see the paragraph before Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let A ≤ M2 be a subalgebra and let ι: A ֒→ M be its natural embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Due to (ii), this embedding induces an embedding s(A) ֒→ s(M2) into s(M2) ∼= 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore, either s(A) ∼= 22 or s(A) ∼= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let p1 : A → M and p2 : A → M be ι followed by the respective projections M2 → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 26 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX First assume that p1 and p2 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then clearly A embeds into M, and therefore it is isomorphic to some subalgebra S of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since M has no non-trivial internal isomorphisms, A needs to coincide with the diagonal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If p1 and p2 are distinct then, using that s is faithful, the morphisms sp1 : s(A) → 2 and sp2 : s(A) → 2 are distinct as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This implies that s(A) ∼= 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Using the adjunction PE ⊣ s we get 4 = |BA(22, s(A))| = |V(E2, A)| and 4 = |BA(22, s(M2))| = |V(E2, M2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So there are exactly four distinct homomorphisms E2 → A and, since ι is a monomorphism, their compositions with ι are also four distinct homomorphisms E2 → M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore every of the former homomorphisms arises in such a way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, the natural embedding E2 ֒→ M2 arises in this way, which implies (0, 1) ∈ A and (1, 0) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' As noted in [22], this leads to A = p1(A) × p2(A), since whenever (a, b), (c, d) ∈ A we also have (a, d) = � (a, b) ∧ (1, 0) � ∨ � (c, d) ∧ (0, 1) � ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ In the remainder of this subsection we show how this theorem relates to the theory of topological functors (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', [1, Chapter VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='21] or [8, Chapter 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In- tuitively speaking, topological functors behave similarly to the forgetful functor Top → Set out of the category of all topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Still, the definitions in- volved are rather technical and the reader not familiar with this topic may skip this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We call a functor F: C → D (1) topological if it is faithful and every F-structured source has an initial lift, (2) essentially topological if it is topological up to categorical equivalence of C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The need for this distinction arises because certain properties of topological functors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' amnesticity [1, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='27], are not preserved under categorical equivalence (this issue is addressed in [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The following is our key observation for the last part of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The forgetful functor U: StoneL → Stone is topological and the Boolean skeleton functor S: A → BA is essentially topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We only need to show that U is topological, which immediately implies that S is essentially topological due to [1, Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='9] together with the fact that S is naturally isomorphic to the dual of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Of course U is faithful since it is the identity on morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now let X ∈ Stone and let (fi : X → U(Xi, vi))i∈I be a U-structured source (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', a collection of continuous maps) indexed by a class I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We define v: X → S(L) by v(x) = � i∈I vi(fi(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Note that this is well-defined, since S(L) is finite and that (X, v) is a member of StoneL, since v−1(S↓) = � i∈I f −1 i (v−1 i (S↓)) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Every fi is now also a morphism in StoneL, which defines a lift of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To show that it is initial, assume there are StoneL-morphisms (gi : (Y, w) → (Xi, vi))i∈I and a continuous NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 27 map g : Y → X with fi ◦ g = gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' All we need to show is that g defines a StoneL- morphism (Y, w) → (X, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To see this simply note that v(g(y)) = � i∈I vi � fi(g(y)) � = � i∈I vi(gi(y)) ≤ w(y), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ We can now easily show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let M be a finite lattice-based algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then M is semi-primal if and only if there is an essentially topological functor s: HSP(M) → BA which satisfies PE ⊣ s ⊣ PM, where E = ⟨0, 1⟩ is the smallest subalgebra of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the previous proposition we showed that if M is semi-primal, then S is essentially topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Conversely, if such an essentially topological s exists, it is faithful by definition and both its adjoints PM and PE are full by [1, Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore, due to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='19, M is semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ In this section we gained an algebraic understanding of all the functors between A and BA appearing on the right-hand side of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Furthermore, we now showed how properties of the Boolean skeleton functor S characterize semi-primality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the next section we investigate how canonical extensions of algebras in A behave under these functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' One of the main results is that the Boolean skeleton functor S may be used to identify canonical extensions of algebras in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Discrete duality and canonical extensions In this section we describe a semi-primal discrete duality similar to the well- known discrete duality between Set and CABA, the category of complete atomic Boolean algebras with complete homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It can be obtained from the finite duality in a similar way to the one of Section 3, except that now we lift it to the level of Ind(Setω L) and Pro(Aω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The members of the latter category are known to be precisely the canonical extensions [33] of members of A (see [21]), and we will provide two new characterizations of this category (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lastly we show that, as in the primal case L = 2, the topological duality from Section 3 can be connected to its discrete version via an analogue of the Stone- ˇCech compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Our first goal is to identify Ind(Setω L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Although it may not be surprising, it will still take some work to prove that it can be identified with the following category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The category SetL has objects of the form (X, v) where X ∈ Set and v: X → S(L) is an arbitrary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A morphism m: (X, v) → (Y, w) is a map X → Y which always satisfies w(m(x)) ≤ v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the context of fuzzy sets, Goguen [34, 35] initiated the study of such categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This research was continued, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', in [5, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In this remark we stick to the notation of [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Given a complete lattice V, the category Set(V) of V-fuzzy sets has objects (X, A) where A: X → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Morphisms (X, A) → (X′, A′) are maps 28 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX m: X → Y which satisfy A′(m(x)) ≥ A(x) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the context of fuzzy set theory, people were mainly interested in the case where V = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' However, we retrieve SetL in the case where V is the order-dual of S(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' ■ Since we are interested in the Ind-completion of Setω L, we will first discuss (fil- tered) colimits in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The category SetL is cocomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The colimit colimi∈I(Xi, vi) of a filtered diagram � fij : (Xi, vi) → (Xj, vj) | i ≤ j � is realized by � (� i∈I Xi)/∼, ¯v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Here, for xi ∈ Xi and xj ∈ Xj, xi ∼ xj ⇐⇒ ∃k ≥ i, j : fik(xi) = fjk(xj) and ¯v([xi]) = � xi∼xj∈Xj vj(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The proof that SetL is cocomplete is completely analogous to the one in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For filtered colimits, on the underlying level of Set we know that X := � i∈I(Xi)/∼ with the canonical inclusions ρi : Xi → X is the colimit of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To see that all the ρi are morphisms in SetL note ¯v(ρi(xi)) = � xi∼xj∈Xj vj(xj) ≤ vi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Given another cocone γi : (Xi, vi) → (Z, u), the unique map g : X → Z is a mor- phism in SetL since, for xi ∈ Xi and xi ∼ xj ∈ Xj we have u � g(ρj(xj)) � = u(γj(xj)) ≤ vj(xj) and thus u � g([xi]) � ≤ � xi∼xj∈Xj vj(xj) = ¯v([xi]), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ We will also make use of the following general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let F: C → D be a functor between categories C and D which both admit filtered colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If F has a right-adjoint G which preserves filtered colimits, then F preserves finitely presentable objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let C ∈ C be finitely presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We want to show that F(C) is finitely presentable in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let colimiDi be a filtered colimit in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then D � F(C), colimiDi � ∼= colimiC � C, G(Di) � ∼= colimiD � F(C), Di � , where the first isomorphism comes from the fact that G preserves filtered colimits and C is finitely presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If X is a finite set, then (X, v) is finitely presentable in SetL for every v: X → S(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' , xn} and let v(xi) = Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then we can clearly identify (X, v) ∼= � 1≤i≤n ({xi}, vSi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since filtered colimits commute with finite limits in Set, it now suffices to show that all ({xi}, vSi) are finitely presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Just like in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='4 we can define the adjunction VS ⊣ CS between SetL and Set for every subalgebra S ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' By Lemma NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3 it now suffices to show that CS preserves filtered colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So let (X, ¯v) be a filtered colimit as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We know that CS(X) = {[xi] | ∃xi ∼ xj ∈ Xj, vj(xj) ≤ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Therefore, for all [xi] ∈ CS we can choose representatives with xi ∈ CS(Xi, vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This yields a bijection between CS(X) and colimCS(Xi, vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ We now have everything at hand to easily prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Ind(Setω L) is categorically equivalent to SetL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since SetL is cocomplete, the inclusion ι: Setω L → SetL has a unique finitary extension ˆι: Ind(Setω L) → SetL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since ι is fully faithful and, by the above corollary, maps all objects to finitely presentable objects in SetL, this extension is also fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To see that it is essentially surjective note that, just like in Set, every member of SetL is the filtered colimit of its finite subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ We now take a closer look at the category Pro(Aω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It is well-known that it consists of the canonical extensions [33] of algebras in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In [21] a description of these canonical extensions as topological algebras can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' But, as in the case of complete atomic Boolean algebras CABA ≃ IP(2), this need not be the only description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the following we apply results of Section 4 to find two easy alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The first one is in terms of (arbitrary) products of subalgebras of L with complete homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let ˆ A be the category with algebras from IPS(L) as objects and complete homomorphisms as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We can essentially repeat our proof of the finite duality from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='6, once we prove the following result analogous to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Let A = � i∈I Si ∈ ˆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Then the complete homomorphisms A → L are precisely the projections (followed by inclusions) in each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3 there is a bijection between A(A, L) and BA(S(A), 2) given by h �→ h|S(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In particular, if h is complete, then so is its restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since S(A) = 2I, the only complete homomorphisms S(A) → 2 are the projections, and they are the restrictions of the respective projections A → L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Pro(Aω) is categorically equivalent to ˆ A Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='5 it suffices to show that SetL is dually equivalent to ˆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This is done completely analogous to the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ The second description of Pro(Aω) is in terms of the Boolean skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The category CAA has as objects algebras A ∈ A which have a complete lattice-reduct and which satisfy S(A) ∈ CABA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The morphisms in CAA are the complete homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Pro(Aω) is categorically equivalent to CAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Using Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8 we show that CAA is categorically equivalent to ˆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Clearly there is a fully faithful inclusion functor ˆ A ֒→ CAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So it suffices to show that this functor is essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In other words, we want to show that every object of CAA is isomorphic to a product of subalgebras of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 30 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX So consider A ∈ CAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since the adjunction S ⊣ P restricts to CABA and CAA, we can use Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='10 to get a complete embedding ηA : A ֒→ P(S(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since S(A) is in CABA it is isomorphic to 2I for some index set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Thus P(S(A)) ∼= P(2I) ∼= LI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We show that A is isomorphic to the direct product of subalgebras � i∈I pri(ηA(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For this it suffices to show that the injective homomorphism ηA maps onto it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' So let α be an element of this product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For each i ∈ I choose ai ∈ A such that pri(ηA(ai)) = α(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since 2I ∼= S(A) ⊆ A all atoms bi ∈ 2I (defined by bi(j) = 1 iff j = i) can be considered as members of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Now define a = � {ai ∧ bi | i ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since A is complete, we have a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' And since ηA is a complete homomorphism we have ηA(a) = α (because pri(ηA(a)) = ηA(ai) = α(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ With the results from this section thus far, it is clear that the chains of adjunc- tions from Section 4 (summarized in Figure 3) have their discrete counterparts, equally defined, between SetL and Set and CAA and CABA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To make the connection between Figure 3 and its discrete counterpart, we finish this section by connecting the respective dualities as indicated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' StoneL � � βL ⊣ (−)♭ � A � � ιc (−)δ ⊢ � SetL � CAA � Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Compactification and canonical extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Here (−)♭ : StoneL → SetL is the forgetful functor with respect to topology and ιc : CAA → A is the obvious inclusion functor (note that both these functors are not full).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor (−)δ takes an algebra to its canonical extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the primal case L = 2, it is well-known that β2 =: β is the Stone- ˇCech compactification (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', [40, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This has been generalized to the Bohr compactification in a (much broader) framework which includes ours in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' However, since things are particularly simple in our setting, we directly show how to define βL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Given (X, v) ∈ SetL, there is a natural way to extend v to the Stone-ˇCech compactification β(X) of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Indeed, since v: X → S(L) can be thought of as a continuous map between discrete spaces, by the universal property of β it has a unique continuous extension ˜v: β(X) → S(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Here, ˜v−1(S↓) is given by the topological closure of v−1(S↓) in β(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Thus, for every morphism f : (X, v) → (Y, w) in SetL, the continuous map βf defines a morphism (β(X), ˜v) → (β(Y ), ˜w) in StoneL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This is due to the observation that whenever x ∈ ˜v−1(S↓) = v−1(S↓), by continuity of βf and the morphism property of f, we have βf(x) ∈ w−1(S↓) = ˜w−1(S↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The functor βL : SetL → StoneL defined on objects by βL(X, v) = (β(X), ˜v) NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 31 and by f �→ βf on morphisms is the dual of the canonical extension functor (−)δ : A → CAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' It suffices to show that βL satisfies the following universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Given (Y, w) ∈ StoneL, every SetL-morphism f : (X, v) → (Y, w) extends uniquely to a StoneL-morphism ˜f : (β(X), ˜v) → (Y, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' On the levels of Set and Stone we get a unique continuous extension ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To show it is a StoneL-morphism, similarly to before, note that if x ∈ v−1(S↓), then by continuity ˜f(x) ∈ f � v−1(S↓) � ⊆ w−1(S↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since w−1(S↓) is closed it equals its own closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ This nicely wraps up this paper by connecting all of its main sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the last section we give a quick summary and discuss some possible directions of future research along similar lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Concluding Remarks and Further Research We explored semi-primality by means of category theory, showing how a variety generated by a semi-primal lattice expansion relates to the variety of Boolean alge- bras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Various adjunctions provide insight into the many similarities between these varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A schematic summary of our results can be found in Figure 6, which also emphasizes once more how close BA and A really are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Setω L Aω SetL CAA StoneL A Set CABA Stone BA Setω BAω Pro Ind Pro Ind Ind Pro Ind Pro Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Summary of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We plan to follow up this research by developing a coalgebraic framework for modal extensions of the many-valued logic corresponding to a semi-primal variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' As mentioned before, from this point of view it is reasonable to assume that L is based on a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' However, it seems plausible that our results generalize to the slightly more general case of semi-primal algebras which possess a coupling in the sense of [26], essentially since Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='8 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2 still apply to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We will now sketch some more potential ways to follow up this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In general, we hope to have set an example in exploring concepts in universal algebra through the lens of (mostly elementary) category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 32 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX For example, other variants of primality could be investigated in a similar man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A finite algebra M is called (1) demi-semi-primal if it is quasi-primal and every internal isomorphism of M can be extended to an automorphism of M (see [53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (2) demi-primal if it is quasi-primal and has no proper subalgebras (see [53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (3) infra-primal if it is demi-semi primal and every internal isomorphism is an automorphism on its domain (see [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' (4) hemi-primal if every operation on M which preserves congruences is term- definable in M (see [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' What is the categorical relationship between BA and the variety gener- ated by an algebra which is quasi-primal or which satisfies one of the properties of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' What about the relationship between distinct variations of primality to each other?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For quasi-primal algebras (and thus, in particular, for algebras satisfying (1), (2) or (3)), there is the duality theorem by Keimel-Werner [41] (which is also a natural duality [17]), possibly a good starting point to a discussion similar to the one presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Hemi-primality seems to have received less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To the best of the authors knowledge, no duality for varieties generated by hemi-primal algebras is known thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Is it possible to obtain a duality for hemi-primal varieties, for example one which stems from a finite dual equivalence using methods similar to our proof of semi-primal duality in Section 3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The Boolean power functor PM: BA → HSP(M) was defined for an arbitrary finite algebra M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the light of our results from Section 4, the following question arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Question 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Under which circumstances does the functor PM have a left-adjoint?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Which information about M can be retrieved from properties of the functors of the form PS with S ≤ M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' If we consider this work as not only comparing varieties but comparing dualities, another range of questions appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' What is the category theoretical relationship between different dual equivalences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, one could consider Priestley duality [52] or Esakia duality [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Lastly, another category theoretical approach to universal algebra, which has not been discussed in this paper, is given by Lawvere theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For example, Hu’s theorem has been analyzed from this angle in [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Of course, one could also try to find out more about other variants of primality in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Question 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' How can semi-primality and other variants of primality be expressed in terms of Lawvere theories?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' NEW PERSPECTIVES ON SEMI-PRIMAL VARIETIES 33 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Some semi-primal FLewalgebras Here we go into more detail in some claims made in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We provide examples of semi-primal FLew-algebras, both chain-based and non chain-based, in- cluding the proof of semi-primality for each one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' All of the examples and their labels are taken from the list [31] by Galatos and Jipsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' For simplicity we only discus FLew-algebras without any idempotent elements other than 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Due to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='16 they are all quasi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' To prove semi-primality, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2, it suffices to describe all subalgebras and argue why there can’t be any non-trivial isomorphisms between then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' We begin with the quasi-primal FLew-chains of five elements R5,1 1,17 to R5,1 1,22 in [31, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='2, row 2] depicted in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 1 a b c a2 R1,5 1,17 1 a b a2 ab R1,5 1,18 1 a a2 c ab R1,5 1,19 1 a b a2 = ab b2 = ac R1,5 1,20 1 a a2 ab b2 = ac R1,5 1,21 1 a b a2 = b2 ac R1,5 1,22 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The quasi-primal FLew-chains of order five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Except for the first one, all algebras depicted in Figure 7 are semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' R1,5 1,17 is not semi-primal because it has isomorphic subalgebras {0, 1, a, c} and {0, 1, a, d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' In the following we show why the other ones are semi-primal by describing the subalgebras other than the obvious ones {0, 1} and {0, 1, a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since isomor- phisms need to be order-preserving, it suffices to note that there are never two subalgebras of the same size in the examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' R1,5 1,18: There are no other subalgebras since {¬a, a2} = {b, c} ⊆ ⟨a⟩ and ¬b = ¬c = a, thus a ∈ ⟨b⟩ and a ∈ ⟨c⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' R1,5 1,19: There is the subalgebra ⟨a⟩ = ⟨b⟩ = {0, 1, a, b} since a → b = a, ¬a = b and ¬b = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since a = ¬c we have a ∈ ⟨c⟩, so c generates the entire algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' R1,5 1,20: There are two different sized subalgebras ⟨a⟩ = ⟨c⟩ = {0, 1, a, c} (since ¬a = c, ¬c = a and a → c = a) and ⟨b⟩ = {0, 1, b} (since ¬b = b → b = b) R1,5 1,21: Note that this algebra corresponds to the �Lukasiewicz-chain �L4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' As thus expected, there is the subalgebra ⟨b⟩ = {0, 1, b}, while b ∈ ⟨a⟩∩⟨c⟩ since a = ¬c, c = ¬a and b = a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' R1,5 1,22: There is the subalgebra ⟨a⟩ = ⟨c⟩ = {0, 1, a, c} (since ¬a = c, ¬c = a and a → c = a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since ¬b = c and ¬c = a we find that b generates the entire algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ To also provide non chain-based examples, we examine the FLew-algebras R6,2 1,11 ([31, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='18, row 4]) and R6,3 1,9 ([31, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='20, row 1]) depicted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' 34 ALEXANDER KURZ, WOLFGANG POIGER, AND BRUNO TEHEUX 1 a b c = a2 d ab R6,2 1,11 1 a b c d = a2 = c2 ab = bc R6,3 1,9 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Two semi-primal FLew-algebras of order six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The two FLew-algebras depicted in Figure 8 are semi-primal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' R6,2 1,11: The only possible candidate for an automorphism of this algebra is the bijection f exchanging c and d (since it needs to be order-preserving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This map, however, is not a homomorphism, as witnessed by the fact that f(a2) = f(c) = d while f(a)2 = a2 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The only other subalgebra other than {0, 1} is ⟨a⟩ = {0, 1, a, b, c} since we have ¬a = b, a2 = c, ¬c = a, a → b = a, a → c = a and b → c = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since this subalgebra is a chain, it does not have any non-trivial isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since ¬d = a we know that d generates the entire algebra, so there are no more subalgebras to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' R6,3 1,9: Again, there is only one possible candidate for an automorphism of this algebra, namely the bijection g exchanging b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This is not a homomorphism because g(b2) = g(0) = 0 while g(b)2 = c2 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' The only other subalgebra except {0, 1} is ⟨a⟩ = {0, 1, a, b, d} since ¬a = b, ¬b = ¬d = a and a → b = a → d = b → d = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' This subalgebra has no non-trivial isomorphisms because it is a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Since c2 = d, the element c generates the entire algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' □ Acknowledgments The second author is supported by the Luxembourg National Research Fund under the project PRIDE17/12246620/GPS.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Studia Logica 105 (2017), 843–872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [20] Davey, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', Haviar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', and Priestley, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Bohr compactifications of algebras and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Applied Categorical Structures 25 (2017), 403–430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [21] Davey, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', and Priestley, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Canonical extensions and discrete dualities for finitely generated varieties of lattice-based algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Studia Logica 100 (2012), 137–161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [22] Davey, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', Schumann, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', and Werner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' From the subalgebras of the square to the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Algebra Universalis 28 (1991), 500–519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [23] Esakia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Topological kripke models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Soviet Mathematics Doklady 15, 1 (1974), 147–151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [24] Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Generalized ”boolean” theory of universal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' part i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Mathematische Zeitschrift 58 (1953), 306–336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [25] Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Generalized ”boolean” theory of universal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' part ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' identities and subdirect sums of functionally complete algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Mathematische Zeitschrift 59 (1953/54), 191–199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [26] Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Semi-primal algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' characterization and normal-decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Mathema- tische Zeitschrift 99 (1967), 105–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [27] Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Automorphisms and functional completeness in universal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Mathema- tische Annalen 180 (1969), 138–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [28] Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Congruence relations and functional completeness in universal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' struc- ture theory of hemi-primals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Mathematische Zeitschrift 113 (1970), 293–308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [29] Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', and Pixley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Semi-categorical algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' semi-primal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Mathe- matische Zeitschrift 83 (1964), 147–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [30] Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=', and Pixley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Semi-categorical algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' Mathematische Zeitschrift 85 (1964), 169–184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content=' [31] Galatos, N.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='lu, bruno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='teheux@uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} +page_content='lu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf'} diff --git a/ztE4T4oBgHgl3EQfyg04/content/tmp_files/2301.05266v1.pdf.txt b/ztE4T4oBgHgl3EQfyg04/content/tmp_files/2301.05266v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc1fa282e1a758498b3c8b12172d5172c24ad517 --- /dev/null +++ b/ztE4T4oBgHgl3EQfyg04/content/tmp_files/2301.05266v1.pdf.txt @@ -0,0 +1,949 @@ +Improving Reliability of Spiking Neural Networks +through Fault Aware Threshold Voltage Optimization +Ayesha Siddique, Khaza Anuarul Hoque +Department of Electrical Engineering and Computer Science +University of Missouri, Columbia, MO, USA +ayesha.siddique@mail.missouri.edu, hoquek@missouri.edu +Abstract—Spiking neural networks have made breakthroughs in +computer vision by lending themselves to neuromorphic hardware. +However, the neuromorphic hardware lacks parallelism and hence, +limits the throughput and hardware acceleration of SNNs on +edge devices. To address this problem, many systolic-array SNN +accelerators (systolicSNNs) have been proposed recently, but their +reliability is still a major concern. In this paper, we first extensively +analyze the impact of permanent faults on the SystolicSNNs. +Then, we present a novel fault mitigation method, i.e., fault-aware +threshold voltage optimization in retraining (FalVolt). FalVolt +optimizes the threshold voltage for each layer in retraining +to achieve the classification accuracy close to the baseline in +the presence of faults. To demonstrate the effectiveness of our +proposed mitigation, we classify both static (i.e., MNIST) and +neuromorphic datasets (i.e., N-MNIST and DVS Gesture) on a +256x256 systolicSNN with stuck-at faults. We empirically show +that the classification accuracy of a systolicSNN drops significantly +even at extremely low fault rates (as low as 0.012%). Our +proposed FalVolt mitigation method improves the performance +of systolicSNNs by enabling them to operate at fault rates of up +to 60%, with a negligible drop in classification accuracy (as low +as 0.1%). Our results show that FalVolt is 2x faster compared +to other state-of-the-art techniques common in artificial neural +networks (ANNs), such as fault-aware pruning and retraining +without threshold voltage optimization. +Index Terms—Spiking neural networks, Stuck-at faults, Systolic +array, Fault mitigation. +I. INTRODUCTION +Spiking neural networks (SNNs) are a promising third +generation of neural networks that ensure high algorithmic +performance at low power. Their hardware acceleration re- +quire specialized architectures such as, SpiNNaker [1], and +TrueNorth [2]. However, these architectures lack parallelism +in each core and efficient dataflows for maximizing the reuse +of weight data. This limits their achievable throughput and +robustness in resource-constrained devices (e.g., battery-driven +autonomous cars). Towards this, leveraging SNNs on massively +parallel hardware accelerators such as systolic arrays has proven +to be an efficient solution [3], [4], [5], [6], [7]. Systolic array +SNN accelerators (systolicSNNs) are inspired by other state-of- +the-art hardware accelerators [8] which support fully parallel +execution of artificial neural networks (ANNs). These accel- +erators have a NxN dense grid of interconnected processing +elements (PEs), which allows efficient parallel processing with +the high spatio-temporal locality. Unlike ANNs, SNNs and their +hardware accelerators are still in a relatively early phase of +adoption [9] and thus ensuring the reliability of systolicSNNs +is still considered a major research challenge. +The systolicSNN hardware chips are manufactured using +nanometer CMOS technologies [10], which require a highly +sophisticated manufacturing process. The imperfections in this +process result in various manufacturing defects ranging from +process variations to permanent faults such as stuck-at faults. +The stuck-at faults affect the output of systolicSNNs in every +execution cycle and hence, lead to significant accuracy loss as +discussed in this paper. Furthermore, the impact of large-scale +failures such as dead synapse faults in SNNs has been thor- +oughly investigated [11], [12]. However, analyzing such failures +in the hardware require a fault model with higher abstraction +to make the simulation traceable. Guo et al. investigated the +fault resilience of SNNs trained with different coding schemes +by using a synaptic stuck-at fault model [13]. El-Sayed et al. +analyzed the effect of these faults in a transistor-level design of +leaky-integrate-and-fire (LIF) neuron [14]. Other state-of-the- +art works focus on bit flips in weight memories [15], [16], [17], +[18]. Conversely, the impact of stuck-at faults on systolicSNNs +has not been investigated. +The stuck-at faults are usually detected using post-fabrication +testing for discarding the faulty manufactured chips. However, +if a high number of manufactured chips are faulty, discarding +them reduces the yield to a large extent. A potential solution +is employing redundant executions (re-execution) to ensure +correct outputs, but it leads to significant latency and energy +overheads [17]. In the current resource-constrained nanoscale +hardware paradigm, where the number of PEs has drastically +increased to meet the robustness requirements of the end users, +it is imperative to maximize the yield with an efficient and +fault-tolerant systolicSNN. Recently, Mehul et al. proposed +an astrocyte self-repair mechanism for stuck-at 0 weights in +SNNs [19]. Other works are either focused on mitigating the +transient faults in SNNs [16], [19] or contemplated permanent +fault mitigation in ANN accelerators [20], [21], [22], [23]. +However, a considerable research gap exists in mitigating the +impact of permanent faults in systolicSNNs. +Novel contributions: In this paper, we present an extensive +stuck-at fault vulnerability analysis and a novel fault mitigation +method i.e., fault-aware retraining through threshold voltage +optimization (FalVolt). FalVolt first sets the weights mapped +to faulty PEs only as zero and then retrains weights mapped +to non-faulty PEs while optimizing the threshold voltage for +each layer to restore the classification accuracy close to its +baseline. The optimized threshold voltage differs from the +arXiv:2301.05266v1 [cs.NE] 12 Jan 2023 + +actual threshold voltage used in initial training. To demonstrate +the effectiveness of our proposed FalVolt mitigation method, we +used both static MNIST [24], and neuromorphic N-MNIST [25] +and DVS128 Gesture [26] datasets. Our results show that +FalVolt can operate at high fault rates of up to 60% with +a negligible impact on the classification accuracy compared +to its baseline. We empirically show that FalVolt takes 2x +fewer retraining epochs, and thus it is 2x faster in restoring +the baseline accuracy compared to other state-of-art techniques +such as fault-aware pruning and retraining. Note, fault-aware +pruning and retraining and threshold voltage optimization have +been conventionally used for ANN fault mitigation [21], [22] +and faster SNN convergence. However, to the best of our +knowledge, this is the first work to employ fault-aware thresh- +old voltage optimization for fault mitigation in SNNs. +The remainder of this paper is structured as follows: Sec- +tion II provides the preliminary information about SNNs and +systolicSNNs. Section III and Section IV present a motivational +case study and the proposed FalVolt mitigation method for +systolicSNNs, respectively. Section V discusses the results +for the fault vulnerability and mitigation. Finally, Section VI +concludes the paper. +II. BACKGROUND +This section provides a brief overview of the state-of-the-art +SNNs and systolicSNNs for better understanding. +Spiking Neural Networks: SNNs are bio-inspired artificial +neural networks. Their working principle can be explained with +a standard LIF model as follows: when the membrane potential +Vt of a presynaptic neuron exceeds a specific threshold voltage +at time t, a post-synaptic spike is fired, and then, Vt relaxes +to the resting state (Vrest < threshold voltage) with a time +constant τ. Vt maintains the resting state for a refractory time +tref before responding to the received spikes. The LIF-based +SNNs learn the presynaptic weights but require manual tuning +of the time constant in training. Furthermore, the time constant +is typically chosen to be the same for all neurons, which limits +the diversity of neurons and, thus, the expressiveness of the +LIF-based SNNs. Recently, Fang et al. proposed to train the +weights along with the time constant through an advanced +LIF model, i.e., parametric leaky integrate-and-fire (PLIF) [27]. +Incorporating the learnable time constants through PLIF-based +SNNs makes the network less sensitive to initial values and +reduces the training time. +Systolic-Array SNN Accelerators: SystolicSNNs exploit +the spatial and temporal parallelisms for which binary spike in- +put, logical 1 or 0 propagate vertically across the systolic array. +As shown in Fig. 1, the spike input is first divided into multiple +time steps and then, all input values in a time step are mapped +on one row of the systolic array. The input binary spikes pass +through a dense NxN grid of interconnected PEs in a clocked +synchronized manner. The filter data is mapped and pre-stored +in the PEs. Fig. 3a shows the design of a standard PE in +systolicSNNs. The PE accumulates 32-bit weight inputs under +1-bit binary spikes on an enable signal. The adder needed for +C = 2 +C +Filters +TS 3 +TS 2 +TS 1 +Input Feature Maps +with All Time Batches +I = 2 +C = 1 +PE +11 +PE +21 +PE +N1 +PE +12 +PE +22 +PE +N2 +PE +1N +PE +2N +PE +NN +TS 2 +TS N +Time Steps +Systolic Array +Mapping of +Images +Mapping of Filters +Time Steps +Spikes +I = 1 +TS 1 +Figure 1: A systolicSNN with faulty processing elements (PEs) +in red color and non-faulty PEs in white color +the accumulation operation in systolicSNNs is cheaper than the +multiplier needed for the multiplier-and-accumulator (MAC) +unit in systolic-array ANN accelerators [4] [28]. The lack of +multipliers renders systolicSNNs energy efficient in comparison +to systolic-array ANN accelerators. The PEs employs an addi- +tion and subtraction selection unit also for processing signed +weights. Furthermore, an internal counter helps in counting the +number of spikes in the inference phase. +III. MOTIVATIONAL CASE STUDY +To motivate the proposed FalVolt mitigation method, we +begin by empirically analyzing the impact of different threshold +voltages on the classification accuracy of a faulty systolicSNN. +To do so, we first train a PLIF-SNN with the MNIST and +DVS128 Gesture datasets. Then, we inject the stuck-at faults +using different fault maps for 30% and 60% PEs in a 256x256 +systolicSNN. Next, we run paralleled retraining simulations +with different threshold voltages. As shown in Fig. 2, we +observe that changing the threshold voltage from 1.0 to 0.55 +and 0.7 values in retraining leads to 99% classification accuracy +with the MNIST dataset when even 30% and 60% PEs are +faulty in a systolicSNN, respectively. However, retraining the +same model with threshold voltage 0.45 and 0.5 leads to +almost 73% and 60% accuracy loss when 30% and 60% PEs +are faulty in a systolicSNN, respectively. In addition, 0.45 +and 0.7 threshold voltages are most suitable for classifying +the DVS128 Gesture dataset with a systolicSNN having 30% +and 60% faults in PEs, respectively. However, retraining the +same model with threshold voltages 0.7 and 0.5 leads to +almost 60% and 55% accuracy loss when 30% and 60% PEs +are faulty in a systolicSNN, respectively. Thus, selecting an +appropriate threshold voltage for retraining the systolicSNN +with high classification accuracy is imperative. Nevertheless, +finding a suitable threshold voltage requires extensive retraining +simulations, which may incur a significant amount of time. +Motivated by this, we propose a novel fault-aware threshold +voltage optimization technique in retraining for fault mitigation. + +0 +50 +100 +0.45 +0.5 +0.55 +0.7 +Accuracy [%] +Threshold Voltage +30% +60% +(a) MNIST classification +0 +50 +100 +0.45 +0.5 +0.55 +0.7 +Accuracy [%] +Threshold Voltage +30% +60% +(b) DVS128 Gesture classification +Figure 2: Stuck-at fault mitigation using different threshold +voltages (Vth), 30% and 60% of the total PEs are faulty in +a 256x256 systolic-array SNN accelerator (systolicSNN) +_ +Fixed-point Adder-Subtractor +Accumulator +Internal Counter +Input +Weight +0 ++ +Add +Sub +_ +Fixed-point Adder-Subtractor +Accumulator +Internal Counter +0 ++ +Pre-sum +Add +Sub +0 +Ctrl +(a) Actual PE +(b) Bypassed PE +Pre-sum +Input +Figure 3: Processing element with actual and bypassed circuitry +IV. PROPOSED FAULT-AWARE THRESHOLD VOLTAGE +OPTIMIZATION (FALVOLT) +Our proposed FalVolt mitigation method improves the re- +liability of systolicSNNs by first setting the input pre-trained +weights which map to the faulty PEs as zero. The fault locations +are determined through post-fabrication tests on a systolicSNN +chip. This initial step is similar to bypassing a PE using a +multiplexer at the hardware level, as shown in Fig. 3b, in +systolicSNNs. With the bypass path enabled, the contribution +of the faulty PEs to the column sum is skipped. However, +bypassing single faulty PE may result in the pruning of multiple +pre-trained weights due to the reuse of systolicSNNs in the +data processing. Therefore, FalVolt next retrains the unpruned +weights while optimizing the threshold voltage for each layer. +The threshold voltage optimization saves the retraining time +by eliminating the need for an exhaustive search for an appro- +priate threshold voltage. It makes SNN less sensitive to initial +values and enhances and speeds up the learning. The optimized +threshold voltage is used for all neurons in a layer to reduce the +retrainable parameters and time. FalVolt optimizes the weights +using the recursive gradient computations during both initial +training and retraining. The weights mapped to faulty PEs are +set as zero at the end of every retraining epoch. However, +the threshold voltage is optimized for each layer during the +retraining only, as discussed below: +Lets consider r as a ratio between the membrane potential +v and threshold voltage V. A neuron fires an output spike o +when v exceeds V. Mathematically, this can be written as: +zt +l = rt +l − 1 and ot +l = +� +1, +if zt +l > 0. +0, +otherwise. +(1) +Here, the notation xt +l represents the parameters of SNN in +the l-th layer of the network at time step t. The discontinuous +gradient ∂o +∂zt +l is approximated with the surrogate function during +Algorithm 1: FalVolt Mitigation Algorithm +Inputs : (i) pre-trained weights: wts; (ii) training data: +trData; (iii) test data: tsData; (iv) fault maps: fmaps; +(v) time steps: T; (vi) max retraining epochs: +trEpochs; (vii) learning rate: η; +Outputs: Accuracy: acc; +1: ind = FindPrunedWeightsIndices (fmaps, wts) +//Find indices of pruning weights from fault maps +2: pWts = SetPrunedWeightsToZero(ind, wts) +//Assign zero to the pruning weights at above indices +3: (pVth, θ) = parameterInnitialization() +//Initialize θ and threshold voltage parameters +4: for epochs = 0 : trEpochs - 1 do +5: +for t = 0 : T - 1 do +6: +for l = 0 : L - 1 do +7: +(nWts) = UpdateWeights (pWts, ts, trData) +//Update weights with backpropagation +8: +(nVth) = UpdateVoltageThresh (pVth, ts, trData) +//Update threshold voltage with backpropagation +9: +end for +10: +L = CalculateLoss(trData) +// Calculate cross entropy loss +11: +θ = θ - η ∆L +//Update network parameter θ +12: +end for +13: +nWts = SetUpdatedWeightsToZero(nWts, ind) +//Assign zero to all pruning weights using indices in Step 1 +14: end for +15: acc = CheckInferenceAccuracy(nWts, tsData) +//Check inference accuracy using new weights +16: return (nWts, nVth, acc); +error-backpropagation in retraining, similar to initial training. +The term +∂o +∂zt +l is expressed mathematically as: +∂ot +l +∂zt +l += γ max(0, 1 − |zt +l|) +(2) +where γ is a constant denoting the maximum value of the sur- +rogate function. During backpropagation, the threshold voltage +V is updated for layer l as follows: +Vl = Vl−1 − η ∆V +(3) +where η represents the learning rate. Here, the gradient of +threshold voltage ∆V for layer l can be computed as: +∆Vl = ∂L +∂Vl += +T −1 +� +t=0 +∂L +∂ot +l +∂o +∂zt +l +∂z +∂Vl += +T −1 +� +t=0 +∂L +∂ot +l +∂o +∂zt +l +(−Vlot−1 +l +− vt +l +V +2 +l +) +(4) +where L represents the cross entropy loss function defined by +the mean square error. Algorithm 1 delineates the proposed +FalVolt mitigation method. Lines 1-2 prunes the pre-trained +weights mapped to the faulty PEs in systolicSNNs. Line 3 +initializes the heavy step function θ and V. Lines 4-5 computes +the un-pruned weights and V with multiple epochs in back- +propagation. The un-pruned weights and V are optimized +in each time-step for every layer in the PLIF-SNN, while +calculating the gradient of loss function (∆L) in Line 10- +11. Line 13 set the weights mapped to faulty PEs as zero +at the end of each training epoch. It is interesting to note +that setting the re-training epochs to zero makes the FalVolt + +Dataset +Run on +GPU +SNN model +configuration +SNN model +generation +Desired Accuracy +Output Accuracy +SNN Fault Vulnerability and Mitigation Framework +Fault +injection +Fault mapping +to systolic array +Structural +Parameters +Systolic Array +Configuration +Fault maps +Fault pruning +with structural +re-training +SNN model +Fault pruning +Figure 4: Experimental setup and tool flow +equivalent to simple fault-aware pruning (FaP). FalVolt returns +new optimized values for the unpruned weights (or the re- +trained model), V for each layer and the improved classification +accuracy. Note, the proposed mitigation needs to be performed +once only for the fabricated chip based on its unique fault map +and thus, helps in avoiding the re-fabrication cost of the chips. +V. RESULTS AND DISCUSSIONS +This section discusses the results obtained from the fault +vulnerability and mitigation analysis of systolicSNNs. +A. Datasets and network architectures +We adopted a static MNIST [24], and two neuromorphic N- +MNIST [25] and DVS128 Gesture [26] datasets in this paper. +Note that the SNN research community widely uses these +datasets for evaluating the performance of SNNs [29], [16]. +As a classifier for N-MNIST and MNIST datasets, we use a +PLIF-based SNN with two times repeated set of convolutional, +batch normalization, spiking neurons, and pooling layers and +also, two times a set of dropout, fully connected, and spiking +neurons layers. The former set is repeated five times with the +same architecture configuration in the classifier for the DVS128 +Gesture dataset. Furthermore, an additional set of convolutional +layer and spiking neurons layer is used for spike encoding the +input images, inspired by [30], in these architectures. We use +the initialization parameters from [27] to achieve the baseline +accuracy i.e., 99% for the MNIST [24] and N-MNIST [25] +datasets, and 97% for DVS128 Gesture [26] dataset, prior +to fault injection in the inference phase. For systolicSNN +inference, we developed a 256x256 grid of PEs in VHDL with +bypass circuitry that incurs only 8% area overhead. +B. Simulation Methodology +Fig.4 illustrates the tool-flow used for fault vulnerability and +mitigation analysis in this paper. First, the SNN models are +trained with their baseline accuracies. Next, the stuck-at faults +are injected into the accumulator outputs of PEs using different +fault maps. Then, the fault pruning is applied by setting +the weights mapped to the faulty PEs as zero. Finally, fault +mitigation through re-training with layer-wise threshold voltage +optimization is employed using Algorithm 1. All simulations +are conducted using NVIDIA GeForce RTX 2080 Ti GPU on +Intel Core i9-10900kF operating at 3.06 GHz with 32 GB RAM. +C. Fault vulnerability analysis +To investigate the stuck-at faults vulnerability in systolic- +SNNs, we extensively analyze their impact by varying the +location of fault bits, the number of faulty PEs, and the size of +the systolic array as follows. +Varying location of fault bits: Before running extensive +simulations for fault mitigation, we first identify the most +vulnerable bits to the stuck-at faults in the PEs of a 256x256 +systolicSNN. For this purpose, we generate the fault maps +such that the stuck-at 0 and stuck-at 1 faults are injected in +different output bit positions of the accumulator inside the PEs. +Note, fault injection with fault maps is a common practice for +analyzing the fault vulnerabilities in systolic arrays [31]. Fault +maps can be generated using post-fabrication testing in a real- +world scenario. It is worth mentioning that we inject faults +in the output of the accumulator, which is the main arithmetic +component of the PEs. As shown in Fig. 5a, our analysis reveals +that stuck-at faults in most significant bits (MSBs) affect the +classification accuracy more than the stuck-at faults in the least +significant bits (LSBs). The reason is that the systolic array is +reused for different layers; therefore, a single unmasked fault +in a PE of a particular layer affects all the connected nodes +in the subsequent layers, decreasing the overall classification +accuracy. We also observe that a stuck-at 1 fault in MSB causes +almost 80% accuracy loss, which is higher than the same fault +in LSB when classifying the MNIST, N-MNIST, and DVS128 +Gesture datasets. It is worth noticing that stuck-at 1 faults are +more perturbing than stuck-at 0 faults in systolicSNN, similar +to systolic array ANN accelerators [20]. +Varying number of faulty PEs: Next, we perform the fault +simulations by considering a random distribution of the stuck- +at faults across a 256x256 systolicSNN. We vary the fault rates +by varying the number of faulty PEs in each experiment and +running each experiment 8 times. The number of faulty PEs +stays the same for all iterations in an experiment. Furthermore, +each iteration uses a distinct fault map. In the following section, +the faults are injected in the higher-order bits (i.e., MSBs) of the +accumulator outputs in PEs to perform the worst-case analysis. +Moreover, the average classification accuracies for all iterations +in an experiment are recorded. As shown in Fig. 5b, our results +demonstrate that even 8 faulty PEs (i.e., 0.012% of total PEs) +can lead to an accuracy drop from 99% to 50%, 99% to 47 +% and 97% to 44% in the MNIST, N-MNIST and DVS128 +Gesture classification, respectively. Hence, the classification of +both static and neuromorphic datasets is prone to stuck-at faults. +Varying size of the systolic array: For further extensive fault +vulnerability study, we analyze the impact of stuck-at faults +across different sizes of NxN systolic arrays i.e., 4x4, 8x8, +16x16, 32x32 and 64x64. As shown in Fig. 5c, our analysis +reveals that stuck-at faults in a small-sized systolic array cause +more accuracy loss as compared to a large-sized systolic array. +For example, 4 faulty PEs units in an 8x8 systolic array (having +16 PEs) lead to 89%, 92% and 93% accuracy loss in the +MNIST, N-MNIST and DVS128 Gesture classification, respec- +tively. However, SNN classification with a 256x256 systolic +array, having the same fault configuration, results in almost +16%, 17%, and 33% accuracy loss only. This is due to the +fact that decreasing the size of the systolic array increases +its chances for re-usability and hence, the reoccurrence of the +permanent faults in every execution cycle. +Our analysis shows that DVS128 Gesture is more vulnerable + +彩彩0 +30 +60 +90 +0 +2 +4 +6 +8 +10 +12 +14 +16 +Accuracy [%] +Fault Bit Location +sa0-MNIST +sa1-MNIST +sa0-NMNIST +sa1-NMNIST +sa0-DVS128Gesture +sa1-DVS128Gesture +(a) Accuracy vs Fault Bit Locations +0 +20 +40 +60 +80 +100 +0 +4 +8 +16 +32 +40 +48 +56 +64 +Accuracy [%] +Number of Faulty PEs +MNIST +N-MNIST +DVS128 Gesture +(b) Accuracy vs number of faulty PEs +0 +20 +40 +60 +80 +100 +16 +64 +256 +1024 +65536 +Accuracy [%] +Total Number of PEs +MNIST +N-MNIST +DVS128 Gesture +(c) Accuracy vs size of systolic array +Figure 5: Stuck-at fault vulnerability analysis of a 256x256 systolic-array based SNN accelerator (systolicSNN). +0 +0.2 +0.4 +0.6 +0.8 +1 +Conv1 +Conv2 +FC1 +FC2 +Threshold +Voltage +Layers +10% +30% +60% +(a) MNIST [24] classification +0 +0.2 +0.4 +0.6 +0.8 +1 +Conv1 +Conv2 +FC1 +FC2 +Threshold +Voltage +Layers +10% +30% +60% +(b) N-MNIST [25] classification +0 +0.2 +0.4 +0.6 +0.8 +1 +Conv1Conv2Conv3Conv4Conv5 FC1 +FC2 +Threshold +Voltage +Layers +10% +30% +60% +(c) DVS128 Gesture [26] classification +Figure 6: Optimized threshold voltage for hidden convolutional and fully connected layers using FalVolt, when 0%, 10%, 30% +and 60% of the total PEs are faulty in a 256x256 systolic-array SNN accelerator (systolicSNN) +to faults when compa red to the MNIST and N-MNIST datasets, +even though their baseline accuracies are the same. As shown in +Fig. 5b, the classification accuracy of DVS128 Gesture remains +comparatively lower than other datasets in the presence of +stuck-at faults. Also, the accuracy loss associated with the +DVS128 Gesture dataset is comparatively higher than other +datasets in Fig. 5c. However, a higher number of stuck-at faults +can render performance penalties unacceptable in all cases. +D. Fault mitigation analysis +In this section, we study the performance of FalVolt and +compare it with the state-of-the-art techniques common for +ANNs. Specifically, we compare FalVolt with fault-aware prun- +ing (FAP) and fault-aware pruning with retraining without +threshold voltage optimization (FaPIT). +Classification accuracy vs. fault rates: For the fault mitigation +analysis, we inject the stuck-at faults using different fault maps +in 10%, 30%, and 60% PEs of a 256x256 systolicSNN and +run paralleled re-training simulations. We employ the proposed +FalVolt mitigation method using Algorithm 1 for 10%, 30%, +and 60% PEs in a 256x256 systolicSNN. Our analysis shows +that optimizing threshold voltage for each hidden convolutional +and fully connected layer helps in achieving baseline accuracy. +Fig. 6 shows the optimized threshold voltage returned from the +FalVolt mitigation method for each hidden layer to achieve the +baseline accuracy for MNIST, NMNIST, and DVS128 Gesture +datasets. For all these datasets, the optimized threshold voltage +for the initial spiking-convolutional and spiking-fully connected +layers is higher than other layers to ensure that the redundant +spikes do not travel to the output layer. +Fig. 7 compares the FalVolt mitigation method with FaP and +FaPIT. We observe that an increased fault rate causes a rapid +accuracy loss in the FaP. FaPIT and FalVolt help in improving +classification accuracy. However, only FalVolt achieves the +baseline classification accuracy in the MNIST, N-MNIST, and +DVS128 Gesture classification with even 60% of the faulty +PEs. This validates the applicability of FalVolt to both static +and neuromorphic datasets. +Classification accuracy vs. number of epochs: FalVolt in- +creases the classification accuracy at the cost of additional +retraining epochs to FaP; however, they are negligible com- +pared to the lifetime of systolicSNNs. As shown in Fig. 8, +FaPVolt is 2x faster than FaPIT. For example, the classification +accuracy of MNIST is as high as 80% with FaPIT using 20 +epochs and converges with baseline accuracy around 25 epochs. +However, the same dataset achieves the baseline accuracy with +FalVolt in 10 epochs, as shown in Fig. 8a. Likewise, FalVolt +achieves the baseline accuracy of NMNIST classification 2x +less number of epochs when compared to FaPIT as shown +in Fig. 8b. Moreover, the classification accuracy of DVS128 +Gesture is as high as 83% with FaPIT using 40 epochs and +converges with baseline accuracy around 50 epochs as shown in +Fig. 8c. However, the same dataset achieves 97% accuracy with +FalVolt around 25 epochs. Since a small change in the base- +line accuracy may cause catastrophic issues in safety-critical +applications; therefore, the epochs for initial training, FaPIT, +and FalVolt algorithms are high to achieve the classification +accuracy close to the baseline. Note, training the large-sized +SNNs itself takes a long time (or a higher number of epochs). +VI. CONCLUSION +This paper extensively analyzes the stuck-at fault vulnera- +bilities of systolicSNNs and proposes a novel fault mitigation +technique ‘fault-aware retraining through threshold voltage +optimization (FalVolt).’ FalVolt uses an optimized threshold +voltage and time steps different from initial training to achieve +classification accuracy close to the baseline. To demonstrate the +effectiveness of FalVolt, we classify the MNIST, N-MNIST, +and DVS128 Gesture datasets on a 256x256 systolicSNN + +0 +20 +40 +60 +80 +100 +10% +30% +60% +Accuracy [%] +Number of faulty PEs +FaP +FaPIT +FalVolt +(a) MNIST [24] classification +0 +20 +40 +60 +80 +100 +10% +30% +60% +Accuracy [%] +Number of faulty PEs +FaP +FaPIT +FalVolt +(b) N-MNIST [25] classification +0 +20 +40 +60 +80 +100 +10% +30% +60% +Accuracy [%] +Number of faulty PEs +FaP +FaPIT +FalVolt +(c) DVS128 Gesture [26] classification +Figure 7: Stuck-at fault mitigation using FaP, FaPIT (using threshold voltage as 1.0) and FalVolt, when 0%, 10%, 30% and 60% +of the total PEs are faulty in a 256x256 systolic-array SNN accelerator (systolicSNN) +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +30 +Accuracy [%] +Retraining Epochs +FaPIT +FalVolt +(a) MNIST [24] classification +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +30 +Accuracy [%] +Retraining Epochs +FaPIT +FalVolt +(b) N-MNIST [25] classification +0 +20 +40 +60 +80 +100 +0 +10 +20 +25 +30 +40 +50 +Accuracy [%] +Retraining Epochs +FaPIT +FalVolt +(c) DVS128 Gesture [26] classification +Figure 8: Performance of FaPIT and FalVolt over different epochs when 30% the total PEs are faulty in a 256x256 systolic-array +SNN accelerator (systolicSNN) +while injecting faults at different rates. 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Hanif et al., “Dependable deep learning: Towards cost-efficient +resilience of deep neural network accelerators against soft errors and +permanent faults,” in IOLTS. +IEEE, 2020, pp. 1–4. + diff --git a/ztE4T4oBgHgl3EQfyg04/content/tmp_files/load_file.txt b/ztE4T4oBgHgl3EQfyg04/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..af2da06cdef088cefb3fc652a72cbf0a3b6508eb --- /dev/null +++ b/ztE4T4oBgHgl3EQfyg04/content/tmp_files/load_file.txt @@ -0,0 +1,661 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf,len=660 +page_content='Improving Reliability of Spiking Neural Networks through Fault Aware Threshold Voltage Optimization Ayesha Siddique, Khaza Anuarul Hoque Department of Electrical Engineering and Computer Science University of Missouri, Columbia, MO, USA ayesha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='siddique@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='missouri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='edu, hoquek@missouri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='edu Abstract—Spiking neural networks have made breakthroughs in computer vision by lending themselves to neuromorphic hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, the neuromorphic hardware lacks parallelism and hence, limits the throughput and hardware acceleration of SNNs on edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' To address this problem, many systolic-array SNN accelerators (systolicSNNs) have been proposed recently, but their reliability is still a major concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' In this paper, we first extensively analyze the impact of permanent faults on the SystolicSNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Then, we present a novel fault mitigation method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', fault-aware threshold voltage optimization in retraining (FalVolt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' FalVolt optimizes the threshold voltage for each layer in retraining to achieve the classification accuracy close to the baseline in the presence of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' To demonstrate the effectiveness of our proposed mitigation, we classify both static (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', MNIST) and neuromorphic datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', N-MNIST and DVS Gesture) on a 256x256 systolicSNN with stuck-at faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' We empirically show that the classification accuracy of a systolicSNN drops significantly even at extremely low fault rates (as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='012%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Our proposed FalVolt mitigation method improves the performance of systolicSNNs by enabling them to operate at fault rates of up to 60%, with a negligible drop in classification accuracy (as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Our results show that FalVolt is 2x faster compared to other state-of-the-art techniques common in artificial neural networks (ANNs), such as fault-aware pruning and retraining without threshold voltage optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Index Terms—Spiking neural networks, Stuck-at faults, Systolic array, Fault mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' INTRODUCTION Spiking neural networks (SNNs) are a promising third generation of neural networks that ensure high algorithmic performance at low power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Their hardware acceleration re- quire specialized architectures such as, SpiNNaker [1], and TrueNorth [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, these architectures lack parallelism in each core and efficient dataflows for maximizing the reuse of weight data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' This limits their achievable throughput and robustness in resource-constrained devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', battery-driven autonomous cars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Towards this, leveraging SNNs on massively parallel hardware accelerators such as systolic arrays has proven to be an efficient solution [3], [4], [5], [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Systolic array SNN accelerators (systolicSNNs) are inspired by other state-of- the-art hardware accelerators [8] which support fully parallel execution of artificial neural networks (ANNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' These accel- erators have a NxN dense grid of interconnected processing elements (PEs), which allows efficient parallel processing with the high spatio-temporal locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Unlike ANNs, SNNs and their hardware accelerators are still in a relatively early phase of adoption [9] and thus ensuring the reliability of systolicSNNs is still considered a major research challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The systolicSNN hardware chips are manufactured using nanometer CMOS technologies [10], which require a highly sophisticated manufacturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The imperfections in this process result in various manufacturing defects ranging from process variations to permanent faults such as stuck-at faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The stuck-at faults affect the output of systolicSNNs in every execution cycle and hence, lead to significant accuracy loss as discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Furthermore, the impact of large-scale failures such as dead synapse faults in SNNs has been thor- oughly investigated [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, analyzing such failures in the hardware require a fault model with higher abstraction to make the simulation traceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' investigated the fault resilience of SNNs trained with different coding schemes by using a synaptic stuck-at fault model [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' El-Sayed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' analyzed the effect of these faults in a transistor-level design of leaky-integrate-and-fire (LIF) neuron [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Other state-of-the- art works focus on bit flips in weight memories [15], [16], [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Conversely, the impact of stuck-at faults on systolicSNNs has not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The stuck-at faults are usually detected using post-fabrication testing for discarding the faulty manufactured chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, if a high number of manufactured chips are faulty, discarding them reduces the yield to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' A potential solution is employing redundant executions (re-execution) to ensure correct outputs, but it leads to significant latency and energy overheads [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' In the current resource-constrained nanoscale hardware paradigm, where the number of PEs has drastically increased to meet the robustness requirements of the end users, it is imperative to maximize the yield with an efficient and fault-tolerant systolicSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Recently, Mehul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' proposed an astrocyte self-repair mechanism for stuck-at 0 weights in SNNs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Other works are either focused on mitigating the transient faults in SNNs [16], [19] or contemplated permanent fault mitigation in ANN accelerators [20], [21], [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, a considerable research gap exists in mitigating the impact of permanent faults in systolicSNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Novel contributions: In this paper, we present an extensive stuck-at fault vulnerability analysis and a novel fault mitigation method i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', fault-aware retraining through threshold voltage optimization (FalVolt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' FalVolt first sets the weights mapped to faulty PEs only as zero and then retrains weights mapped to non-faulty PEs while optimizing the threshold voltage for each layer to restore the classification accuracy close to its baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The optimized threshold voltage differs from the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='05266v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='NE] 12 Jan 2023 actual threshold voltage used in initial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' To demonstrate the effectiveness of our proposed FalVolt mitigation method, we used both static MNIST [24], and neuromorphic N-MNIST [25] and DVS128 Gesture [26] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Our results show that FalVolt can operate at high fault rates of up to 60% with a negligible impact on the classification accuracy compared to its baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' We empirically show that FalVolt takes 2x fewer retraining epochs, and thus it is 2x faster in restoring the baseline accuracy compared to other state-of-art techniques such as fault-aware pruning and retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Note, fault-aware pruning and retraining and threshold voltage optimization have been conventionally used for ANN fault mitigation [21], [22] and faster SNN convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, to the best of our knowledge, this is the first work to employ fault-aware thresh- old voltage optimization for fault mitigation in SNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The remainder of this paper is structured as follows: Sec- tion II provides the preliminary information about SNNs and systolicSNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Section III and Section IV present a motivational case study and the proposed FalVolt mitigation method for systolicSNNs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Section V discusses the results for the fault vulnerability and mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Finally, Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' BACKGROUND This section provides a brief overview of the state-of-the-art SNNs and systolicSNNs for better understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Spiking Neural Networks: SNNs are bio-inspired artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Their working principle can be explained with a standard LIF model as follows: when the membrane potential Vt of a presynaptic neuron exceeds a specific threshold voltage at time t, a post-synaptic spike is fired, and then, Vt relaxes to the resting state (Vrest < threshold voltage) with a time constant τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Vt maintains the resting state for a refractory time tref before responding to the received spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The LIF-based SNNs learn the presynaptic weights but require manual tuning of the time constant in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Furthermore, the time constant is typically chosen to be the same for all neurons, which limits the diversity of neurons and, thus, the expressiveness of the LIF-based SNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Recently, Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' proposed to train the weights along with the time constant through an advanced LIF model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', parametric leaky integrate-and-fire (PLIF) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Incorporating the learnable time constants through PLIF-based SNNs makes the network less sensitive to initial values and reduces the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Systolic-Array SNN Accelerators: SystolicSNNs exploit the spatial and temporal parallelisms for which binary spike in- put, logical 1 or 0 propagate vertically across the systolic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 1, the spike input is first divided into multiple time steps and then, all input values in a time step are mapped on one row of the systolic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The input binary spikes pass through a dense NxN grid of interconnected PEs in a clocked synchronized manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The filter data is mapped and pre-stored in the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 3a shows the design of a standard PE in systolicSNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The PE accumulates 32-bit weight inputs under 1-bit binary spikes on an enable signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The adder needed for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='C = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Filters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='TS 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='TS 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='TS 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Input Feature Maps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='with All Time Batches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='I = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='C = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='PE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='PE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='PE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='N1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='PE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='PE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='PE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='PE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='1N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='PE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='2N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='PE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='NN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='TS 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='TS N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Time Steps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Systolic Array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Mapping of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Mapping of Filters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Time Steps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Spikes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='I = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='TS 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Figure 1: A systolicSNN with faulty processing elements (PEs) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='in red color and non-faulty PEs in white color ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='the accumulation operation in systolicSNNs is cheaper than the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='multiplier needed for the multiplier-and-accumulator (MAC) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='unit in systolic-array ANN accelerators [4] [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The lack of multipliers renders systolicSNNs energy efficient in comparison to systolic-array ANN accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The PEs employs an addi- tion and subtraction selection unit also for processing signed weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Furthermore, an internal counter helps in counting the number of spikes in the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' MOTIVATIONAL CASE STUDY To motivate the proposed FalVolt mitigation method, we begin by empirically analyzing the impact of different threshold voltages on the classification accuracy of a faulty systolicSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' To do so, we first train a PLIF-SNN with the MNIST and DVS128 Gesture datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Then, we inject the stuck-at faults using different fault maps for 30% and 60% PEs in a 256x256 systolicSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Next, we run paralleled retraining simulations with different threshold voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 2, we observe that changing the threshold voltage from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='55 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='7 values in retraining leads to 99% classification accuracy with the MNIST dataset when even 30% and 60% PEs are faulty in a systolicSNN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, retraining the same model with threshold voltage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='45 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='5 leads to almost 73% and 60% accuracy loss when 30% and 60% PEs are faulty in a systolicSNN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' In addition, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='45 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='7 threshold voltages are most suitable for classifying the DVS128 Gesture dataset with a systolicSNN having 30% and 60% faults in PEs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, retraining the same model with threshold voltages 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='7 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='5 leads to almost 60% and 55% accuracy loss when 30% and 60% PEs are faulty in a systolicSNN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Thus, selecting an appropriate threshold voltage for retraining the systolicSNN with high classification accuracy is imperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Nevertheless, finding a suitable threshold voltage requires extensive retraining simulations, which may incur a significant amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Motivated by this, we propose a novel fault-aware threshold voltage optimization technique in retraining for fault mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='7 Accuracy [%] Threshold Voltage 30% 60% (a) MNIST classification 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='7 Accuracy [%] Threshold Voltage 30% 60% (b) DVS128 Gesture classification Figure 2: Stuck-at fault mitigation using different threshold voltages (Vth),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 30% and 60% of the total PEs are faulty in a 256x256 systolic-array SNN accelerator (systolicSNN) _ Fixed-point Adder-Subtractor Accumulator Internal Counter Input Weight 0 + Add Sub _ Fixed-point Adder-Subtractor Accumulator Internal Counter 0 + Pre-sum Add Sub 0 Ctrl (a) Actual PE (b) Bypassed PE Pre-sum Input Figure 3: Processing element with actual and bypassed circuitry IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' PROPOSED FAULT-AWARE THRESHOLD VOLTAGE OPTIMIZATION (FALVOLT) Our proposed FalVolt mitigation method improves the re- liability of systolicSNNs by first setting the input pre-trained weights which map to the faulty PEs as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The fault locations are determined through post-fabrication tests on a systolicSNN chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' This initial step is similar to bypassing a PE using a multiplexer at the hardware level, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 3b, in systolicSNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' With the bypass path enabled, the contribution of the faulty PEs to the column sum is skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, bypassing single faulty PE may result in the pruning of multiple pre-trained weights due to the reuse of systolicSNNs in the data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Therefore, FalVolt next retrains the unpruned weights while optimizing the threshold voltage for each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The threshold voltage optimization saves the retraining time by eliminating the need for an exhaustive search for an appro- priate threshold voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' It makes SNN less sensitive to initial values and enhances and speeds up the learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The optimized threshold voltage is used for all neurons in a layer to reduce the retrainable parameters and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' FalVolt optimizes the weights using the recursive gradient computations during both initial training and retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The weights mapped to faulty PEs are set as zero at the end of every retraining epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, the threshold voltage is optimized for each layer during the retraining only, as discussed below: Lets consider r as a ratio between the membrane potential v and threshold voltage V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' A neuron fires an output spike o when v exceeds V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Mathematically, this can be written as: zt l = rt l − 1 and ot l = � 1, if zt l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' (1) Here, the notation xt l represents the parameters of SNN in the l-th layer of the network at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The discontinuous gradient ∂o ∂zt l is approximated with the surrogate function during Algorithm 1: FalVolt Mitigation Algorithm Inputs : (i) pre-trained weights: wts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' (ii) training data: trData;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' (iii) test data: tsData;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' (iv) fault maps: fmaps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' (v) time steps: T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' (vi) max retraining epochs: trEpochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' (vii) learning rate: η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Outputs: Accuracy: acc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 1: ind = FindPrunedWeightsIndices (fmaps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' wts) //Find indices of pruning weights from fault maps 2: pWts = SetPrunedWeightsToZero(ind,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' wts) //Assign zero to the pruning weights at above indices 3: (pVth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' θ) = parameterInnitialization() //Initialize θ and threshold voltage parameters 4: for epochs = 0 : trEpochs - 1 do 5: for t = 0 : T - 1 do 6: for l = 0 : L - 1 do 7: (nWts) = UpdateWeights (pWts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' trData) //Update weights with backpropagation 8: (nVth) = UpdateVoltageThresh (pVth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' trData) //Update threshold voltage with backpropagation 9: end for 10: L = CalculateLoss(trData) // Calculate cross entropy loss 11: θ = θ - η ∆L //Update network parameter θ 12: end for 13: nWts = SetUpdatedWeightsToZero(nWts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' ind) //Assign zero to all pruning weights using indices in Step 1 14: end for 15: acc = CheckInferenceAccuracy(nWts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' tsData) //Check inference accuracy using new weights 16: return (nWts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' nVth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' acc);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' error-backpropagation in retraining, similar to initial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The term ∂o ∂zt l is expressed mathematically as: ∂ot l ∂zt l = γ max(0, 1 − |zt l|) (2) where γ is a constant denoting the maximum value of the sur- rogate function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' During backpropagation, the threshold voltage V is updated for layer l as follows: Vl = Vl−1 − η ∆V (3) where η represents the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Here, the gradient of threshold voltage ∆V for layer l can be computed as: ∆Vl = ∂L ∂Vl = T −1 � t=0 ∂L ∂ot l ∂o ∂zt l ∂z ∂Vl = T −1 � t=0 ∂L ∂ot l ∂o ∂zt l (−Vlot−1 l − vt l V 2 l ) (4) where L represents the cross entropy loss function defined by the mean square error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Algorithm 1 delineates the proposed FalVolt mitigation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Lines 1-2 prunes the pre-trained weights mapped to the faulty PEs in systolicSNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Line 3 initializes the heavy step function θ and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Lines 4-5 computes the un-pruned weights and V with multiple epochs in back- propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The un-pruned weights and V are optimized in each time-step for every layer in the PLIF-SNN, while calculating the gradient of loss function (∆L) in Line 10- 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Line 13 set the weights mapped to faulty PEs as zero at the end of each training epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' It is interesting to note ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='that setting the re-training epochs to zero makes the FalVolt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Run on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='GPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='SNN model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='SNN model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Desired Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Output Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='SNN Fault Vulnerability and Mitigation Framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Fault ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='injection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Fault mapping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='to systolic array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Structural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Systolic Array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Fault maps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Fault pruning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='with structural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='re-training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='SNN model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Fault pruning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Figure 4: Experimental setup and tool flow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='equivalent to simple fault-aware pruning (FaP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' FalVolt returns new optimized values for the unpruned weights (or the re- trained model), V for each layer and the improved classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Note, the proposed mitigation needs to be performed once only for the fabricated chip based on its unique fault map and thus, helps in avoiding the re-fabrication cost of the chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS This section discusses the results obtained from the fault vulnerability and mitigation analysis of systolicSNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Datasets and network architectures We adopted a static MNIST [24], and two neuromorphic N- MNIST [25] and DVS128 Gesture [26] datasets in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Note that the SNN research community widely uses these datasets for evaluating the performance of SNNs [29], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' As a classifier for N-MNIST and MNIST datasets, we use a PLIF-based SNN with two times repeated set of convolutional, batch normalization, spiking neurons, and pooling layers and also, two times a set of dropout, fully connected, and spiking neurons layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The former set is repeated five times with the same architecture configuration in the classifier for the DVS128 Gesture dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Furthermore, an additional set of convolutional layer and spiking neurons layer is used for spike encoding the input images, inspired by [30], in these architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' We use the initialization parameters from [27] to achieve the baseline accuracy i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', 99% for the MNIST [24] and N-MNIST [25] datasets, and 97% for DVS128 Gesture [26] dataset, prior to fault injection in the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' For systolicSNN inference, we developed a 256x256 grid of PEs in VHDL with bypass circuitry that incurs only 8% area overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Simulation Methodology Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='4 illustrates the tool-flow used for fault vulnerability and mitigation analysis in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' First, the SNN models are trained with their baseline accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Next, the stuck-at faults are injected into the accumulator outputs of PEs using different fault maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Then, the fault pruning is applied by setting the weights mapped to the faulty PEs as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Finally, fault mitigation through re-training with layer-wise threshold voltage optimization is employed using Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' All simulations are conducted using NVIDIA GeForce RTX 2080 Ti GPU on Intel Core i9-10900kF operating at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='06 GHz with 32 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Fault vulnerability analysis To investigate the stuck-at faults vulnerability in systolic- SNNs, we extensively analyze their impact by varying the location of fault bits, the number of faulty PEs, and the size of the systolic array as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Varying location of fault bits: Before running extensive simulations for fault mitigation, we first identify the most vulnerable bits to the stuck-at faults in the PEs of a 256x256 systolicSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' For this purpose, we generate the fault maps such that the stuck-at 0 and stuck-at 1 faults are injected in different output bit positions of the accumulator inside the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Note, fault injection with fault maps is a common practice for analyzing the fault vulnerabilities in systolic arrays [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Fault maps can be generated using post-fabrication testing in a real- world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' It is worth mentioning that we inject faults in the output of the accumulator, which is the main arithmetic component of the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 5a, our analysis reveals that stuck-at faults in most significant bits (MSBs) affect the classification accuracy more than the stuck-at faults in the least significant bits (LSBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The reason is that the systolic array is reused for different layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' therefore, a single unmasked fault in a PE of a particular layer affects all the connected nodes in the subsequent layers, decreasing the overall classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' We also observe that a stuck-at 1 fault in MSB causes almost 80% accuracy loss, which is higher than the same fault in LSB when classifying the MNIST, N-MNIST, and DVS128 Gesture datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' It is worth noticing that stuck-at 1 faults are more perturbing than stuck-at 0 faults in systolicSNN, similar to systolic array ANN accelerators [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Varying number of faulty PEs: Next, we perform the fault simulations by considering a random distribution of the stuck- at faults across a 256x256 systolicSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' We vary the fault rates by varying the number of faulty PEs in each experiment and running each experiment 8 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' The number of faulty PEs stays the same for all iterations in an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Furthermore, each iteration uses a distinct fault map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' In the following section, the faults are injected in the higher-order bits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', MSBs) of the accumulator outputs in PEs to perform the worst-case analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Moreover, the average classification accuracies for all iterations in an experiment are recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 5b, our results demonstrate that even 8 faulty PEs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='012% of total PEs) can lead to an accuracy drop from 99% to 50%, 99% to 47 % and 97% to 44% in the MNIST, N-MNIST and DVS128 Gesture classification, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Hence, the classification of both static and neuromorphic datasets is prone to stuck-at faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Varying size of the systolic array: For further extensive fault vulnerability study, we analyze the impact of stuck-at faults across different sizes of NxN systolic arrays i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=', 4x4, 8x8, 16x16, 32x32 and 64x64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 5c, our analysis reveals that stuck-at faults in a small-sized systolic array cause more accuracy loss as compared to a large-sized systolic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' For example, 4 faulty PEs units in an 8x8 systolic array (having 16 PEs) lead to 89%, 92% and 93% accuracy loss in the MNIST, N-MNIST and DVS128 Gesture classification, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, SNN classification with a 256x256 systolic array, having the same fault configuration, results in almost 16%, 17%, and 33% accuracy loss only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' This is due to the fact that decreasing the size of the systolic array increases its chances for re-usability and hence, the reoccurrence of the permanent faults in every execution cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Our analysis shows that DVS128 Gesture is more vulnerable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='彩彩0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Accuracy [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Fault Bit Location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='sa0-MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='sa1-MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='sa0-NMNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='sa1-NMNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='sa0-DVS128Gesture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='sa1-DVS128Gesture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='(a) Accuracy vs Fault Bit Locations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Accuracy [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Number of Faulty PEs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='N-MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='DVS128 Gesture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='(b) Accuracy vs number of faulty PEs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='1024 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='65536 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Accuracy [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Total Number of PEs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='N-MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='DVS128 Gesture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='(c) Accuracy vs size of systolic array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Figure 5: Stuck-at fault vulnerability analysis of a 256x256 systolic-array based SNN accelerator (systolicSNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='8 1 Conv1 Conv2 FC1 FC2 Threshold Voltage Layers 10% 30% 60% (a) MNIST [24] classification 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='8 1 Conv1 Conv2 FC1 FC2 Threshold Voltage Layers 10% 30% 60% (b) N-MNIST [25] classification 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='8 1 Conv1Conv2Conv3Conv4Conv5 FC1 FC2 Threshold Voltage Layers 10% 30% 60% (c) DVS128 Gesture [26] classification Figure 6: Optimized threshold voltage for hidden convolutional and fully connected layers using FalVolt, when 0%, 10%, 30% and 60% of the total PEs are faulty in a 256x256 systolic-array SNN accelerator (systolicSNN) to faults when compa red to the MNIST and N-MNIST datasets, even though their baseline accuracies are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 5b, the classification accuracy of DVS128 Gesture remains comparatively lower than other datasets in the presence of stuck-at faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Also, the accuracy loss associated with the DVS128 Gesture dataset is comparatively higher than other datasets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, a higher number of stuck-at faults can render performance penalties unacceptable in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Fault mitigation analysis In this section, we study the performance of FalVolt and compare it with the state-of-the-art techniques common for ANNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Specifically, we compare FalVolt with fault-aware prun- ing (FAP) and fault-aware pruning with retraining without threshold voltage optimization (FaPIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Classification accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' fault rates: For the fault mitigation analysis, we inject the stuck-at faults using different fault maps in 10%, 30%, and 60% PEs of a 256x256 systolicSNN and run paralleled re-training simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' We employ the proposed FalVolt mitigation method using Algorithm 1 for 10%, 30%, and 60% PEs in a 256x256 systolicSNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Our analysis shows that optimizing threshold voltage for each hidden convolutional and fully connected layer helps in achieving baseline accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 6 shows the optimized threshold voltage returned from the FalVolt mitigation method for each hidden layer to achieve the baseline accuracy for MNIST, NMNIST, and DVS128 Gesture datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' For all these datasets, the optimized threshold voltage for the initial spiking-convolutional and spiking-fully connected layers is higher than other layers to ensure that the redundant spikes do not travel to the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 7 compares the FalVolt mitigation method with FaP and FaPIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' We observe that an increased fault rate causes a rapid accuracy loss in the FaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' FaPIT and FalVolt help in improving classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, only FalVolt achieves the baseline classification accuracy in the MNIST, N-MNIST, and DVS128 Gesture classification with even 60% of the faulty PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' This validates the applicability of FalVolt to both static and neuromorphic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Classification accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' number of epochs: FalVolt in- creases the classification accuracy at the cost of additional retraining epochs to FaP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' however, they are negligible com- pared to the lifetime of systolicSNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 8, FaPVolt is 2x faster than FaPIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' For example, the classification accuracy of MNIST is as high as 80% with FaPIT using 20 epochs and converges with baseline accuracy around 25 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, the same dataset achieves the baseline accuracy with FalVolt in 10 epochs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Likewise, FalVolt achieves the baseline accuracy of NMNIST classification 2x less number of epochs when compared to FaPIT as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Moreover, the classification accuracy of DVS128 Gesture is as high as 83% with FaPIT using 40 epochs and converges with baseline accuracy around 50 epochs as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, the same dataset achieves 97% accuracy with FalVolt around 25 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Since a small change in the base- line accuracy may cause catastrophic issues in safety-critical applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' therefore, the epochs for initial training, FaPIT, and FalVolt algorithms are high to achieve the classification accuracy close to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Note, training the large-sized SNNs itself takes a long time (or a higher number of epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' CONCLUSION This paper extensively analyzes the stuck-at fault vulnera- bilities of systolicSNNs and proposes a novel fault mitigation technique ‘fault-aware retraining through threshold voltage optimization (FalVolt).’ FalVolt uses an optimized threshold voltage and time steps different from initial training to achieve classification accuracy close to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' To demonstrate the effectiveness of FalVolt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' we classify the MNIST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' N-MNIST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' and DVS128 Gesture datasets on a 256x256 systolicSNN 0 20 40 60 80 100 10% 30% 60% Accuracy [%] Number of faulty PEs FaP FaPIT FalVolt (a) MNIST [24] classification 0 20 40 60 80 100 10% 30% 60% Accuracy [%] Number of faulty PEs FaP FaPIT FalVolt (b) N-MNIST [25] classification 0 20 40 60 80 100 10% 30% 60% Accuracy [%] Number of faulty PEs FaP FaPIT FalVolt (c) DVS128 Gesture [26] classification Figure 7: Stuck-at fault mitigation using FaP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' FaPIT (using threshold voltage as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0) and FalVolt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' when 0%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 10%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' 30% and 60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='of the total PEs are faulty in a 256x256 systolic-array SNN accelerator (systolicSNN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Accuracy [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Retraining Epochs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='FaPIT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='FalVolt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='(a) MNIST [24] classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Accuracy [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Retraining Epochs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='FaPIT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='FalVolt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='(b) N-MNIST [25] classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Accuracy [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Retraining Epochs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='FaPIT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='FalVolt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='(c) DVS128 Gesture [26] classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='Figure 8: Performance of FaPIT and FalVolt over different epochs when 30% the total PEs are faulty in a 256x256 systolic-array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='SNN accelerator (systolicSNN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='while injecting faults at different rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Our results show that even 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='012% faulty PEs in a systolicSNN leads to significant accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' However, FalVolt improves the performance of systolicSNNs by enabling them to operate at fault rates of up to 60%, with a negligible drop in the classification accuracy (as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' Furthermore, our results show that FalVolt is 2x faster when compared to state-of-the-art techniques, such as fault-aware pruning without threshold voltage optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf'} +page_content=' REFERENCES [1] E.' metadata={'source': 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